Burdens and Standards of Proof forInference to the Best Explanation
Floris BEXa,1 and Douglas WALTON b
Argumentation Research Group, University of Dundee, United KingdombCentre for Research in Reasoning, Argumentation and Rhetoric (CRRAR), Universityof Windsor, CanadaaAbstract. In this paper, we provide a formal logical account of the burden of proof
and proof standards in legal reasoning. As opposed to the usual argument-basedmodel we use a hybrid model for Inference to the Best Explanation, which uses
stories or explanations as well as arguments. We use examples of real cases toshow that our hybrid reasoning model allows for a natural modeling of burdens
and standards of proof.
Keywords. burden of proof, inference to the best explanation, arguments, stories

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 oflogically modeling the legal burden of proof and proof standards have been proposed[6], [12]. These approaches both presuppose some type of frameworks for defeasible
argumentation, in which arguments (or argument graphs) are constructed by
performing consecutive reasoning steps from the evidence to the facts in issue.
When talking about the facts of a criminal case, [2] have argued that argumentative
approaches such as [6],[12] 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 of arguments are individual facts
in issue. In a real case, however, these facts will be related to each other in various
ways (e.g. causally, temporally, motivationally) and these relations may also be thesubject of argumentative reasoning. [9] have shown that using explanatory stories isclosest to how legal decision makers actually think about a case and [8] have argued
that reasoning in trials involves the relative plausibility of the various explanations of
the evidence. It is argued that this is not only true for criminal but also for civil trials, inwhich the parties have to provide competing “theories” about the facts of the case.2
In sum, the reasoning performed in legal trials involves arguments as well as
explanations, at least when reasoning about the facts of the case. [2] propose a hybrid
theory of Inference to the Best Explanation (IBE), which consists of a combination of1

Corresponding Author.
[8] cite a number of civil cases that involve competing explanations of the events. Examples areAnderson 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

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 using arguments
based on evidence. Ultimately, the alternative explanations in a case should becompared. In this comparison, the burden of proof and standards of proof play animportant role. For example, even if the prosecution’s explanation of guilt in a criminalcase is the “best” explanation (according to some rational standard) it may still notmeet the legal “beyond a reasonable doubt” standard, resulting in the acquittal of the
defendant. Similarly, when the parties’ alternative explanations are equally good, theburden 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. [8] discuss how the burden of
proof influences the process 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 isentirely informal and not specified in exact detail. [7] gives a formal model ofabductive 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.
In this paper we make a first tentative step in logically modeling reasoning withthe burden of proof and proof standards in IBE, using [2]’s hybrid theory of IBE as ourmodel of reasoning. Thus, we want to explore how the different types of burdens andstandards of proof may be modeled for reasoning that is not purely argument-based.
The rest of this paper is organized as follows. Section 1 contains a summary of [2]’shybrid theory. Section 2 briefly discusses different types of legal burdens and standardsof proof. In section 3, we model these burdens and standards of proof in the hybrid
theory and provide examples of a civil case (section 3.1) and a criminal case (section
3.2). Section 4 concludes the paper and makes some proposals for further research.

1. A Hybrid Theory of Inference to the Best Explanation
In [2], the authors propose a hybrid theory for reasoning with arguments, stories andcriminal evidence. The basic idea of this hybrid theory is that the propositions to be
explained, the explananda, are causally explained by different stories, alternative
accounts of what happened in the case. These stories can then be reasoned about usingarguments. For example, arguments can be used to support the story with evidence orto reason about the relative plausibility of the stories.
The formal hybrid theory HT = (ET, CT) is a combination of a causal-abductive
theory CT and an evidential argumentation theory ET. The logic L of this theory is acombination of the inference rules of classical logic and a modus ponens inference rule
for the connective ⇒ (defeasible implication). Object-level rules in CT and ET areformalized 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 anevidential rule used in arguments and ⇒C denotes a causal rule used in explanations.
In the abductive theory CT = (H, T, F), T is a set of causal rules, H is the set of
hypotheticals, ground literals occurring in the antecedent of some rule in T and F is theset of explananda. The basic idea of abductive inference is that if we have a rule cause→ effect in T and we observe effect, we are allowed to infer cause as a hypotheticalexplanation of the effect. Such an explanation can be a single proposition but it canalso be a causally connected story consisting of chains of causal rules. Thus, given F
and T we need to abductively infer some hypotheticals such that together with T they
explain F, that is, the explananda follow from the hypotheticals and the causal theory.
More precisely, S = Hi ∪ Ti, where Hi ⊆ H and Ti ⊆ T, is an explanation for a set of
explananda F iff for each f ∈ F: S ⊢ f. Furthermore, we require that S is consistent andthat 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
∪ KA is a knowledge base, where KE is a consistent set evidence and KA is a set ofcommonsense assumptions. The logic for ET is similar to the ASPIC logic [10], which
integrates ideas on rule-based argumentation and structured arguments (e.g. [12])within [3]‘s abstract approach. Evidential arguments can be built by taking evidence orassumptions from K and rules from R as premises and chaining applications ofdefeasible modus ponens into tree-structured arguments, where each node in the tree isthus an element 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 each other if they have an opposite (intermediate) conclusion. AR1 undercuts AR2
if there is a conclusion ¬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 someassumption (from KA) in AR2. For a collection of arguments and their binary defeat
relations, the dialectical status of the arguments can be determined: arguments can beeither justified, which means that they are not defeated by other justified arguments,
overruled, which means that they are defeated by other justified arguments, ordefensible, which means that they are neither justified nor overruled.
In IBE, multiple explanations should be generated and compared according tocriteria that express the degree to which they conform to the evidence and theirplausibility. These criteria are defined using the argumentation theory. Arguments
based on evidence can be used to show that an 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 anargument 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 Aand ¬s is a conclusion of A, where s ∈ S). Arguments may also be used to reason about
the plausibility of an explanation, as the validity and applicability of causal rules canbecome the subject of an argumentation 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 an argument A which is not based on evidence(i.e. s ∈ S and ¬s is a conclusion of A, where A’s premises {a1,…,an} ⊆ KA). Here,arguments about the plausibility of explanations are based on assumptions from KA, asreasoning about plausibility is done using commonsense knowledge about how theworld generally works. Note that for the criteria, only arguments which are notoverruled support or contradict an explanation: if an argument based on evidence isitself 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 the comparison of explanations. In principle, the higher the evidential support
and the lower evidential contradiction and implausibility, the better the explanation.
This comparison of explanations can be further specified if desired: [2] define more

criteria for judging the quality of explanations, such as completeness (whether the story
has all its required elements in order to be coherent), and [9] also provide other criteriasuch as the exclusivity and specificity of explanations. Furthermore, we could attachdifferent weights to evidence so that particular pieces of evidence give a higher degreeof support to explanations than others. In addition, we could also distinguish between
explanations supported by defensible arguments and explanations supported byjustified arguments. For current purposes, however, the three criteria are sufficient.
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.

2. Burdens of Proof and Proof Standards
Allocation of burden of proof tells each side in a dispute how strong its argumentationneeds to be in order to be successful in prevailing over contention of the other side. [4]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
achieved by one side to win an argument. On their account, burden of proof has twofunctions ([4], p. 156): to act as a move filter at local moves in a dialogue and to act asa termination criterion that determines the winner at the end of the dialogue.
In their logical account, [12] define three kinds of burden of proof in terms of theirframework for defeasible argumentation. 3 The first type of burden of proof is theburden of persuasion. It is set by law at the opening stage of a trial, 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 examined([12] define it as the task of making sure there is a winning argument for one’s claim).
The burden of persuasion does not shift from the one side to the other during 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 onsome specific issue during the trial. If the evidence offered at any point by one side
does not meet this burden, the issue can be decided as a matter of law against this side,and that is the end of the trial.
The third type of burden is called the tactical burden of proof. It is a hypotheticalassessment 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 tacticalburden can shift back and forth between the parties any number of times during the trial,
because it depends on who has the winning argument at a particular point in the trial.
Not all sources in law agree, but in general the tactical burden is the only one of thethree of the three burdens that can be properly said to shift during the course of a trial.
[12] (p. 227) argue that the distinction between the burden of production and tacticalburden 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 of law,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 the domain of common law, there are four main proof standards for factual3

The burdens of claiming and contesting, which are also discussed by [12], assume an explicit
dialogical context and will hence not be discussed here.

issues called scintilla of evidence, preponderance of evidence, clear and convincing
evidence, and beyond reasonable doubt ([6], p. 241). The scintilla of evidence proof
standard is met if “even the slightest amount of relevant evidence exists on an issue”([5], p. 1464). The preponderance of evidence standard is met by “evidence that has themost 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” ([5], p. 1301).
The preponderance of evidence standard is often compared to a balance, where theevidence 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 ishighly probable or reasonably certain” ([5], p. 636). This standard is supposed to behigher than that of preponderance of the evidence, but not as high as the highest
standard in law, that of evidence beyond a reasonable doubt. Finally, the beyond
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 thepurposes of providing computational models of the standards useful in AI and law. A
definition expressed in, for example, terms of the conviction of the jury that the charge
against the defendant is true ([5], p. 1380) is not very helpful for getting an idea of how
this standard should be represented in a normative model of rational argumentation.
It is debatable whether such a precise definition of proof standards can be given.
Take, for example, the standard of beyond a reasonable doubt. According to
McCormick on Evidence ([13], p. 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 of reasonable doubt itself needs no definition([13], p. 447), the reason being that any definition might have to be so subtle andtechnical that there would be dangers of misunderstandings if judges were to instructjuries with it. This judicial climate of opinion poses an apparently insurmountablechallenge for any attempt to provide a computational model of the proof standard.
However, as [14] have shown, even though there is a well-settled maxim supported by
judicial wisdom that the beyond reasonable standard is not quantifiable by assigningprobability values to it, it does not follow that this standard of doubt is not open toprecise analysis based on a computational argumentation model.
[6] show how proof standards can be analyzed in the formal Carneades system asfollows. For there to be a Scintilla of Evidence (SE), there should be at least one
applicable argument4 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 the claim) 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 of applicable pro arguments exceeds some threshold α, and thedifference 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 is satisfied and the maximum weight of theapplicable con arguments is less than some threshold γ. Notice that here the thresholds
4

Roughly, an argument is considered applicable if its premises are not defeated and there is no exception
to the inference.

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 not easily done.

3. Burden of Proof and IBE
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 [8], who provide an informal overview of IBE in trials. Note that we
mainly concern ourselves with the factual part of trials, that is, the evidence and thefacts which we might infer from this through evidential reasoning. The legal reasoning,which is intertwined with this reasoning about the facts,5 is not shown in detail and we
simply assume that the legal conclusion follows from the established facts in some way.
In the hybrid theory, having the burden of persuasion for an explanation S meansthat at the end 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 theother explanations, so good that the other explanations 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 an evidential argument for the claim on which theburden rests. This claim may be, for example, an element of one’s own explanation or
the negation of an element of the opponent’s explanation. The burden of production isessentially the same as in [12], because of the similarity between the hybrid theory’s(evidential) argumentation theory and [12]’s framework. Finally, the tactical burdenmeans that one should gauge if one’s explanation is currently the best and if it trumps
the other explanations by a particular margin, dependant on the standard of proof.
Standards of proof may also be formalized in the hybrid theory. Definitions ofstandards 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 inthemselves. Following [6], such margins or standards will not be given fixed values but
rather they will be left open. Now, an explanation S meets the Scintilla of Evidence(SE) standard if there is a 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 each alternative explanation S’. That is, all else being
equal S is either supported by more evidence (es(S) > es(S’)) or contradicted by lessevidence (ec(S) < ec(S’)). This way of comparing explanations is similar to that in [1].
However, a key difference is that in [1], 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 just more 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 high evidential 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 muchbetter 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
5
For example, once we have established that it was in fact John who killed Harry, we still need todetermine whether the killing was manslaughter or murder.

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 very strong and muchstronger 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 [8] argue, a plausible explanation consistent withinnocence 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 the evidential support of the competing explanations is not tied to extra
requirements (beyond those set for the CCE standard), as these explanations only haveto be consistent with the evidence.
3.1. Inference to the best explanation in civil trials
In this section we give an example of IBE in a civil trial. In the case, which is based on
the Summary 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 theconnection between the brake pedal and the brakes. Debris kicked up from the surfaceof the highway (road junk) struck a 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, the plaintiffs 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 a looseness in the driveshaft and had asked thetruck 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’srecords about the repairs on the truck (they state that the repairmen did not repair thejoint). Figure 1 shows this explanation (and defendant’s alternative, see below).
Here,arrows denote causal links,
arrows denote evidential (argumentative)
links and thearrow denotes evidential contradiction. White boxes are part of theabductive theory CT (p or d denotes whether events are part of plaintiffs’ ordefendant’s explanation) and gray boxes are part of the argumentation theory ET.
Premises with italic text denote evidence in KE.
truck dealer’s records

plaintiff’s expert

p: truck dealer did not repair driveshaft
d: there wasdebris on the roadwitnesses

d: debris struck
the driveshaft
defendant’s expert

p, d: driveshaft broke

p, d:
crash

driveshaft rotates
plaintiff’s expert

Figure 1: two explanations in the Anderson v. Griffin case.

Defendant, the truck dealer, now has the tactical burden of proof: if he does notquestion plaintiffs’ explanation or provide a better explanation for the crash fromwhich it does not follow that he had been negligent, the jury will rule for plaintiffs.

Defendant gives such an alternative explanation, claiming that debris struck thedriveshaft properly. That there was debris on the road follows from statements made bywitnesses. Defendant could also have denied the fact that he did not repair thedriveshaft (i.e. attacked the plaintiff’s explanation), but seeing 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 explanationwhilst defendant only needs to cast sufficient doubt on this explanation, which he hasdone by providing 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 tomeet the burden of production, i.e. produce evidence so that a fact-finder candifferentiate between explanations. Thus plaintiffs now have the tactical burden of
proof and they produce evidence to improve their explanation: an expert witness whostates that the crash was caused by the fact that defendant did not repair the driveshaft.
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 thedriveshaft 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 supports his own explanation with an expert testimony. It was
proposed that the accident had been caused by debris on the highway that might havebeen yanked up and against the driveline by chains hanging from the truck.
Both explanations now have equal support, so the tactical burden again shifts backto plaintiffs, who decide to attack defendant’s explanation. The plaintiffs argued that apiece of road junk 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).
Now plaintiffs’ explanation is slightly better than defendant’s so the Preponderance ofEvidence standard seems to have been met. However, in the case the jury ruled for the
defendant; for some reason, they must have found that the attacking argument based onplaintiffs’ expert was not convincing enough.
3.2. A Criminal Case with Two Competing Stories
As an example of a criminal case we discuss Jackson v. Virginia: 443 U.S. 307. The
case concerns 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 andJackson stayed in contact. Witnesses testified that on the day the crime was committed,
Jackson had been drinking while shooting 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 in Cole’s car the sheriff testified that hehad 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 in sexual activity.
The same evening, Jackson drove from Virginia to North Carolina. A day and a halflater, 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.
The prosecution proposed an explanation SP from which it followed that Jackson
murdered Cole. The explanation first recounted the events at the diner and then,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) andpolice 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 the trial was whether there was sufficient evidence toprove 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. Thus, the prosecution
gave an argument A1 that Jackson knowingly (and hence intentionally) performed hisact, using the generalization “if a person who is proficient with firearms shoots
someone else at close range, the shooter intended to kill the victim” (more formally r1:
x is proficient with firearms ∧ x shoots y at close range ⇒E x intended to kill y).
The prosecution met the burden of persuasion and production for their explanation:
SP had sufficient evidential coverage and no evidential contradiction and there was nocompeting explanation. Jackson had the tactical burden which he could meet by casting
doubt on SP. Under the BRD standard, this doubt may be created by proposing asufficiently plausible alternative explanation consistent with the evidence. Jacksonpresented 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, andthen reloaded the weapon. When Cole attempted to take the gun away from him, itwent off during the ensuing struggle. He claimed that he had fled without seeking help
for Cole because he was afraid. Later, during the trial he claimed that he had acted inself-defense. He also offered the argument A2 that, as the State’s own evidence (i.e. thepolice officers’ testimonies) showed, he had been too intoxicated to form the specificintent necessary to make him guilty of the crime of first-degree murder. In other words,
he undercut r1, thus taking away the support this argument gave to SP.
The tactical burden then shifted to the prosecution, who had to defend theargument A1 (for Jackson’s intentions) by giving a counterargument to A2 (that he wasintoxicated) as well as show that Jackson’s explanation SJ is implausible. First, the factthat Jackson drove without mishap from Virginia to North Carolina was taken to be atodds with his argument of extreme intoxication at the time of the killing. Thus, A2 wasconsidered overruled and A1 again justified. It was further argued that SJ contained
implausible elements. First, Jackson said to the police officer’s he was going engage insexual activity with Cole but later supposedly resisted Cole’s sexual advances.
Furthermore, it was found implausible that Cole first willingly removed part of her
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.
Now SP’s quality is restored (as A1 again supports it) and SJ’s implausibility isdemonstrated and thus the BRD standard is met, as is evident from the trial record.

4. Conclusion
In this paper we have shown how notions of burden of proof and proof standards canbe incorporated in a formal model of IBE, namely [2]’s hybrid theory of stories andarguments. We have also given the first example of a civil case in the hybrid theory,
which shows that at least some civil cases lend themselves well to being analyzed in
the theory. In 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 withlegal rules and exceptions). It would then be interesting to see how this combination of(hybrid) reasoning about the facts and the law would influence the modeling of theburden 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 thesestandards. [6] and [12] argue that for a standard of proof to be met, one position has tobe stronger (by a certain margin) than another and the criteria for the quality of
explanations can be used to give a fine-grained analysis of why one explanation isstronger than another. In future research, the additional criteria defined in [2] and [9]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 thesame way as [11] 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 the context of the actual case. In some cases, for instance, [9]’s criterion of
specificity might play an important role whilst in other cases, such as the Jackson case,
it is sufficient that the best explanation outlines what happened in abstract terms. Being
able to give reasons for priorities between explanations would allow using the criteria
as they are intended, namely as reasons for why one explanation is better than another.

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