Many sources of information provide uncertain information. Such information may come with probabilistic estimations of how likely specific events are (think of a weather report), in which case we deal with (precise or first order) probabilistic uncertainty. However, often probabilistic information is missing, or the probabilities are not known precisely, in which case we deal with higher-order uncertainty (in short, HOU). HOU occurs when the underlying probability distribution is not or only partially known.¹ We illustrate the role of HOU with two examples.

Example 1 (ComArg). The ComArg conference is to be held during December 2023. We have the following information concerning the question whether ComArg will be held hybrid (see Figure 1, left).

The answer to the question whether the next ComArg conference will be held hybrid is uncertain. Moreover, one cannot attach a precise probability to it: the best that can be said is that it has at least the likelihood 0.7 (given statements 1 and 3). We are dealing with HOU in contradistinction to mere first order uncertainty: in contrast to the question how likely a COVID wave in autumn is, the question how likely it is that ComArg will be held hybrid has no precise answer.

Example 2 (Ellsberg, 1961). Suppose an urn contains 30 red balls, and 60 non-red balls, among which each ball is yellow or black, but we do not know the distribution of yellow and black balls. The question of whether a randomly drawn ball is red is one of first order uncertainty since it comes with the (precise) probability of ⅓. The question whether it is yellow is one of HOU since the available probabilistic information does not lead to a precise probabilistic estimate. See Figure 1 (right) for an illustration.

Since our environments come with many sources of uncertain information, both quantifiable and not, it is not surprising that human reasoning is well-adjusted to dealing with such situations. What is more, human reasoning distinguishes the two types of uncertainty by treating them differently. For example, in Example 2 people are more willing to bet on drawing a red ball than on drawing a yellow ball in a game in which one wins if one bets the right color. This phenomenon is known as ambiguity (or uncertainty) aversion. The distinction can be traced back both to the psychological and neurological level. For instance, different types of psychological or other medical problems are associated with a compromised decision making under first order uncertainty, but not under HOU [e.g., gambling problems in Brevers et al. (2012), obsessive-compulsive disorder in Zhang et al. (2015), pathological buying issues in Trotzke et al. (2015)] and vice versa (e.g., Parkinson's disease in Euteneuer et al., 2009). This shows that different causal mechanisms are related to the human capacities of reasoning with the two types of uncertainties. Similarly, on the neurological level differences can be traced, though it is still an open issue whether the two uncertainty types have separate or graded representations in the brain [see De Groot and Thurik (2018), to which we also refer for a recent overview on both the psychological and neurological literature].

What the discussion highlights is that a formal model of human reasoning should pay special attention to both types of uncertainties and provide a framework that can integrate mixed reasoning processes, such as in our Examples 1 and 2. The same can be stated for AI for the simple reason that in many applications artificial agents will face sources of uncertain information.

When reasoning with uncertain information, we infer defeasibly, that is, given new (and possibly more reliable) information we may be willing to retract inferences. As forcefully argued on philosophical grounds in Toulmin (1958), reasoning is naturally studied as a form of argumentation. Similarly, the cognitive scientists Mercier and Sperber developed an argumentative theory of human reasoning (Mercier and Sperber, 2017). Dung's abstract argumentation theory (Dung, 1995) provides a unifying formal framework for an argumentative model of defeasible reasoning and has been widely adopted by now both in the context of symbolic AI and to provide explanatory frameworks in the context of human cognition (Saldanha and Kakas, 2019; Cramer and Dietz Saldanha, 2020). Several ways of instantiating abstract argumentation with concrete formal languages and rule sets have been proposed, such as ASPIC+ (Modgil and Prakken, 2014), assumption-based argumentation (Dung et al., 2009), and logic-based argumentation (Besnard and Hunter, 2001; Arieli and Straßer, 2015).

It would therefore seem advantageous for the theoretical foundations of HCAI to combine formal argumentation with a representation of first and higher-order uncertainty. This paper will propose such a formal framework.

Several formal models of this type of reasoning are available: from imprecise probabilities (Bradley, 2019) to subjective logic (Jøsang, 2001) and probabilistic argumentation (Haenni, 2009). However, the link to the leading paradigm of computational argumentation, namely Dung-style argumentation semantics, is rather loose.

Probabilistic argumentation with uncertain probabilities is comparatively understudied in formal argumentation. Works by Hunter and Thimm (Hunter and Thimm, 2017; Hunter, 2022) focus on precise probabilities. Our framework generalizes aspects of such settings to include a treatment of HOU. Also, in contrast to them, we will utilize Dung argumentation semantics in the context of probabilistic argumentation. Hunter et al. (2020) equip arguments with a degree of belief as well as disbelief, notions that can also be expressed in Haenni's framework and will be considered in our study of argument strength. A framework that considers imprecise probabilities is presented by Oren et al. (2007). It utilizes subjective logic in the context of a dialogical approach for reasoning about evidence. Similarly, Santini et al. (2018) label arguments in abstract argumentation with opinions from subjective logic. In contrast, our study focuses on structured argumentation.

Mainly starting with the seminal (Ellsberg, 1961), HOU has been intensively studied in the context of decision theory. As has been shown there, human reasoning with HOU may lead to violations of axioms of subjective expected utility theory (as axiomatized in Savage, 1972), leading to several alternative accounts [e.g., maxmin expected utility in Gilboa and Schmeidler (2004) or prospect theory in Kahneman and Tversky (1979)]. In this paper, we omit utilities, values, and practical decision making and concentrate instead on reasoning in the epistemic context of belief formation and hypothesis generation. As we will show, even without utilities HOU gives rise to interesting and challenging reasoning scenarios.

In this paper we integrate reasoning with HOU in abstract argumentation. For achieving this goal, several key questions have to be answered:

Our work takes as the starting point the theory of probabilistic argumentation developed in Haenni (2009). The framework is enhanced by (1) a structured notion of argument in the style of logical argumentation, (2) argumentative attacks, (3) several notions of argument strength [based on notions of degree of support and degree of possibility presented in Haenni (2009)], and (4) a study of Dung-style argumentation semantics. This way, we obtain a generalization of both (some forms of) logical argumentation (Besnard and Hunter, 2018) and probabilistic argumentation in the tradition of Hunter and Thimm (2017).

The structure of the paper is as follows. In Section 2, we introduce knowledge bases and arguments. In Section 3, we discuss the application of argumentation semantics and study rationality postulates relative to the attack form used. Section 4 presents the empirical study on argument strength. We provide a discussion and conclusion in Section 5. In the Appendix (Supplementary material), we provide proofs of our main results, some alternative but equivalent definitions, and details on our empirical study.

Our reasoning processes never start from void, but we make use of available information when building arguments. This available information is encoded in a knowledge base. In our initial Example 1, we had two types of information available:

More generally we will follow this rough distinction in probabilistic information that gives rise to defeasible assumptions, on the one hand, and factual constraints, on the other hand. Constraints are taken for granted, either because a reasoner is convinced of their truth, or otherwise committed to them in the reasoning process (e.g., they may be supposed in an episode of hypothetical reasoning³).

Altogether a knowledge base consists of the following components:

For a sentence ϕ we let ∥ϕ∥ be the set of states s∈states(Vp) for which s⊨ϕ. For some s∈states(Vp), we denote by ŝ the conjunction ∧{ϕ∣α(ϕ)=1,ϕ∈Vp}∪{¬ϕ∣α(ϕ)=0,ϕ∈Vp}. The reader finds an overview of the notation used in this paper in Table 1.

We may also use propositional variables for which no (direct) probabilistic information is considered. We collect these in Vl (the logical variables) and require Vl∩Vp=∅. By allowing logical variables as well as probabilistic ones, we can unite logical (where Vp=∅) and probabilistic reasoning (where Vl=∅) and can involve both systems seamlessly, following the approach by Haenni (2009). Constraints will typically relate probabilistic with non-probabilistic information and therefore they are based on atoms from both sets, Vp and Vl.

In the following we write Γ⊢Cϕ as an abbreviation of Γ∪C⊢ϕ, where Γ⊆sent(Vl∪Vp). It will also be useful to collect all states that are consistent with C and that support a formula ϕ∈sent(Vl∪Vp) in ∥ϕ∥C={s∈states(Vp)∣ŝ⊬C¬ϕandŝ⊢Cϕ}. Similarly, we write s⊨Cϕ in case s∈∥ϕ∥C.

We summarize the above discussion in the following definition:

Definition 1 (Knowledge Base). A knowledge base 𝕂 is a tuple 〈〈Vp,Vl〉,A,C,ℙ〉 for which

Example 3 (Ex. 1 cont.). Let us return to our example. It can be modeled by the knowledge base

where P(p) = 0.7, p stands for a COVID-Wave to happen and q for the conference to by held hybrid. Our defeasible assumptions A are {p, ¬p} (“there will (not) be a COVID-Wave”). Our set of factual constraints is C={p→q}. Table 2 shows the state space induced by Vp.

Given a knowledge base 𝕂=〈A,C,ℙ〉, a natural way of thinking about arguments is in terms of support-conclusion pairs:

Definition 2 (Argument). Given a knowledge base 𝕂=〈A,C,ℙ〉, an argument a for 𝕂 is a pair 〈Sup(a), Con(a)〉, where

We write Arg(𝕂) for the set of all arguments based on 𝕂.

Example 4 (Ex. 3 cont.). In our example we can form the argument hybrid = 〈{p}, q〉 for the conference to be held hybrid, the argument wave = 〈{p}, p〉 for there being a COVID-wave, and noWave = 〈{¬p}, ¬p〉 for there being no wave.

When considering the question of how strong an argument a = 〈Γ, ϕ〉 is, a naive approach is to simply measure the probabilistic strength of the support. In the simple case of our example and the argument hybrid = 〈{p}, q〉 this would amount to P(p) = 0.7, the same as for the argument wave, whereas noWave would only have a strength of 0.3. However, there are some subtleties which motivate a more fine-grained analysis. To show this, we enhance our example as follows.

Example 5 (ComArg2). We also consider another conference, ComArg2, for which we know that it will be held hybrid (symbolized by q′) if and only if(!) a COVID-wave breaks in autumn. Our enhanced knowledge base is ℂ𝕆𝕍𝕀𝔻′=〈〈{p},{q,q′}〉,A:{p,¬p},C′:{p→q,p↔q′}〉. We now added also p ↔ q′ to the set of constraints C. We can now also consider the additional argument hybrid′ = 〈{p}, q′〉 for ComArg2 to be held hybrid.

Let us analyse this observation in more formal terms. We write s⊨C◇ϕ iff s∈∥⊤∥C\∥¬ϕ∥C. This means that ϕ is possible in s in view of the constraints in C. Similarly, we write ∥◇ϕ∥C for the set of states ∥⊤∥C\∥¬ϕ∥C.

Fact 1. Let a ∈ Arg(𝕂) and Con(a)⊢Cϕ. Then (1) ∥Sup(a)∥C=∥∧Sup(a)∥C⊆∥Con(a)∥C⊆∥ϕ∥C, and (2) ∥Con(a)∥C⊆∥◇Con(a)∥C⊆∥◇ϕ∥C.

In our Example 5 we have the validities for the different states shown in Table 3. Following Observation 1, hybrid = 〈{p}, q〉 is stronger than hybrid′ = 〈{p}, q′〉. The reason seems to be that despite having the same support, the “space of possibility” for q is larger than the one for q′: {s₁, s₂} vs. {s₂}. From the probabilistic perspective, the support for p seems to be located in [0.7, 1] while the one for q′ is exactly 0.7.

Following this rationale, the strength of the support of an argument is measured relative to a lower and upper bound: the lower bound is the cautious measure of how probable the support is in the worst case, the upper bound considers the best case scenario in which states in which the conclusion holds have maximal probability mass. As we will see, the central idea for modeling argument strength in this paper is by means of functions that map arguments to [0, 1] (their strength) by aggregating the worst and best case support.

Before discussing two complications, we shortly summarize the ideas so far. Arguments are support-conclusion pairs. When considering the strength of an argument a = 〈Sup, ϕ〉 it is advisable not only to consider the probabilistic strength of its support Sup, but also to consider the probabilistic support for the possibility of its conclusion ◇ϕ. A measure of argument strength is expected to aggregate the two.

In many scenarios it will be advantageous or unavoidable to work with families of probability functions, instead of a unique probability function. These are cases in which the probabilistic information concerning the probabilistic variables in Vp is incomplete or it stems from various sources, each providing an individual probability function. The following example falls in the former category.

Example 6 (Three conferences). Peter and Mary are in the steering committee of ConfB, ConfP, and ConfM. Their votes have different weights for the decision making of the respective committees. Both of their positive votes are sufficient but not necessary for ConfB to be held hybrid. For ConfP the decision relies entirely on Peter's vote, and for ConfM it relies entirely on Mary's vote.

Altogether our knowledge base is given by 𝕂=〈〈{p1,p2},{q1,q2,q3}〉,A,C,ℙ:{Pμ∣μ∈[0,⅓]}〉, where A=sent(Vp) and C={p1→q1,p2→q1,p1↔q2,p2↔q3}. Moreover, in this case the probabilities for our defeasible assumptions p₁ and p₂ are not precise. They are expressed by means of a family of probability functions (see Table 4).⁵

Given an argument 〈Sup, ϕ〉, a cautious way to consider the worst case probabilistic support is by considering infP∈ℙ(P(∥Sup∥C)). Following Haenni, we refer to this measure as the degree of support of an argument. For the best case probabilistic support, on the other hand, we consider supP∈ℙ(P(∥◇ϕ∥C)). We refer to this measure as the degree of possibility of an argument. An overview for the current example can be found in Table 5. Before formally defining the two discussed measures, we have to still consider one more complication, however, which will discuss in the next section.

Consider the following example:

Example 7 (Witnesses). 1. According to witness 1 p ∧ q is the case.      p₁ → p ∧ q

We may model this scenario with the knowledge base 𝕂=〈〈Vp:{p1,p2},Vl:{p,q}〉,A:sent(Vp),C:{p1→p∧q,p2→p∧¬q},ℙ={P}〉 where P assigns the probabilities as depicted in Table 6.

In this case s₄ is incompatible with the set of constraints C of our knowledge in 𝕂 and the probabilities have to be updated. We follow Haenni (2009) by using a Bayesian update on ∧C and letting

Similarly, where ℙ is a family of probability functions, we let ℙC={PC∣P∈ℙ}. When calculating the degrees of support and degrees of possibility of an argument we will consider ℙC instead of ℙ.

Definition 3 (Degree of Support and Degree of Possibility, (Im)Precision). Given a knowledge base 𝕂=〈A,C,ℙ〉 and an argument a = 〈Sup, ϕ〉 for 𝕂,

Fact 2. Let a, b ∈ Arg(𝕂).

As discussed above, we expect a measure of argument strength to aggregate the two measures of degree of support and degree of possibility.

Definition 4 (Argument strength function). Let 𝕂=〈A,C,ℙ〉 be a knowledge base. A measure of argument strength for 𝕂 is a function str:Args(𝕂) → [0, 1] that is associated with a function π : Θ → [0, 1] for which Θ = {(n, m) ∈ [0, 1]² ∣n ≤ m} and str(a) = π(dsp(a), dps(a)).

As discussed above, we have two underlying measures which can serve as input for a measure of argument strength: the degree of support and the degree of possibility (recall Definition 4): str(a) = π(dsp(a), dps(a)) where π : {(n, m) ∈ [0, 1]² ∣n ≤ m} → [0, 1]. As for π there are various straight-forward options. We list a few in Table 7. Support and possibility reflect the lower and upper probabilistic bounds represented by dsp and dps, while mean represents their mean. Boosted support follows the idea underlying Observation 1 according to which an argument c with dsp(c) < dps(c) should get a “boost” as compared to an argument d for which dsp(d) = dps(d) = dsp(c). The factor m ≥ 1 determines the magnitude of the boost, the lower m the more the lower bound dsp is boosted (where for m = 1 the boosted support is identical to dps). Convex combination follows a similar idea by letting the strength of an argument a be the result of a convex combination of dsp(a) and dps(a), where the parameter α determines how cautious an agent is: the higher α the less epistemic risk an agent is willing to take (where for α = 1 the convex combination is identical to dsp(a)). Precision mean is a qualification of mean in that it also considers the precision of an argument as a marker of strength (see Pfeifer, 2013). The precision of an argument a is given by 1−(dps(a)−dps(a)): the closer dsp(a) and dps(a) the more precise is a. The precision mean of an argument is the result of multiplying its mean with its precision. We note that this measure is in tension with the intuition behind Observation 1 in that it would measure the strength of hybrid higher than that of hybrid′, unlike boosted support or convex combination.

Clearly, some of the measures coincide for specific parameters (the proof can be found in Appendix A):

Proof: Items 1 and 2 are trivial. We show Item 3. We have, on the one hand, bstm(a)=dsp(a)+dps(a)-dsp(a)m=dsp(a)-dsp(a)m+dps(a)m=(1-1/m)·dsp(a)+(1-(1-1/m))·dps(a)=convex1-1/m(a). On the other hand, convex_α(a) = α·dsp(a) + (1 − α)dps(a) = dsp(a)+dps(a) − dps(a)·α − dsp(a)+dsp(a)·α = dsp(a) + (dps(a) − dsp(a))·(1 − α) = bst_{¹/(1 − α)}(a).     □

Example 8. In Table 8, we apply the different argument strength measures to Examples 1 and 6.

We now analyse the different strength measures in view of several properties, some of which may be considered desiderata.⁶ Table 9 offers an overview on which properties are satisfied for which measures.

The following properties specify various ways the degrees of support and/or possibility are related to argument strength in terms of offering sufficient resp. necessary conditions. For the following properties let a ⊑ b iff dsp(a) ≤ dsp(b) and dps(a) ≤ dps(b). Let ⊏ be the strict version of ⊑, i.e., a ⊏ b iff a ⊑ b and b ⋢ a.

Fact 4. Let a, b be precise arguments (so, prec(a) = prec(b) = 1). If Precision holds for str, then: str(a) ≤ str(b) iff a ⊑ b.

The following criteria present various ways of considering precision a sign of argument quality. For instance, Pfeifer (2013) considers precision a central marker of strength.

Finally, we offer some criteria that relate arguments to other arguments in a logical way.

Before studying these properties for our different notions of argument strength, we observe some logical relations between some of them.

Proposition 1. For any argument strength measure str we have:

Proof: Ad 1. Trivial. Ad 2. Concerning R-weakening and L-weakening, observe that if a and b fulfill the requirements of the left hand side of R-weakening resp. of L-weakening, then b ⊒ a. So, by Weak epistemic sufficiency, str(a) ≤ str(b).     □

Example 9 (Violation of properties for dsp, dps and mean.). An argument a with prec(a) = 0 is such that dsp(a) = 0 and dps(a) = 1. Clearly, neutrality is violated for dsp and dps. Such an argument also violates moderation for dps.

To illustrate other violations we give an example similar to Example 6. Let 𝕂=〈〈Vp:{p1,p2},Vl:{q1,q2,q3,q4}〉,A:sent(Vp),C:{(p1∧p2↔q1),¬(p1∨p2)→q2,p1→q4},ℙ:{P}〉 with the probabilities as in Table 10. We note that mean(a₁) > mean(a₂) [resp. dsp(a₁) > dsp(a₂)] while dps(a₂) > dps(a₁) illustrating a violation of lower compensation for dps. Since prec(a₂) = 0.1 < 1 = prec(a₁) this also gives a counter-example for precision compensation and necessity, for dps. For a counter-example for upper compensation and dsp consider arguments a₃ and a₅: dsp(a₃) < dsp(a₅) and mean(a₅) ≤ mean(a₃), while dps(a₅) < dps(a₃). A counter-example for strict epistemic sufficiency and dps is given in view of dps(a₂)≯dps(a₃), although a₃ ⊏ a₂.

Strict epistemic sufficiency and precision necessity for dsp is violated in view of hybrid and hybrid′ in Example 5, where dsp(hybrid) = dsp(hybrid′) while hybrid ⊐ hybrid′ and prec(hybrid) < prec(hybrid′).

Consider 𝕂=〈A:{p},C:∅,ℙ:{P}〉 where P(p) = 0.5 and the arguments a : 〈{p}, p〉 and b : 〈∅, ⊤〉. Then dsp(a) = 0.5 = dps(a) = mean(a) and prec(a) = 1, while dsp(b) = 0, dps(b) = 1, mean(b) = 0.5 and prec(b) = 0. The example represents a counter-example for (i) precision necessity for str ∈ {dps, mean}, (ii) strict precision sufficiency for str ∈ {dps, mean} and (iii) precision sufficiency for dps.

Example 10 (Violation of properties for precision mean.). In Table 8, we have dsp(hybrid) = 0.7, dps(hybrid) = 1, while precMean(hybrid) = 0.595 (see Table 8). This shows that precMean does not satisfy domain restriction. Note that wave ⊑ hybrid and precMean(wave) = 0.7. So, we also have a counter-example for weak and strict epistemic sufficiency, as well as for R-weakening.

We consider the knowledge base 𝕂=〈〈Vp:{p1,p2},Vl:{q1,q2,q3,q4}〉,A:sent(Vp),C:{¬(p1∨p2)→q1,(¬p2∨p1)→q2,¬p2→q3,(p1∧p2)→¬(q3∨q4),¬p1∧p2→q4,¬(p1∨p2)→¬q4},ℙ:{P}〉 with the probabilities and arguments in Table 11. For a counter-example for upper (resp. lower) compensation consider a₁ and a₂ (resp. a₃). The arguments a₃ and a₅ also provide a counter-example for precision necessity since precMean(a₃) > precMean(a₅) while prec(a₅) = 0.75 > prec(a₃) = 0.6.

Example 11 (Violation of lower compensation, Boosted support, and Convex combination). In the knowledge base of Table 11 we have a counter-example for lower compensation and bst_m for m = 1.5. Note that bst_m(a₂) > bst_m(a₄) and mean(a₂) ≤ mean(a₄) while dsp(a₄) > dsp(a₂). In view of Fact 3 the example applies equally to convex_α for α = ⅓.

Example 12 (Epistemic Risk Tolerance). We note that, in the example of Table 11, dps(a₂) > dps(a₁) [resp. mean(a₂) > mean(a₁)] (resp. bst_1.5(a₂) > bst_1.5(a₁)), while dsp(a₂) < dsp(a₁), demonstrating epistemic risk tolerance for dps [resp for mean] (resp. for bst_1.5 and convex_⅔). For precMean we consider arguments a₁ and a₃.

Argumentation semantics aim at providing a rationale for selecting arguments for acceptance in discursive situations in which arguments and counter-arguments are exchanged. Some requirements are, for instance, that a selection does not contain conflicting arguments, or that a selection is such that any counter-argument to one of its arguments is attacked by some argument in the selection. In this section, we will gradually introduce new notions and observations based on a list of problems. Ultimately the critical discussion will lead to an improved account to be introduced in Section 3.3. In order to define argumentation semantics we first need a notion of argumentative defeat.

Definition 5 (defeat types). Let 𝕂 be a knowledge base, str a strength measure, and a, b ∈ Arg(𝕂).

Lemma 1. Suppose Weak Epistemic Sufficiency holds for str. Let a, b ∈ Arg(𝕂).

Proof: Ad 1. Suppose Con(a)⊢CCon(b) and Sup(a) = Sup(b). By Fact 2, dsp(a) = dsp(b) and dps(b) ≥ dps(a). By Weak Epistemic Sufficiency, str(a) ≤ str(b). Ad 2. This is a special case of item 1 since Sup(a) = Sup(@(Sup(a))) and Con(@(Sup(a)))⊢CCon(a). Ad 3. In this case Con(@(Sup(b)))⊢CCon(@(Sup(a))). By Fact 2, dps(a) ≥ dps(b) and dsp(a) ≥ dsp(b). By Weak Epistemic Sufficiency, str(@(Sup(a))) ≥ str(@(Sup(b))). Ad 4. Suppose a undercuts b. In order to show that a undercuts′ b we only have to show that str(a) ≥ str(@(Sup(b))). Since str(a) ≥ str(b) this follows with Item 2.     □

We are now in a position to define argumentation frameworks and subsequently argumentation semantics.

Definition 6 (AF). An argumentation framework based on a knowledge base 𝕂 is a pair 〈Arg(𝕂), def〉 where def is a (non-empty) set of defeat-types (as in Definition 5) for a given measure of argument strength str.