1

Burdens and Standards of Proof for Inference
to the Best Explanation: Three Case Studies
Floris Bex1 and Douglas Walton2
Abstract.
In this paper, we provide a formal logical model of evidential reasoning with proof standards and burdens of proof
that enables us to evaluate evidential reasoning by comparing stories on either side of a case. It is based on a hybrid
inference model that combines argumentation and explanation, using inference to the best explanation as the central
form of argument. The model applied to one civil case and two criminal cases. It is shown to have some striking
implications for modeling and using traditional proof standards like preponderance of the evidence and beyond
reasonable doubt.

Keywords.
Inference to the best explanation, beyond reasonable doubt, evidence supporting a story, argumentation, shifting ofburden of proof, critically questioning a story, plausible reasoning.

1. Introduction
In legal trials, the burden of proof and its associated standards of proof determine how strong
a party‘s position needs to be in order to prevail. In AI and Law, various ways of logicallymodeling the legal burden of proof and proof standards have been proposed (Gordon and
Walton, 2009; Prakken and Sartor, 2009). These approaches both presuppose some type offrameworks for defeasible argumentation, in which arguments (or argument graphs) areconstructed by performing consecutive reasoning steps from the evidence to the facts in issue.
When talking about the facts of a criminal case, however, Bex and colleagues (Bex 2011, Bex etal. 2010) have argued that argumentative approaches such as the ones mentioned above need to
be expanded to include reasoning with stories or explanations, that is, alternative accounts about
what (might have) happened in the case. In a purely argument-based approach the conclusions ofarguments are individual facts in issue. In a real case, however, these facts will be related to eachother in various ways (e.g. causally, temporally, motivationally) and these relations may also be
the subject of argumentative reasoning. Pennington and Hastie (1993) have shown that using
explanatory stories is closest to how legal decision makers actually think about a case. Pardo andAllen (2007) have persuasively argued that reasoning in trials involves evaluating the relativeplausibility of the various explanations of the evidence. We build a formal and computational
argumentation model that is used to implement Pardo and Allen‘s contention that inference to thebest explanation can be applied to civil as well as criminal trials in which the parties provide
competing ―theories‖ about the facts of the case.3

1

Argumentation Research Group, School of Computing, University of Dundee, UK.
Centre for Research in Reasoning, Argumentation and Rhetoric, University of Windsor, Canada.
3
Pardo and Allen (2007) cite a number of civil cases that involve competing explanations of the events. Examples
are Anderson v. Griffin, 397 F.3d 515 [7th Cir. 2005], which is discussed in section 3.1, and Los Angeles v.
Alameda Books, Inc., 535 U.S. 425, 437–38 [2002]).
2

2

Bex and colleagues proposed a hybrid theory of inference to the best explanation (IBE) thatconsists of a combination of abductive causal reasoning with explanations and defeasible
evidential argumentation. In this hybrid theory, hypothetical explanations are constructed
through abductive reasoning and these explanations can then be supported and attacked usingarguments based on evidence. As the basis for weighing the strength or weakness of evidence,
the alternative explanations in a case are compared. In this comparison, burdens and standards of
proof play an important role (Bex and Walton 2010). For example, even if the prosecution‘s
explanation of guilt in a criminal case is the ―best‖ explanation (according to some rational
standard) it may still not meet the legal ―beyond a reasonable doubt‖ standard, resulting in theacquittal of the defendant. Similarly, when the parties‘ alternative explanations are equally good,the burden of persuasion influences which of these alternatives should be chosen.
At the moment, the hybrid theory does not include a notion of burden of proof nor does it say
how various proof standards may be met (apart from Bex and Walton 2010, of which the current
paper is an extension). Pardo and Allen (2007) discuss how the burden of proof influences theprocess of IBE in legal trials and they also provide ideas on how standards of proof may be met
by an explanation. However, their theory of IBE is informal and not specified in the level ofexact detail that is needed for a formal computational model. Building on our previous work
(Bex and Walton 2010), in this paper we logically model reasoning with the burden of proof in
inference to the best explanation, using Bex and colleagues‘ (2011, 2010) hybrid theory of IBE
as our start point. The aim is to explore how the different types of burdens of proof (e.g. burdenof persuasion, burden of production) may be modeled. We also show in a distinctive way how
proof standards should be modeled in the hybrid theory. We provide extensive examples of both
criminal cases and a civil case.
The rest of this paper is organized as follows. Section 2 contains a summary of the hybrid
theory. Section 3 briefly discusses different types of legal burden of proof and proof standards.
In section 4, we model these burdens of proof and proof standards in the hybrid theory. In theremaining sections we provide three examples of how to analyze and evaluate evidential
reasoning in a civil case and two criminal cases. The third case in particular raises someinteresting problems and has some surprising implications concerning the beyond reasonable
doubt standard.

2. A Hybrid Theory of Inference to the Best Explanation
As its basic structure, our formal account of evidential reasoning will represent evidence assomething presented to a trier of fact as data, often perceptually or by testimony, and thenexplained one way or another, leading to a conclusion of some sort. According to Josephson andJosephson (1994, 14), an abductive inference has the following general form.
D is a collection of data.
H explains D.
No other hypothesis can explain D as well as H does.
Therefore H is probably true.
Here, the hypothesis H may be a single proposition but it can also be an ordered sequence of
events, a story. Various critical questions can then be asked (see below). This suggest that an
explanatory story in IBE is defeasible, that is, it holds tentatively as an instance of plausible
reasoning by placing a burden on an opponent to critically question it. Josephson (2001) has

3offered a formal model of abductive inference and an indication as to how the standard of
beyond a reasonable doubt may be modeled, but does not digress any further on the subject. Partof the problem with analyzing the notion of an explanation in a precise logical model suitable tosupport its implementation in fields like artificial intelligence was that, until Bex et al.‘s hybrid
theory (Bex 2011, Bex et al. 2010), structures such as argument diagrams were lacking a basic
notion of an ordered sequence of events and actions. By showing how to build such a structured
notion of a connected story that makes sense as a script underlying an explanation, our theory
moves IBE ahead.
Bex and colleagues (Bex 2011, Bex et al. 2010) propose a formal hybrid theory for IBE,
combining arguments and stories. In this hybrid theory, the data that is to be explained, theexplananda, are causally explained by different stories, alternative accounts of what happened in
the case. These stories can then be reasoned about using evidential arguments. The formal hybrid
theory HT = (ET, CT) is thus a combination of a causal-abductive theory CT and an evidential
argumentation theory ET. The logic L of this theory is a combination of the inference rules ofclassical logic and a modus ponens inference rule for the connective (defeasible implication).
Object-level rules in CT and ET are formalized 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 (e.g. p
E q, which stands for ‗p is evidence forq‘) and C denotes a causal rule used in explanations (e.g. p C q, which stands for ‗p causesq‘).
In the abductive theory CT = (H, T, F), T is a set of causal rules, H is the set of hypotheticalevents or hypotheticals and F is the set of explananda. As in traditional models of abductivemodel-based reasoning (Console and Torasso 1991), the explains relation between a story andthe 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). Thus, given F we need to abductively infer some hypotheticals and some causal
rules such that they explain F, that is, the explananda follow from the hypotheticals and thecausal theory. More precisely, a story S = Hi Ti, where Hi H and Ti T, is an explanation fora set of explananda F iff for each f F: S |~ f (where |~ denotes defeasible entailment).
Furthermore, we require that S is consistent and that S is minimal w.r.t. set-inclusion.
In the argumentation theory ET = (R, K), R is a set of evidential rules and K = KE U KA is aknowledge base, where KE is a consistent set evidence and KA is a set of commonsense
assumptions. The logic for ET is similar to the ASPIC+ framework (Prakken 2010), which
integrates ideas on rule-based argumentation and structured arguments (e.g. Prakken and Sartor
1997) within Dung‘s (1995) abstract approach. Evidential arguments can be built by taking
evidence or assumptions from K and rules from R as premises and chaining applications ofdefeasible modus ponens into tree-structured arguments, where each node in the tree is thus anelement of K, a rule from R or the result of an application of the defeasible modus ponens to one
or more other nodes.
An argument AR1 can defeat another argument AR2 in various ways. AR1 and AR2 rebut eachother if they have an opposite (intermediate) conclusion. AR1 undercuts AR2 if there is aconclusion ri in AR1 and an application of defeasible modus ponens to ri in AR2. Finally, AR1
undermines AR2 if AR1 has a conclusion that is the opposite of some assumption (from KA) in
AR2. For a collection of arguments and their binary defeat relations, the dialectical status of thearguments can be determined: informally, arguments can be either justified, which means thatthey are not defeated by other justified arguments, overruled, which means that they are defeated4by other justified arguments, or defensible, which means that they are neither justified nor
overruled.4
Stories and arguments can now be combined in various ways: arguments based on evidence
can be used to support or attack stories or attack stories and arguments can also attack each other.
Figure 1 visualizes the various ways in which stories and arguments interact as diagrams akin tothe ones used in argumentation (cf. Freeman 1991, Reed and Rowe 2004). Here, arrows
denote causal links,
arrows denote evidential (argumentative) links and thearrow
denotes an attack of an argument (i.e. contradiction). White boxes are part of the abductive
theory CT and gray boxes are part of the argumentation theory ET.

Story 1

event

event

causalgeneralization

event
evidential
generalization
evidential
generalization

explanandum
Story 2

event

event

causalgeneralization
Alternative stories explaining an
explanandum
event

source ofevidence
An evidential argument with an
event as its conclusion

event

Arguments attacking a causal
link and an event in a story

An argument attacking anevidential link in anotherargument

Combining stories and arguments

Figure 1: Visualizing the hybrid theory

In IBE, multiple explanations should be generated and compared according to criteria that
express the degree to which they conform to the evidence and their plausibility. These criteria aredefined using the argumentation theory. Arguments based on evidence can be used to show thatan explanation is consistent or inconsistent with the evidence. More specifically, evidential
support for an explanation S (denoted as es(S)) is the number of sources of evidence from KE that
support S through an argument A (i.e. e KE is a premise of A and s S is a conclusion of A)
and evidential contradiction for an explanation S (denoted as ec(S)) is the number of sources ofevidence from KE that contradict S through an argument A (i.e. e KE is a premise of A and s isa conclusion of A, where s S). Arguments may also be used to reason about the plausibility ofan explanation, as the validity and applicability of causal rules can become the subject of anargumentation process. Thus, the third criterion for judging an explanation S is implausibility(denoted as impl(S)), which stands for number of elements in S explicitly contradicted by anargument A which is not based on evidence (i.e. s S and s is a conclusion of A, where A‘spremises {a1,…,an} KA). Here, arguments about the plausibility of explanations are based onassumptions from KA, as reasoning about plausibility is done using commonsense knowledge
4Note that these informal definitions have been formalized in various ways, see e.g. (Jakobovits and Vermeir
1999).

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about how the world generally works. Note that for the criteria, only arguments which are notoverruled support or contradict an explanation: if an argument based on evidence is itself
defeated, the evidence does not support the explanation.
The above criteria can be used to judge the quality of explanations and thus they allow for thecomparison of explanations. According to Josephson and Josephson (1994, 14), the evaluation ofan abductive inference should be carried out by asking one or more of the following criticalquestions.
1. How decisively does H surpass the alternatives?
2. How good is H by itself as an explanation, independently of the alternatives?
3. How much confidence there is that all plausible explanations have been considered?
4. How good are the original data?
5. How can the cost of being wrong be weighed against the benefit of being right?
6. How strong is the need is to come to a conclusion at all, as opposed to collecting more
information?
Now, question 1, 2 and 4 can be directly modeled in the formal theory. Question 2 can be
answered by analyzing an explanation‘s evidential support and plausibility: the higher theevidential support and the lower evidential contradiction and implausibility, the better the
explanation. Plausibility can be further specified if desired: Bex (2011) defines more criteria for
judging the quality of explanations using story schemes and Pennington and Hastie (1993) also
provide other criteria such specificity. Furthermore, we could attach different weights toevidence so that particular pieces of evidence give a higher degree of support to explanations
than others. Thus we can answer question 1: if one explanation is supported by ―better‖evidence, it surpasses the other explanation. Finally, how ―good‖ the evidence in itself is (e.g.
how reliable a witness is) can be tested separately using the critical questions for the arguments
based on the evidence (question 4). For current purposes, however, the three criteria aresufficient. Because the comparison of explanations is influenced by the standard of proof,
comparing explanations according to these standards is discussed in detail in section 3.

3. Burdens of Proof and Proof Standards
Allocation of burden of proof tells each side in a dispute how strong its argumentation needs to
be in order to be successful in prevailing over contention of the other side. Farley and Freeman(1995) presented a computational model of dialectical argumentation that has the notion of
burden of proof as its key element. They defined it as the level of support that must be achievedby one side to win an argument. On their account, burden of proof has two functions (Farley andFreeman, 1995, p. 156): to act as a move filter at local moves in a dialogue and to act as atermination criterion that determines the winner at the end of the dialogue.
In their logical account, Prakken and Sartor (2009) define three kinds of burden of proof wellknown in common law systems in terms of their framework for defeasible argumentation.5 The
first type of burden of proof is the burden of persuasion. It is set by law at the opening stage of atrial, and determines which party has to prove and what proof standard has to be met, that is,which side has won or lost the case at the end of the trial once all the arguments have been
5

The burdens of claiming and contesting, which are also discussed by Prakken and Sartor (1997), assume an explicit
dialogical context and will hence not be discussed here.

6

examined. Prakken and Sartor (2009) define it as the task of making sure there is a justified
argument for one‘s claim. The burden of persuasion does not shift from the one side to the otherduring the trial. The second type of burden is called the burden of production, or sometimes
burden of producing evidence. The burden of production, which is like the burden of persuasion
assigned by law, specifies which party has to offer an argument based on evidence on somespecific issue during the trial. If the evidence offered does not meet the standard set for thisburden, the issue can be decided as a matter of law against this side.
The third type of burden is called the tactical burden of proof. It is a hypothetical assessment
made at a given point by each party to try to determine whether they will win or lose the case if
no further arguments are put forward at that point. A tactical burden can shift back and forth
between the parties any number of times during the trial, because it depends on who has thewinning argument at a particular point in the trial. Not all sources in law agree, but in general thetactical burden is the only one of the three of the three burdens that can be properly said to shiftduring the course of a trial. (Prakken and Sartor, 2009, 227) argue that the distinction between
the burden of production and tactical burden of proof is usually not clearly made in common law,and is usually not explicitly considered in civil law countries, but is relevant for both systems oflaw, because it is induced by the logic of the reasoning process.
Burden of proof rests on the prior notion that there can be different standards of proof. In thedomain of common law, there are four main proof standards for factual issues called scintilla ofevidence, preponderance of evidence, clear and convincing evidence, and beyond reasonable
doubt (Gordon and Walton, 2009, 241). The scintilla of evidence proof standard is met if ―even
the slightest amount of relevant evidence exists on an issue‖ (Garner, 1990, 1464). The
preponderance of evidence standard is met by ―evidence that has the most convincing force,superior evidentiary weight that […] is still sufficient to incline a fair and impartial mind to one
side of the issue rather than the other‖ (Garner, 1990, 1301). The preponderance of evidence
standard is often compared to a balance, where the evidence on one side has greater probative
weight than the evidence on the other side. Clear and convincing evidence is ―evidence
indicating that the thing to be proved is highly probable or reasonably certain‖ (Garner, 1990,
636). This standard is supposed to be higher than that of preponderance of the evidence, but notas high as the highest standard in law, that of evidence beyond a reasonable doubt. Finally, thebeyond reasonable doubt standard is used to determine guilt in criminal cases, and is often
equated with the presumption that the defendant is innocent.
Law defines the standards using cognitive terminology, for example by using criteria of
whether an attempt at proof is credible, or convincing to the mind examining it. These cognitive
descriptions, although they are useful in law for a judge to instruct the jury on what the burden of
proof is in the case, are not precise enough to serve the purposes of providing computationalmodels of the standards useful in AI and law. A definition expressed in, for example, terms ofthe conviction of the jury that the charge against the defendant is true (Garner, 1990, 1380) is notvery helpful for getting an idea of how this standard should be represented in a normative modelof rational argumentation.
It is debatable whether such a precise definition of proof standards can be given. Take, forexample, the standard of beyond a reasonable doubt. According to McCormick on Evidence(Strong, 1992, 447), ―The term reasonable doubt is almost incapable of any definition which will
add much to what the words themselves imply‖. Courts have held that the legal concept ofreasonable doubt itself needs no definition (Strong, 1992, 447), the reason being that any
definition might have to be so subtle and technical that there would be dangers of

7

misunderstandings if judges were to instruct juries with it. This judicial climate of opinion poses
an apparently insurmountable challenge for any attempt to provide a computational model of theproof standard. However, as Tillers and Gottfried (2006) have shown, even though there is awell-settled maxim supported by judicial wisdom that the beyond reasonable standard is notquantifiable by assigning probability values to it, it does not follow that this standard of doubt is
not open to precise analysis based on a computational argumentation model.
Gordon and Walton (2009) show how proof standards can be analyzed in the formal
Carneades system as follows. For there to be a scintilla of evidence (SE), there should be at least
one applicable argument6 for a claim. For the preponderance of evidence (PE) standard, SE
should be satisfied and the maximum weight assigned to an applicable pro argument (for theclaim) is greater than the maximum weight of an applicable con argument (against the claim).
For clear and convincing evidence (CCE), PE should be satisfied, the maximum weight ofapplicable pro arguments exceeds some threshold α, and the difference between the maximum
weight of the applicable pro arguments and the maximum weight of the applicable con
arguments exceeds some threshold β. Finally, for beyond reasonable doubt (BRD), CCE issatisfied and the maximum weight of the applicable con arguments is less than some threshold γ.
Notice that here the thresholds α, β and γ are left open, and not given fixed numerical values.
Doing this would involve quantifying the proof standards, which, as was argued above, is noteasily done.

4. Burden of Proof and Inference to the Best Explanation
In this section, we will give an indication of how the burden of proof and proof standards might
influence the process of IBE in legal trials. For a large part, we draw inspiration from 0, whoprovide an informal overview of IBE in trials. Note that we mainly concern ourselves with thefactual part of trials, that is, the evidence and the facts which we might infer from this throughevidential reasoning. The legal reasoning, which is intertwined with this reasoning about thefacts,7 is not shown in detail and we simply assume that the legal conclusion follows from theestablished facts in some way.
In the hybrid theory, having the burden of persuasion for an explanation S means that at theend of the trial S should be accepted as the correct explanation of what happened in the case.
Note that just having the best explanation is not always enough: in order to satisfy, for example,
a BRD proof standard S should be much better than the other explanations, so good that the otherexplanations do not even raise a (reasonable) doubt whether S happened. This will be further
discussed below. The burden of production may be met by providing evidence that ultimately
supports the explanation on which the burden rests. So, for example, one can give evidential
arguments for elements of one‘s own explanation or the negation of elements of the opponent‘s
explanation. Finally, the tactical burden means that one should gauge if one‘s explanation iscurrently the best and if it trumps the other explanations by a particular margin, dependent on thestandard of proof.

6

Roughly, an argument is considered applicable if its premises are not defeated and there is no exception to theinference.
7
For example, once we have established that it was in fact John who killed Harry, we still need to determine
whether the killing was manslaughter or murder.

8Standards of proof may also be formalized in the hybrid theory. Definitions of standards of
proof require us to indicate not only when one explanation is stronger or better than another but
also by which margin they are better and how good they are in themselves. Following (Gordon
and Walton, 2009), such margins or standards will not be given fixed values but rather they willbe left open. Now, an explanation S meets the scintilla of evidence (SE) standard if there is ajustified supporting argument based on evidence (es(S) ≥ 1). An explanation S meets the
preponderance of evidence (PE) standard if it meets the SE standard and it is better than eachalternative explanation S’. That is, all else being equal S is either supported by more evidence(es(S) > es(S’)) or contradicted by less evidence (ec(S) < ec(S’)). This way of comparing
explanations is similar to that in (Bex et al. 2007). However, a key difference is that in (Bex et al.
2007), if two explanations have the same evidential support and contradiction, the explanation
with the lowest implausibility is best. While comparing explanations on their relative plausibility
is in principle perfectly rational, we cannot say that S meets a formal proof standard if it is justmore plausible than S’.
For clear and convincing evidence (CCE), an explanation S should be good in itself as well as
much better than each competing explanation S’. In order to be good, S should have a highevidential support (es(S) > α, where α is some threshold) and low evidential contradiction (ec(S)< β, where β is some threshold). In order to be much better than any alternative S’, S should have
either significantly higher evidential support (es(S) – es(S’) > γ, where γ is some threshold) or
significantly lower evidential contradiction (ec(S’) – ec(S) > δ, where δ is some threshold).
Finally, an explanation meets the beyond a reasonable doubt (BRD) standard if it is strong andmuch stronger than its competing explanations (i.e. it meets the CCE standard with highthreshold α and a low threshold β), and each competing explanation S’ is very weak, so weak as
to be highly implausible. As Pardo and Allen (2007) argue, a plausible explanation consistent
with innocence creates a reasonable doubt, so for each competing explanation S’, either theevidential contradiction or the implausibility should be high (ec(S’) > α or impl(S’) > β, where α
and β are thresholds) if the ―guilt‖ explanation is to meet the standard of proof. Note that theevidential support of the competing explanations is not tied to extra requirements (beyond those
set for the CCE standard), as these explanations only have to be consistent with the evidence.

5. The Case of Anderson v. Griffin
In this section we give an example of IBE in a civil trial. In the case, which is based on theSummary of Case of Anderson v. Griffin (397 F.3d 515), the driveshaft suddenly broke on a
tractor-trailer truck proceeding down an interstate highway, severing the connection between thebrake pedal and the brakes. Debris kicked up from the surface of the highway (road junk) strucka pickup truck behind the tractor-trailer. The pickup truck crashed into a part of the tractor-trailer
and a car following the pickup truck struck the wreckage from the collision between the two
trucks, injuring the two people in the car. Plaintiffs, the two people in the car, sued the truck
dealer, who (supposedly) was responsible for the technical maintenance of the trailer. Now, theplaintiffs should propose a coherent explanation from which it follows that the dealer had been
negligent. Three weeks earlier, the trucking company who owned the tractor-trailer had noticed alooseness in the driveshaft and had asked the truck dealer to tighten the driveshaft. The dealer
tightened all the joints except for the middle one, which broke. This first explanation is
supported by the truck dealer‘s records about the repairs on the truck (they state that the
repairmen did not repair the joint). Figure 2 shows this explanation (and defendant‘s alternative,9

see below). In the figure, p and d denote whether an event is part of plaintiff‘s explanation,
defendant‘s explanation or both.
Truck driver’s records.

Plaintiff’s expert.

The truck dealer did not
repair the driveshaft (p).
There was debris
on the road (d).

Testimony of
witnesses.

The driveshaft
broke (p,d).
Debris struck thedriveshaft (d).

Testimony ofdefendant’s expert.

The crashoccurred (p,d).

Driveshaft rotates
at high speed.

Testimony ofplaintiff’s expert.

Figure 2: Two explanatory stories in the Anderson v. Griffin case

Defendant, the truck dealer, now has the tactical burden of proof: if he does not question
plaintiffs‘ explanation or provide a better explanation for the crash from which it does not followthat he had been negligent, the jury will rule for plaintiffs. Defendant gives such an alternative
explanation, claiming that debris struck the driveshaft properly. That there was debris on the roadfollows from statements made by witnesses. Defendant could also have denied the fact that hedid not repair the driveshaft (i.e. attacked the plaintiff‘s explanation), but as his own records state
he did not repair the slip yoke, this might not be a strong argument.
Plaintiffs now have the burdens of persuasion and production for their explanation whilst
defendant only needs to cast sufficient doubt on this explanation, which he has done byproviding a reasonable alternative which is at least as good as plaintiffs‘ explanation. If a verdict
were to be given now, the judgment would go against the party with the burden of persuasion, in
this case the plaintiffs, because they have failed to meet the burden of production, i.e. produce
evidence so that a fact-finder can differentiate between explanations. Thus plaintiffs now havethe tactical burden of proof and they produce evidence to improve their explanation: an expert
witness who states that the crash was caused by the fact that defendant did not repair thedriveshaft.
The tactical burden now shifts to defendant, as plaintiffs‘ explanation is slightly better(because it is supported by more evidence). Defendant can now, for example, have his own
expert deny the causal link between the dealer‘s failure to repair the driveshaft and the crash or
he may want to question the plaintiffs‘ expert‘s veracity, undercutting the support the testimony
gives to plaintiffs‘ explanation. However, defendant chooses to support his own explanation withan expert testimony. It was proposed that the accident had been caused by debris on the highwaythat might have been yanked up and against the driveline by chains hanging from the truck. This
explanation is also shown in Figure 2 (bottom sequence of events).
Both explanations now have equal support, so the tactical burden again shifts back toplaintiffs, who decide to attack defendant‘s explanation. The plaintiffs argued that a piece of roadjunk would be highly unlikely to strike the driveshaft with enough force to break it, because of
the speed at which the driveshaft rotates (27 times a second). This argument is shown in the10sequence of grey boxes at the bottom of Figure 2. Now plaintiffs‘ explanation is slightly better
than defendant‘s so the preponderance of evidence standard seems to have been met. However,in the case the jury ruled for the defendant. For some reason, they must have found that theattacking argument based on plaintiffs‘ expert was not convincing enough.
The sequence of dialogue in which the burdens of proof shift from one side to the other issummarized in figure 3. The shifts are indicated by the darkened boxes.
Plaintiff presents
plausible and wellsupported story about
truck dealer’s negligent
repair of driveshaft.

Plaintiff
providesexpert witness
testimonyevidence toshow his
explanation isbetter.

Tactical burden
shifts to defense
to question story
or provide analternative one.

Plaintiff’s
story is now
better, and so
the tacticalburden now
shifts back to
defendant.

Plaintiff argues thatdebris would be
unlikely to break thedriveshaft because it
rotates at high speed.

Defendant offers
plausible andwell-supported
story of debris
striking
driveshaft.

Defendant
provides expert
witness
testimonysupporting story
about debris
yanked up byroad chains.

Tactical burden
now shifts to
defendant’s side.

Plaintiff now hasburden of
production andmight lose if hecannot prove his
explanation isbetter.

Tactical burden
has now shiftedto plaintiff toquestion story orprovide a betterone.

Preponderance ofevidence standard
now appears to be
met by plaintiff.

Figure 3: Sequence of Shifting Burdens in the Anderson v. Griffin case

6. The Case of Jackson v. Virginia
As an example of a criminal case we discuss Jackson v. Virginia: 443 U.S. 307. The caseconcerns the death of Mary Cole, who had been a member of staff at the county jail where she
had befriended James Jackson, an inmate. After his release, Cole and Jackson stayed in contact.
Witnesses testified that on the day the crime was committed, Jackson had been drinking whileshooting at targets with his revolver. Later that day, Cole and Jackson drove to a diner where
they were seen drinking by two police officers. As the two were preparing to leave the diner inCole‘s car the sheriff testified that he had offered to keep Jackson's revolver until he sobered up,
but that Jackson had said this would be unnecessary since he and Cole were about to engage insexual activity. The same evening, Jackson drove from Virginia to North Carolina. A day and ahalf later, Cole‘s body was found in a secluded parking lot, naked from the waist down, herslacks beneath her body; Jackson was arrested a few days later.
Jackson was convicted of first-degree murder by the Virginia court and sentenced to 30 years
in the penitentiary. Then he appealed to the Supreme Court arguing that the beyond reasonable
doubt standard had not been met. The case went to the Supreme Court (Jackson v. Virginia 443

11

U.S. 307 (1979)) because of the raising of the question whether there was sufficient evidence tojustify a rational trier of fact to find guilt beyond a reasonable doubt. His argument was that theevidence was insufficient to support the finding that he had intended to kill Mary Cole. Jackson
had appealed to a District Court in Virginia, claiming that there was no evidence ofpremeditation. This Court found that there was some evidence that he had intended to kill Cole,and listed this evidence as follows (page 312). He reloaded his gun after firing warning shots. He
had time to reload his gun. He shot the victim not once but twice. The court also supported theconclusion of the state trial judge that he was not so intoxicated that he was incapable ofpremeditation. A problem posed, as expressed in the Supreme Court notes, is whether a small
amount of evidence, called a ―mere modicum of evidence‖, is by itself adequate to rationally
support a conviction to the standard of beyond a reasonable doubt (page 320).
The prosecution in the Supreme Court case proposed an explanation SP from which it
followed that Jackson murdered Cole. The explanation first recounted the events at the diner andthen, without mentioning a specific motive, argued that Jackson intended to kill Cole, that heshot her with his revolver, killing her, and that he then drove to North Carolina. The story was
supported by witness statements (on Jackson shooting and drinking) and police officers‘
statements, as well as expert medical evidence that Cole had been shot twice at close range with
Jackson's revolver. Because Jackson admitted he had shot Cole, the main factual dispute at thetrial was whether there was sufficient evidence to prove Jackson‘s intention to kill Cole.
Evidence that he had so intended was that Jackson had fired two shots at Cole at close range,
shots that were predictably fatal given that he was a person experienced in the use of firearms.
Figure 4 shows this argument supporting the prosecution‘s story Sp.
J and C were at
the diner.

J possessed thenecessary intent
before the killing.

J killed C.

J drove to
North Carolina.

The shots werepredictably fatal.

J shot C
twice.

If a person who is
proficient with
firearms shoots
someone else at
close range, theshots arepredictably fatal.

J is aperson who
is proficientwithfirearms.

Figure 4: The Argument that Jackson Intended to Shoot ColeThus, the prosecution gave an argument A1 that Jackson knowingly (and hence intentionally)
performed his act.
Teleological reasoning can be used to establish intent by drawing an inference from premises
concerning facts of a case to a conclusion that an intention exists (Walton and Schafer, 2006). A
sequence of teleological reasoning leads from a set of circumstances in a case to a hypothesis

12that postulates the existence of a motive. To see how this was done in (Walton and Schafer,2006), we have to put this scheme into a broader argumentation framework by introducing theargumentation scheme for practical reasoning. This scheme can be used to show how an agent
reasons from a goal (or intention) to action, and also how an agent can reason abductively froman action to the postulation of an intention. How this scheme can be so used will be illustrated n
the next case (the Beard case) below.
The prosecution met tactical burden for their explanation: SP had sufficient evidential
coverage and no evidential contradiction and there was no competing explanation. Jackson had
the tactical burden which he could meet by casting doubt on SP. Under the BRD standard, thisdoubt may be created by proposing a sufficiently plausible alternative explanation consistent
with the evidence. Jackson presented the following story SJ. After they left the diner, Cole had
made sexual advances towards Jackson. When he had resisted her sexual advances, she had
attacked him with a knife. He defended himself by firing warning shots into the ground, and thenreloaded the weapon. When Cole attempted to take the gun away from him, it went off during
the ensuing struggle. He claimed that he had fled without seeking help for Cole because he wasafraid. Later, during the trial he claimed that he had acted in self-defence. He also offered theargument A2 that, as the State‘s own evidence (i.e. the police officers‘ testimonies) showed, he
had been too intoxicated to form the specific intent necessary to make him guilty of the crime offirst-degree murder. In other words, he attacked SP.
The tactical burden then shifted to the prosecution, who had to defend the argument A1 (forJackson‘s intentions) by giving a counterargument to A2 (that he was intoxicated) as well asshow that Jackson‘s explanation SJ is implausible. First, the fact that Jackson drove without
mishap from Virginia to North Carolina was taken to be at odds with his argument of extreme
intoxication at the time of the killing. This counterargument, together with A1 and A2 are shownin figure 5.
J possessed thenecessary intent
before the killing.

J was incapable ofpremeditation.

The shots werepredictably fatal.

J shot C
twice.

If a person who is
proficient with
firearms shoots
someone else at
close range, theshots arepredictably fatal.

J is aperson who
is proficientwithfirearms.

J wasintoxicated.

J was not so
intoxicated as to
be incapable ofpremeditation.

J later drove
without mishapfrom Virginia to
North Carolina.

In his statement afterarrest, J claimed hehad not been drunk,even though he had
been “pretty high”.

Figure 5: Prosecution Arguments Attacking Jackson’s Story about No Premeditation

Thus, A2 was considered overruled and A1 again justified.

13

It was further argued that SJ contained implausible elements. First, Jackson said to the police
officer‘s he was going engage in sexual activity with Cole but later supposedly resisted Cole‘ssexual advances. Furthermore, it was found implausible that Cole first willingly removed part ofher clothing and then attacked him with a knife when he resisted her advances, even though he
was armed with a loaded revolver that he had just demonstrated he knew how to use. The wholestory in figure 6 is implausible, and many aspects of it could be questioned.
J’s statements to police
officers that he wasgoing to engage insexual activity with C.

J was uninterested insexual activity with C.

Gun went off
in struggle.

Normally, women do not nakedly attack
men with a knife when such men are
uninterested in sexual activities.

C was interested insexual activity with J.

J and C
struggled.

J reloaded.

C willingly removed
part of her clothing.

J resisted C’s
advances.

J fired warning
shots.

C attacked Jwith knife.

ex

J unintentionally
shot C.

Testimony by
C’s relatives.

On the day of thekilling, J spent agreat deal of timeshooting at targets
with his revolver.

pl

ai

ns

C knew J was armed
with a loaded
revolver that heknew how to use.

Figure 6: The Full Sequence of Jackson’s Story Questioned by an Argument

First, consider the argument based on the testimonies of the relatives. From this evidence we canconclude that Cole knew Jackson was armed with a loaded revolver. This attacks the story: if she
knew he had a gun, it would be foolish to attack him with a knife so this event is not very
plausible. Furthermore, the conclusion also shows that Jackson‘s case as a whole iscontradictory: the fact that Cole knew he was armed follows not only from the relatives‘
testimonies but is also explained by the event `Jackson fired warning shots‘ in Jackson‘s story.
The second case of implausibility concerns not single causal links or events, but rather thestory as a whole. Jackson claims he was uninterested in Cole and that hence she took off her
clothes and attacked him with a knife. This is implausible because it does not fit any plausible
story scheme (Bex 2011)8 or script (Schank and Abelson 1977), a general sequence of events as
we expect them to normally happen. As an example of such a story scheme, consider the wellknown restaurant script, which describes a pattern of events that a typical story about a visit to arestaurant contains (e.g. ordering, eating, paying). Now, if a story deviates from such a script weusually consider it to be implausible (for example, a man enters a restaurant, tells the waiter heneeds a hamburger as soon as possible because he is very hungry, receives his hamburger from
8

Story schemes are formalised in (Bex 2011).

14

the waiter, stands up from his seat and removes his pants, dances the tango as he moves aroundthe floor, offers the waiter his pants in exchange for the hamburger and starts reciting the
Declaration of Independence from memory when confronting a policeman called to therestaurant deal with the disturbance). In the case of Jackson, it is the genders of the actors thatare the wrong way around (women do not usually try to force men to have sex, at least notphysically) and the sequence of events also makes no sense (first Cole undresses, then she grabs
a knife; from where, one might wonder).
Now SP‘s quality is restored (as A1 again supports it) and SJ‘s implausibility is demonstrated
and thus the beyond reasonable standard is met, as is evident from the trial record.

7. The Beard Case
This case (US v. Beard, 354 F. 3d 691 - Court of Appeals, 7th Circuit 2004) was an appeal of an
earlier conviction in which John Beard was found guilty of carrying a gun in relation to a drug
offense, a violation of the law. Police were watching a parking lot in which two cars were parked
side-by-side. Beard got out of one of the cars, entered the other car for a few minutes, and thenreturned to his car. Both cars left the scene and were later stopped by the police. In Beard's car
police found drugs in a secret compartment behind the rear seat and a loaded derringer in theclosed center console of the front seat, concealed under some papers. In the other car they foundcash. Beard did not own the car. He had borrowed it eight months before from its owner, and had
been seen driving it from time to time during this period. However, he was not the only person
who used the car during that period. The papers concealing the derringer did not seem to havebelonged to Beard either. They did not pertain to him.
Let's look at the arguments on both sides. The argument given on the prosecution side can be
set up as follows. It depends on premises that express some generally known characteristics ofactivities during drug dealing cases.
Premise 1: The gun was being carried during a drug offense.
Premise 2: Drug dealers bring guns to a deal to instil fear in their business associates, or ifthey feel the need for protection.
Conclusion: It is reasonably clear that the gun was Beard‘s.
This argument is a fairly weak one, but it carries some weight, because of the general knowledge
from previous cases about how drug dealers generally conduct their operations. It tells us that
drug dealers have reasons to carry a gun during a drug offense, and it can be inferred from this as
an interim conclusion that drug dealers commonly bring a gun along during a drug deal. Theseassumptions suggest the conclusion that Beard took the derringer along during this drug deal andconcealed it in the console of the car.
The issue was whether the derringer belonged to Beard. Let's look at the argument that it
belonged to Beard using a standard argument mapping representation in figure 7. The ultimate
conclusion, represented at the top left, is the statement that it is reasonably clear that the gun
belonged to Beard. To the right of the diagram at the top we have a linked argument with two
premises. One premise is the statement that the gun was being carried during a drug offense. The
other is the statement that drug dealers commonly bring a gun along during a drug deal. Thislatter statement has been inserted as an implicit premise, as indicated by the fill in the text box
being shaded (in yellow). Another implicit premise, the statement that drug dealers have reasons

15

to carry a gun along during a drug offense, supports the implicit premise above it. This statement
is in turn supported by the arguments below it. Note that this way of representing the argument
makes it depends on a number of assumptions in such a way that the argument is not very strong.
Not only does it depend on two implicit premises, but it also depends on generally acceptedknowledge about how the way things go in a drug deal. It depends on your knowledge about how
things can be generally expected to go in a script that, at least in general outline, follows astereotypical pattern. Two parties exchange something for money. The transaction is illegal, andlarge amounts of money are involved. Hence the situation is a dangerous one, and as we all know
from Hollywood movies, it can easily erupt into gunfire.
It is reasonably clear thatthe gun was Beard’s.

Beard took the derringer along during this drug
deal and concealed it in the console of the car.

The gun was being carried
during a drug offense.

Drug dealers bring guns to a deal toinstill fear in their business associates,
or if they feel the need for protection.

There is generally accepted knowledge about how
drug dealers generally conduct their operations.

Drug dealers commonly bring a
gun along during a drug deal.

Drug dealers have reasons tocarry a gun during a drug offense.

Citation of five cases aboutcarrying a gun in drug offenses.

Figure 7: Argument Map of the Argumentation Concluding the Gun was Beard’s

Adopting this way of modeling the argumentation in the case, we could also examine theargument on the other side. Conceivably one of the others who used the car during the eightmonth period placed the gun there. However, there doesn‘t seem to be any evidence supportingthe conclusion that some other person left a gun in the console. The argument on the other side
appears to be nonexistent. So looking at the case from a standard argumentation point of view,
we have a weak argument on one side and no argument on the other. But there is a different way
to analyze and evaluate the argument by using inference to the best explanation.
Here we put forward another hypothesis giving an analysis of the logical structure in thedecision. The hypothesis is that it was decided by inference to the best explanation of the facts.
The choice had to be made between three explanations of the facts of the case. What needed to
be explained was how the gun got into the console between the seats in the car. One of the threepossible explanations is that the derringer was the property of the car‘s owner, and that she had
the loaded gun in the console when she lent Beard the car. This explanation was ruled out by the
court. We are not told why, but possibly it was because the car owner had no previous criminal
convictions and was not related to the drug trade. The second explanation is that Beard put the16derringer in the console because he had the goal of instilling fear in a business associate or thegoal of protection during a dangerous drug deal. This seems to be a fairly plausible explanation,
because it had been decided by the previous court that Beard was guilty of conducting the sale ofdrugs, and it is known from previous cases that there is common knowledge about how drug
dealers conduct these exchanges. It is known these exchanges can be highly dangerous, and that
therefore the drug dealers have reasons for being armed.
Next we show how this explanation is based on the use of abductive reasoning to draw
inferences to agents presumed goals and intentions based on evidence of actions and speech
attributed to the agent. In forward practical reasoning (see e.g. Walton, Reed and Macagno,
2008, 323) an agent has a goal, sees that there is an action that is a means of fulfilling to orcontributing to the goal, and concludes that he ought to perform the action. Backward practicalreasoning is an abductive use of practical reasoning in which we observe the actions and speechof an agent, an attempt to explain them by hypothesizing that the agent carried them out in orderto realize some goal or intention (Bex et al. 2009).
In figure 8 it is shown how an explanation can be given to support the hypothesis that Beard
put the derringer in the console. The explanation is based on practical reasoning applied to an
explanation (Bex et al., 2009) and it postulates that Beard had a goal in mind, and that having anavailable gun was a means required to fulfill this goal. The explanation is based on the argumentthat drug dealers can use a gun to achieve their goal of instilling fear in their associates. The
argument thus provides evidence for the causal link between Beard wanting to instil fear and him
bringing a gun, and in turn this provides an explanation for Beard‘s supposed action of putting
the derringer in the console of the car.
Beard had the goalof instilling fear in abusiness associateor the goal ofprotection.

Practical
Reasoning

These goals require
an available gun.

Beard put thederringer inthe console.

Drug dealers bring guns to a deal if theywish to instill fear in a business asociate
or if thet feel the need for protection.

From previous cases there iscommon knowledge about how
drug deal exchanges work.

Figure 8: The Story that Beard Put the Gun in the Console

The third explanation is that one of the other people who used car during the eight-month period
placed the derringer in the console. This explanation is possible, but as pointed out in the court
notes, it is not very plausible that someone would place a loaded gun in the console, cover it withpapers to conceal it, and then forget about it. If the defence had investigated further to collectevidence about who borrowed the car during that period, and what reasons they might have had

17for placing a loaded gun in the console, and then later not removing it, such an investigation
could have given at least some reason to think that this explanation has some slight plausibility.
However, the defence did not present any such evidence, nor did they put forward an argument
giving reasons to suppose that this third explanation could be even slightly plausible. Therefore,
according to the interpretation of the argumentation in the case put forward here, the decision
was arrived at on the basis of evaluating and comparing these three explanations. The first
explanation was ruled out as implausible. The second explanation was shown to be plausible,
given what is known about the general practices of drug dealers during an exchange, even though
there was no evidence to support the explanation other than that given by this generalknowledge.
The remarks quoted above suggest that the hypothesis that someone who borrowed the car
placed a loaded gun in the console and covered with papers to conceal it is not a very lively
possibility. As will be suggested in some remarks quoted from the court transcript below, it
seems implausible that someone would conceal a gun in the console of a car and then forget
about it.
No one supposed that the derringer was the property of the car's owner - that she hid a loaded
gun in the center console when she lent Beard the car. Hence there remain only two possible
explanations of the facts. One is the explanation that Beard placed the gun in the console. The
other is the explanation that someone who borrowed the car from Beard placed it in the console.
The remarks in the court transcript compare the two explanations.
[Quote begins] Since others besides himself used the car during the eight months that he possessed it, conceivably
the gun was left there by one of these users, but that is highly unlikely. It would mean that someone who borrowed
the car from Beard placed a loaded gun in the console, covered it with papers to conceal it, and then — what? Forgot
about it? That is possible, but it was not so lively a possibility as to compel a reasonable jury to acquit Beard [Quoteends].

The problem for the defence is that it provided no evidence to support the alternative story, andwithout such support it is inherently implausible. The court‘s remarks make this clear by asking aquestion. Is it plausible that someone borrowed the car from Beard, placed a loaded derringer inthe console, covered with papers to conceal it, and then forgot about it? As these remarks
suggest, this sequence of actions is just not very plausible as a story.
If someone takes the trouble to conceal
something, he is unlikely to forget about it.

Someone otherthan Beardborrowed thecar.

The borrower
placed a loaded
gun in the console
in the car.

The borrower
covered the gun
with papers toconceal it.

The borrower
forgot that theloaded gun wasin the console
in the car.

A loaded gun is a significant
object that anyone would be
unlikely to forget and leavein a car.

There are potential negative
consequences of someone(police) finding the gun.
Argument from
Negative
Consequences

Figure 9: The Story that the Car Borrower Left the Gun in the Console

18

The third explanation could possibly have been given some plausibility by the collecting ofadditional evidence about the particulars of the case. However the defence did not present any
evidence of this sort. Also, the argument posed in the question above, ―Is it plausible thatsomeone borrowed the car from Beard, placed a loaded derringer in the console, covered withpapers to conceal it, then forgot about it?‖ could have been turned against Beard‘s side. The
defence could have asked the question ―Is it plausible that Beard placed a loaded derringer in the
console, covered with papers to conceal it, then forgot about it?‖ The tactic would have turned
the argument on its head. But the defence did not use this clever argument either. For thesereasons, the best explanation of the three was the second one. Hence it can be conjectured thatthe process of reasoning that led to the conclusion that the gun belonged to Beard can only beadequately represented by using inference to the best explanation.
At the appeal, Beard's lawyer offered no explanation for the presence of the derringer in thecar. The reason suggested for this failure by the trial notes (written by Judge Posner) is worth
quoting.
[Quote begins] The lawyer seems to have thought that since the government had the burden of proof, and Beard wasprivileged not to testify (and he did not testify), it was irrelevant that the jury was given no alternative to thegovernment explanation as to whose gun it was [Quote ends].

The explanation given by the court of the approach of the defence is also worth quoting, because
it cites a number of precedents and writings of legal scholars showing that the argumentation in acase like this needs to be evaluated by looking at competing explanations, and depends on
evaluating the plausibility of alternative explanations against the plausibility of an explanationthat has been put forward.
[Quote begins] "[T]he plausibility of an explanation depends on the plausibility of the 693*693 alternative
explanations." Spitz v. Commissioner, 954 F.2d 1382, 1384 (7th Cir. 1992). And so, "realistically, a jury called upon
to decide guilt must compare the prosecution's version of the incident giving rise to the case with the defence
version." Sandoval v. Acevedo, 996 F.2d 145, 150 (7th Cir.1993); see Ronald J. Allen, "Factual Ambiguity and a
Theory of Evidence", 88 Nw. L.Rev. 604, 611 (1994); Allen, "The Nature of Juridical Proof," 13 Cardozo L.Rev.
373, 409-40 (1991); Allen, "A Reconceptualization of Civil Trials," 66 B.U.L.Rev. 401 (1986). Confidence in a
proposition, such as Beard's guilt, is created by excluding alternatives and undermined by presenting plausible
alternatives. See United States v. Tucker, 716 F.2d 576, 580 (9th Cir.1983); United States v. Reyes, 302 F.3d 48, 56(2d Cir.2002) [Quote ends].

Thus the problem in the case is that the defence failed to carry out a reasonable search forevidence that would support its explanation of the case. As the trial notes put the point, thisobjective of raising sufficient doubt against the given explanation ―is not satisfied just by looking
for ways of poking holes in the government's case‖. In other words, asking critical questions
about the government's case is not enough to refute their argument. What is being said is thatsince the government has provided an explanation, to mount an adequate defence the other side
must put forward a competing explanation that is at least plausible enough to raise reasonable
doubt about the government's explanation. Since the defence failed to do this, and indeed did notprovide any evidence that supported any explanation at all, the jury was left with no alternative
to the government's explanation of what happened in the case.
As the court‘s remarks make clear, the defence offered no alternative argument for theconclusion that the gun belonged to somebody else who had used the car. The defence lawyer
might have thought that, since the prosecution had to prove beyond reasonable doubt, and since
the prosecution's argument was relatively weak, no counterargument against it was needed.
This case was an appeal, and it was noted that the jury hung at Beard's first trial. So it isinteresting to take into account that the jury at Beard's first trial did not find that he was guilty of19carrying a gun in relation to his drug offense beyond a reasonable doubt. In the court notes it isstated that the jury could reasonably have acquitted Beard on this charge. Indeed, the basis of theappeal was that the evidence was so thin that no reasonable jury could have found Beard guilty
beyond a reasonable doubt. And it was also emphasized that the standard applied in judging theappeal was not one of whether the defendant was guilty beyond a reasonable doubt, but one of―whether a reasonable jury could have found he was guilty beyond reasonable doubt‖. Several
cases were cited to support this interpretation (United States v. Powell, 469 U.S. 57, 67, 105
S.Ct. 471, 83 L.Ed.2d 461 (1984); United States v. Morris, 349 F.3d 1009, 1013 (7th Cir.2003);
United States v. Capozzi, 347 F.3d 327, 337 (1st Cir.2003)).
This case is interesting from an argumentation point of view because it suggests that theargumentation in a criminal case is based on competing explanations, and on the comparativeevaluation of the acceptability and plausibility of these explanations. It also tells us something
about the meaning of the how the expression ‗beyond reasonable doubt‘ is applied as a proof
standard. This was a criminal case, and so, as is often emphasized, the defendant had to beconvicted by evidence strong enough to meet the beyond reasonable doubt standard. As stated in
Beard‘s first trial, the evidence the gun belonged to him was very slim. Indeed, there was no
circumstantial evidence that the gun belonged to Beard. All the evidence was about how things
normally go in drug deal. So how could it be decided by the jury on this basis that Beard was
guilty of carrying a gun in relation to his drug offense beyond a reasonable doubt? It does notseem to be logically possible that the issue could have been decided on the basis of theprosecution providing an argument for this claim that was so strong that it met the standard of
beyond a reasonable doubt. Some other interpretation of the logic of the argumentation in thedecision is called for.
The really interesting problem posed by the case is if you look at the argumentation in thecase in the standard way by analyzing an exclusively as an argument without recourse to
explanation, or to inference to the best explanation, it appears that the evidence used to convict
did not meet the standard of beyond reasonable doubt. Judge Posner's comment that the standard
should not be seen as one of finding guilt beyond reasonable doubt, but rather one of whether areasonable jury could have found guilt beyond a reasonable doubt, seems to soften thestrangeness of this finding. However, it does not solve the problem altogether. We still need toask how a reasonable jury could have found guilt beyond a reasonable doubt when the evidence
was so slim. There is still a clash between the judge's instruction that the jury has to find guilt tothe standard of beyond reasonable doubt and the capability for the jury to come to a decision thatthe defendant is guilty based on such slim evidence. If you look at the beyond reasonable doubt
standard of proof as a criterion that has to be met by a chain of argument that mounts up so muchevidence that the conclusion has to be proved without leaving any possibility for reasonable
doubt, the standard is surely not met in a case like that of Beard. The only way to analyze thestructure of the reasoning in such a case in a realistic way is to look at it, instead, as based on acomparing of explanations for the facts in the case.

8. Conclusions
In this paper we have shown how notions of burden of proof and proof standards can beincorporated in a formal model of IBE, namely Bex et al.‘s (2010) hybrid theory of stories andarguments. We have also given the first example of a civil case in the hybrid theory, whichshows that at least some civil cases lend themselves well to being analyzed in the theory. In20

addition to adding to the research on the burden of proof, we have thus also looked at how the
hybrid theory may be expanded. An interesting further expansion of the theory could allow one
to perform legal reasoning (e.g. reasoning with legal rules and exceptions). It would then beinteresting to see how this combination of (hybrid) reasoning about the facts and the law would
influence the modeling of the burden of proof and proof standards.
Even though a precise, quantitative definition of proof standards cannot reasonably be
expected, the hybrid theory is a good tool for analyzing and modeling these standards. Gordon
and Walton (2009) and Prakken and Sartor (2009) argue that for a standard of proof to be met,
one position has to be stronger (by a certain margin) than another and the criteria for the qualityof explanations can be used to give a fine-grained analysis of why one explanation is stronger
than another. In future research, the additional criteria defined in (Bex et a., 2010) and(Pennington and Hastie, 1993) may also be used to further analyze the standards.
Another possible avenue for future research in this respect is to expand the hybrid theory to
make it possible to reason about the (relative) strength of explanations, in the same way as
Prakken and Sartor (1997) allow one to give reasons for priorities between arguments. The
criteria are not hard-and-fast rules for which explanation is the best; this often depends on thecontext of the actual case. In some cases, for instance, Pennington and Hastie‘s (1993) criterionof specificity might play an important role whilst in other cases, such as the Jackson case, it issufficient that the best explanation outlines what happened in abstract terms. Being able to givereasons for priorities between explanations would allow using the criteria as they are intended,
namely as reasons for why one explanation is better than another.
Perhaps the most controversial aspect of our paper is the way we have modeled the beyond
reasonable doubt standard of proof using the hybrid theory. We can broadly sum up in outline
the position we have taken with respect to modeling argumentation meeting the BRD standard asfollows. An explanation of the facts of a case meets the beyond reasonable doubt standard if (a)it is a plausible explanation and (b) the competing explanations (if there are any) are implausible.
Criterion (a) means that the explanation selected not only fits together into a coherent story, but
is also supported by evidential arguments and meets other criteria of a good explanation.
Criterion (b) means that each of the competing explanations is so weak that it fails to raise areasonable doubt. The tactical burden of proof shifts back and forth during the trial as each side
offers arguments to support its own explanation and arguments to weaken the explanation
offered by the other side. At the end, the prosecution has to fulfill its burden of persuasion set
according to the beyond reasonable standard, or otherwise the defence wins.
Our analyses of these cases throws new light on how the logic of reasoning with burden of
proof in both civil and criminal cases can be modeled by an IBE framework that reaches an
outcome by comparatively evaluating competing explanations. It also tells us something about
the meaning of the how the expression ‗beyond reasonable doubt‘ can be applied as a proof
standard in the framework. In the Beard case it did not seem to be logically possible that theissue could have been decided in the way it was on the standard argumentation approach that
requires the prosecution to provide an argument for the charge that was so strong that it met thestandard of beyond a reasonable doubt. However, we have argued for a different model of thelogic of the reasoning in this and comparable cases that may not justify the decision, but can atleast give a more robust and realistic account of how the reasoning behind it works.

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