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In AI & law research several formal models of burdens and standards of proof have been proposed [4, 9, 5, 1, 13, 6, 12, 7]. In this paper we focus on the burden of persuasion, which is the burden to prove a fact to a specified degree, on the penalty of failing to establish that fact in the proceedings. Proof standards for the burden of persuasion differ depending on the type of case. For example, in Anglo-American jurisdictions, in civil cases the proof standard usually is «preponderance of evidence» while in criminal cases it is «beyond reasonable doubt». The burden of persuasion is verified at the end of a proceeding, after all evidence has been provided, so its verification is purely a matter of inference, since no further evidence can be adduced. Accordingly, in [12] we logically characterised the burden of persuasion as the task to make sure that in the final stage of the proceedings there exists a justified argument for the claim. We claimed that to model this, a Dung-style argumentation logic can be used, that is, a logic that generates a set of arguments with a binary relation of defeat [3], such as our logic of [11]. We also proposed that proof standards for the burden of persuasion can be incorporated in the defeat relation between arguments for conflicting conclusions. However, we did not formalise the latter proposal. A first aim of this paper is to provide such a formalisation. A second aim of this paper is to reconsider [9]’s claims that shifts in the burden of persuasion cannot be modelled in a Dungean semantics. To model such shifts, [11]’s argument game for Dung’s so-called grounded semantics was in [9] adapted to allow for role switches between a plaintiff and a defendant, depending on who has the burden of persuasion for a claim. To date no Dungean semantics for the adapted game has been found, which is a theoretical drawback.

An alternative model of reasoning with burdens and standards of proof is the Carneades argumentation framework [5, 6]. In Carneades, arguments for and against statements can be constructed and to each statement a separate standard of proof can be assigned. A statement is acceptable if it satisfies its standard of proof. Until recently, it was an open question whether Carneades could be given a Dungean semantics but Bas van Gijzel [14] translated Carneades into Dung’s frameworks via the ASPIC+ framework of [10]. In light of these results, it would seem at first sight that simply adopting Carneades solves the semantic problem noted in [9] and in addition provides a suitable formalisation of proof standards. However, we will argue that Carneades’ treatment of proof burdens and proof standards is not fully satisfactory. Accordingly, we will present a way to model shifts of the burden of persuasion within ASPIC+ by changing its defeat relation, thus retaining Dungean semantics. A crucial element in our analysis is a distinction between default and inverted burdens of persuasion.

In this section we review Dung’s [3] abstract argumentation frameworks and the ASPIC+ framework of [10]. An abstract argumentation framework (AF) is a pair [Args,Def], where Args is a set of arguments and Def ⊆ Args×Args is a binary relation of defeat. A semantics for AFs returns sets of arguments called extensions, which are internally coherent and defend themselves against attack. A key notion is that of an admissible set of arguments: S ⊆ Args is admissible if it is conflict-free (no argument in S defeats an argument in S) and for all A ∈ S: if B ∈ Args defeats A, then some C ∈ S defeats B. Each extension in maximal in some sense; different semantics define different notions of maximality. Since their differences do not matter for our purposes, we will not present Dung’s semantics here. Relative to a semantics, an argument is justified on the basis of an AF if it is in all extensions of the AF returned by the semantics, it is overruled if it is defeated by a justified argument, and it is defensible if it is neither justified nor overruled.

The ASPIC+ framework [10] gives structure to Dung’s arguments and defeat relation, integrating and generalising work of [8, 15, 11] and others. It assumes an unspecified logical language L with a contrariness relation − and defines arguments as inference trees formed by applying strict or defeasible inference rules of the form φ1, . . . , φ n → φ and φ, . . . , φ n ⇒ φ, interpreted as ‹if the antecedents φ1, . . . , φn hold, then without exception, respectively presumably, the consequent φ holds›. In order to focus on the essence, we present a simplified version, with a symmetric instead of arbitrary contrary relation and with just two types of premises instead of four.

In ASPIC+ argumentation systems are applied to knowledge bases to generate arguments and counterarguments. Combining these with an argument ordering results in argumentation theories, which generate Dung-style AFs.

In this section we give a conceptual analysis of the burden of persuasion, arguing for a distinction between default and inverted burdens of persuasion. This distinction will be the basis for our formal proposal in the following sections.

Any legal dispute has a main claim defended by one party and opposed by another party. Usually the burden of persuasion is on the main claim. We call this the default burden of persuasion. It is by default inherited by any proposition advanced in support of the main claim (either directly as a premise or conclusion of a subargument, or indirectly as an element of an argument that attacks a counterargument). However, this default inheritance is blocked if an explicit burden is assigned to an element of a counterargument of the opposing side. This we call an inverted burden of persuasion. Once a burden is inverted, then new burdens explicitly assigned to elements in the defending side’s counter-counterarguments are also of the inverted type, and so on. As a matter of fact, in both civil- and common law jurisdictions the burden of persuasion is never inverted in criminal cases while it can be inverted in civil cases.

The difference between default and inverted burdens is only relevant when they are not met. If a default burden of persuasion for Φ is not met, then Φ is merely considered unprovable while if an inverted burden for Φ is not met, the opposite, that is −Φ, is considered to hold, even if the arguments pro and con Φ have equal weight. Thus, when the evidence is balanced, the two burdens lead to different outcomes: if there is a default burden on Φ, a legal reasoner may well remain agnostic on the matter, while if there is an inverted burden on Φ, he must accept −Φ as legally justified. In other words, an inverted burden on Φ generates a presumption of −Φ in case the evidence is balanced.

Since in criminal cases the burden of persuasion is never inverted, if arguments for selfdefence are provided, then the prosecution has to remove any reasonable doubt on this issue. Assume first that the prosecution attempts to do so with an argument for ￢ selfdefence. Then, on our analysis, if it outweighs any argument for selfdefence beyond reasonable doubt, then ￢ selfdefence is justified and selfdefence is overruled. If the evidence is balanced or the arguments for ￢ selfdefence are stronger but not beyond reason-able doubt, then both statements are defensible. Finally, if the evidence for selfdefence is stronger by any degree, then selfdefence is justified and ￢ selfdefence is overruled. Assume next that the prosecution does not provide an argument for ￢ selfdefence but an argument attacking a premise, subargument or defeasible inference of the accused’s argument for selfdefence. Then our analysis applies likewise: if the prosecution’s argument outweighs its target beyond reasonable doubt, then the argument for selfdefence is overruled and the argument for pubishment is justified.

Assume also that (as in Italian law) the burden of persuasion with respect to negligence is inverted, that is, the doctor has the inverted burden to prove ￢ negligence. If both arguments for and against negligence are provided, then our analysis implies the following. If the argument for ￢ negligence outweighs the argument for negligence, then ￢ negligence is justified. However, if the evidence is balanced, then since the burden is an inverted one, instead negligence is justified and ￢ negligence is overruled. A fortiori this also holds if an argument for negligence is stronger.

Assume next that the burden of persuasion concerning negligence is not inverted. Then the outcome should be different when the evidence on negligence is balanced: then all of negligence, ￢ negligence and compensation should be defensible.

In Carneades [5, 6] an argument has a set of premises P, a set of exceptions E and a conclusion c, which is either pro or con a statement. Unlike in ASPIC+, all arguments are elementary, that is, they contain a single inference step; they are combined in recursive definitions of applicability of an argument and acceptability of its conclusion. In essence, an argument is applicable if (1) all its premises are given as a fact or are else an acceptable conclusion of another argument and (2), none of its exceptions is given as a fact or is an acceptable conclusion of another argument. A statement is acceptable if it satisfies its proof standard. Facts are stated by an audience, which also provides numerical weights of each argument plus thresholds for argument weights and differences in argument weights. Three of Carneades’ proof standards are then defined as follows:

Let us consider Carneades in light of our above conceptual analysis. We first argue that its notions of burdens and standards of proof do not adequately model their legal counterparts. To start with, since Carneades models proof burdens and proof standards as orthogonal concepts, it does not model that in the law proof standards are relative to a proof burden in that first a proof burden is assigned to either Φ or ￢Φ and then the proof standard is only assigned to the statement that has the burden of proof. Among other things, this leads to the following problem. In [6] the burden of persuasion for a statement Φ is said to be met if Φ is acceptable in the final stage of a dispute. However in our Example 3.1 this implies that if selfdefence is acceptable in the final stage, then the accused has fulfilled his burden of persuasion for selfdefence, while the law instead assigns the burden of persuasion for ￢ selfdefence to the prosecution.

Next, in order to respect our conceptual analysis of Example 3.1, a Carneades user has to represent the example differently depending on the stage of a dispute (as also noted by [6]). To model that initially the accused has the burden of production for selfdefence, this statement has to be modelled as an exception. Consider the following arguments:

• A₁ with P = {crime}, E = {selfdefence} and a conclusion pro punishment

• A₂ with P = any, E = ∅ and a conclusion pro selfdefence

• A₃ with P = any, E = ∅ and a conclusion con selfdefence (= pro ￢ selfdefence)

Let us assume that with A₂ (and before A₃ has been moved) the accused has fulfilled its burden of production. Then A₁ has to be modified by making ￢ selfdefence an ordinary premise:

• A’₁ with P = {crime, ￢selfdefence}, E = ∅ and a conclusion pro punishment

If not, then if neither for selfdefence nor for ￢ selfdefence the proof standard is satisfied (so there remains reasonable doubt on this issue), then A₁ is acceptable since its exception selfdefence is not acceptable. Clearly, this is counterintuitive, so A₁ has to be modified to A’₁: then if there remains reasonable doubt on selfdefence, A’₁ is not acceptable. However, then the problem is that attempts by the prosecution to meet its burden of persuasion for punishment by attacking a premise of A₂ without constructing an argument for ￢ selfdefence cannot be modelled. So [6] representation method does not fully respect our analysis of Example 3.1.

We now formalise our idea in [12] that proof standards for the burden of persuasion can be incorporated in the defeat relation. A stronger argument should strictly defeat a weaker rebuttal only if the degree to which it is stronger satisfies the applicable proof standard; otherwise both arguments should defeat each other. For example, if the standard is ‹preponderance of evidence›, then A already strictly defeats B if A is just a little bit stronger than B, while if the standard is ‹beyond reasonable doubt›, A strictly defeats B only if A is very much stronger than B. We combine this with a formal account of the difference between default and inverted burdens of persuasion. For simplicity we only consider the proof standards preponderance and beyond reasonable doubt.

For ease of comparison we retain Carneades’ weights and thresholds. They are used here just as a language to express natural-language judgements about argument strength. For example, in practice such judgements are often not just about whether one argument is stronger than another but about whether the difference in strength is sufficient to let one argument defeat another. Such a comparison can be expressed in our proposal as w(A) > w(B) + β, without having to specify or calculate with numbers. In the examples we use numbers for ease of illustration only, as shorthand for such comparative judgements; no further meaning should be read into the numbers.

Our method still produces a set of arguments plus a binary defeat relation, so it still generates an abstract argumentation framework according to Definition 2.7, which has Dungean semantics. We now assume various other input elements besides a knowledge base, namely, a given explicit allocation of inverted proof burdens to statements, plus a given assignment of a proof standard to each statement that has a proof burden. The new definition of successful rebuttal considers whether the burden of persuasion is on Φ or on −Φ, and it replaces the reference to the argument ordering ≤ with an expression in terms of the weights of the rebutting arguments. Since these expressions assume numerical argument weights and thresholds for these weights, we now also assume these as given.

The idea behind the definition is twofold (i) Shifts in the burden of persuasion are captured since arguments that attempt to fulfill an inverted burden defeat their target only if they are stronger than their target. (ii) Proof standards are modelled by incorporating [6]’s β threshold in our definition of defeat. Note that with the preponderance standard (2) and (3) are equivalent, since in that case β = 0. Note also that if there are no other relevant arguments and assuming for beyond reasonable doubt that A exceeds the threshold α and B does not meet the threshold γ, then A for p strictly defeats B for –p just in case A meets the proof standard for p according to Definition 4.1. Let us illustrate the definition with our two running examples. For simplicity we assume that weights range from 0 to 1, and that beyond reasonable doubt is satisfied iff the weight of an argument exceeds 0.9. Then for this standard β = 0.8 (recall that for preponderance it is 0). In our criminal case, we have dbop(punishment) and dbop(¬selfdefence). Consider again the arguments A₁,A₂,A₃ but now in their obvious ASPIC+ format. Assume first w(A₂) = 0.05, w(A₃) = 0.95. Then w(A₂) + β = 0.85, and 0.85 < 0.95 so A₃ defeats A₂ while A₂ does not defeat A₃. So ¬selfdefence and punishment are justified, as we want. Assume next w(A₂) = 0.3, w(A₃) = 0.7. Then w(A₂) + β = 1.1, so while A₃ defeats A₂, now A₂ also defeats A₃. So all of ¬selfdefence, ¬selfdefence and punishment are justified, as we want. Assume finally w(A₂) = 0.6, w(A₃) = 0.4. Then w(A₂) + β = 1.4, so A₂ now defeats A₂ while A₃ does not defeat A₂. So selfdefence is justified while ¬selfdefence and punishment are overruled, as we want.

In our civil case, we have ibop(¬negligence). Furthermore, our idea is that we have dbop(compensation) since this is plaintiff’s main claim, while we don’t have dbop(negligence) since the inverted burden is on the opposite. Consider the following arguments (leaving implicit what are the facts and rules):

arguments A₄ and B₂ rebut each other on whether the doctor was negligent. Assume first w(A₄) = 0.₄, w(B₂) = 0.6. Then B₂ defeats A₄ by clause (1) of definition 5.2, while A₄ does not defeat B₂ by clause (3). So ￢negligence is justified while negligence and compensation are overruled, as we want. Assume next w(A₄) = 0.6, w(B₂) = 0.4. Then A₄ defeats B₂ by clause (3) while B₂ does not defeat A₄ by clause (1). So negligence and compensation are justified while ￢negligence is overruled, as we want. Assume finally w(A₄) = w(B₂) = 0.5. Then A₄ defeats B₂ by clause (3) while B₂ does not defeat A₄ by clause (1). So again negligence and compensation are justified while ￢negligence is overruled, as we want.

Assume next that the burden of persuasion with respect to negligence is not inverted. Then the plaintiff has the default burden of persuasion for both compensation and negligence. Then the outcome only differs in the third case, that is, when w(A₄) = w(B₂) = 0.5. Then A₄ defeats B₂ by clause (2) while B₂ defeats A₄ by clause (3). So all statements are defensible, as we want. In sum, our formal proposal adequately models our conceptual analysis of Section 3, as illustrated by our two running examples. One issue is left to be discussed, namely, how an explicit allocation of inverted burdens and proof standards determines burdens and standards for other statements. Let us add the following arguments to the ones above in our civil example (see Figure 1):

As we said above, we want that plaintiff’s main claim compensation has a default burden of persuasion. Then, for instance, since A₁ is a subargument simply supporting plaintiff’s main claim and no explicit inverted burden was assigned to injury, that statement should also have a default burden. On the other hand, negligence should not have a default burden even though A₄ also is a subargument supporting plaintiff’s main claim. The reason is that A₄ is rebutted by B₂ and the conclusion of B₂ has an inverted burden: then negligence should have no burden. Let us next look at A₃ and B₅. The latter argument rebuts a subargument for negligence and therefore indirectly supports B₂ for ￢ negligence, which has an inverted burden. Therefore, riskyOperation should also have an inverted burden. This in turn implies that ￢riskyOperation should have no burden. Now we can also say more about, for instance, B₄ and C₂. Since B₄ supports an argument for an inverted burden, badCirculation should also have an inverted burden, so ￢badCirculation should have no burden. All these default inferences can be overridden by explicit inverted burden, but in our example these were not given.

In [9] similar ideas were formalised in terms of an argument game, but in this paper we are focussing on Dung-style semantics, so this route is not open to us. Since a formalisation is not trivial and our space is limited, we leave this issue for future research.

In this paper we have proposed a formalisation of reasoning with burdens and standards of proof, overcoming some shortcomings of earlier work of ourselves and the Carneades system of [5, 6]. Our main contribution is a Dungean semantics for argumentation with shifting burdens of persuasion, while we have also shown how Carneades’ proof standards can be modelled within our semantics. A key ingredient in our model was a distinction between default and inverted burdens of persuasion.

In future research we wish to formalise our sketch in Section 5 of how explicit proof burdens determine implicit burdens.We also want to incorporate [7]’s idea to make proof burdens relative to legal rules. Finally, we aim to investigate whether our proposal respects the consistency and closure postulates of [2]. In [10] these postulates were proven for ASPIC+ as summarised above in Section 2 but since we changed its definition of defeat, satisfaction of the postulates has to be investigated anew.

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Henry Prakken. Department of Information and Computing Sciences, Utrecht University and Faculty of Law, University of Groningen, The Netherlands.

 

Giovanni Sartor. CIRSFID, University of Bologna and European University Institute, Law Department, Florence, Italy. Giovanni Sartor was supported by the ALIAS (Addressing Liability in Automated Systems) project, cofinanced by EUROCONTROL on behalf of the SESAR Joint Undertaking.

 

This article is republished with permission of IOS Press, the authors, and JURIX, Legal Knowledge and Information Systems from: Kathie M. Atkinson (ed.), Legal Knowledge Systems and Information Systems, JURIX 2011: The Twenty-Fourth Annual Conference, IOS Press, Amsterdam et al.

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