Deductive and abductive argumentation based on information graphs

1.Introduction

In the legal and forensic domains, reasoning about evidence plays a central role in the rational process of proof [2,4]. To aid in this process, various graph-based tools exist that allow domain experts to make sense of a mass of evidence in a case, such as mind maps [23,35], argument diagrams [6,23] and Wigmore charts [40]. Because of their informal nature, these tools typically do not directly allow for formal evaluation using AI techniques such as computational argumentation [13]. Hence, we wish to formalise and disambiguate analyses performed using such tools in a manner that (1) allows for formal evaluation and that (2) adheres to principles from the literature on reasoning about evidence [2,4,17,25], while (3) allowing inference to be performed and visualised in a manner that is closely related to the way inference is performed and visualised by domain experts using such tools.

As we described in previous work [39], principles from the literature on reasoning about evidence state that inference is often performed using domain-specific generalisations [2,4,6], also called defaults [25,32], which capture knowledge about the world in conditional form. A distinction can be made between causal generalisations (e.g. ‘fire typically causes smoke’) and evidential generalisations (e.g. ‘smoke is evidence for fire’) [4,25]. In the current paper, we also consider generalisations that are neither causal nor evidential; examples are abstractions [4,12] and mere statistical correlations. Inference can be performed in a deductive or forward fashion, where from a generalisation (e.g. ‘fire typically causes smoke’) and its antecedent (fire), the consequent (smoke) is strictly or defeasibly inferred, and in an abductive [12,17] or backward fashion, where from a causal generalisation or an abstraction and by affirming the consequent (smoke), the antecedent (fire) is defeasibly inferred. Note that the term ‘deduction’ is not consistently used in the literature, as it can either mean strict inference, in which the consequent universally holds given the antecedents (e.g. [20]) or defeasible inference, in which the consequent tentatively holds given the antecedents (e.g. [33]). To cover both meanings, in this paper ‘deduction’ is used as an umbrella term for both defeasible ‘forward’ inference and strict ‘forward’ inference.

Pearl [25, p. 264] argued that people generally consider it difficult to express knowledge using only causal generalisations, and in an empirical study, van den Braak and colleagues [36] found that while there are situations in which subjects consistently choose either causal or evidential modelling techniques, there are also many examples in which people use both types of generalisations in their reasoning. For instance, subjects often considered testimonies to be evidential, whereas a motive for committing an act is considered a cause for committing that act. This discussion illustrates that in formal accounts of reasoning about evidence, it is important to allow for causal and evidential generalisations [4]. Moreover, in this paper we show that it is important to also allow for abstractions and other types of generalisations, as these allow domain experts to be more expressive in stating their knowledge. The need for including these types of generalisations will become apparent from the examples we consider and the conceptional analysis of reasoning about evidence we provide.

When performing analyses using aforementioned tools such as mind maps, domain experts naturally mix the different types of generalisations and perform both deductive and abductive inferences, where the used generalisations and the inference type (deduction, abduction) are typically left implicit. Hence, in previous work [39] we set out to formalise analyses performed using these tools by providing a precise account of the interplay between the different types of inferences and generalisations and the constraints on performing inference we need to impose in terms of the information graph (IG) formalism. In this paper, we propose an extension of the IG-formalism, where in addition to causal and evidential generalisations we now also allow for abstractions and introduce a category of generalisations termed ‘other’, consisting of generalisations that are neither causal nor evidential nor abstraction such as aforementioned mere correlations, thereby increasing the expressivity of the IG-formalism. We particularly focus on identifying conditions under which performing inference with abstractions can lead to undesirable results. Specifically, care should be taken that no version of an event at a lower level of abstraction is inferred if an alternative version of this event at a lower level of abstraction was already previously inferred. Hence, we extend on the constraints imposed by Pearl’s C–E system [25] which say that, in performing inference, care should be taken that no cause for an effect is inferred in case an alternative cause for this effect was already previously inferred. Moreover, in this paper we also consider exceptional circumstances under which the constraints of Pearl’s C–E system should not be imposed, namely in case enabling conditions [11] are provided under which a generalisation may be used in performing inference. Based on these constraints and our conceptional analysis of reasoning about evidence, we define how deductive and abductive inference may be performed with IGs. Most existing formalisms that allow both inference types with causal and evidential information, abstractions, and other types of information are logic-based (e.g. [4,5,12,20]); instead, we opt for a graph-based formalism to remain closely related to the way analyses are visualised using aforementioned graph-based tools.

The information specified in an IG serves as a source of information that can be used to facilitate the construction of AI systems for which formal semantics are defined. In earlier work [39], we investigated the application of our IG-formalism in facilitating the construction of Bayesian networks (BNs) [16], graphical models of joint probability distributions. In this paper, we instead focus on argumentation, where we propose an argumentation formalism based on IGs that allows for both deductive and abductive argumentation [38]. Previous work on abduction includes work on formal logical models of abductive reasoning (e.g. [12,17]) and the work of Kakas and colleagues on abductive logic programming [19]. However, to the best of our knowledge, our proposed formalism is one of the first formalisms that models combined abductive and deductive reasoning in a formalism for structured argumentation. The closest to the current paper is Bex’s integrated theory of causal and evidential arguments [5], which is based on the ASPIC+ framework [20]. In Bex’s integrated theory, the roles of generalisation and inference are not separated; instead, causal and evidential inferences are defined and arguments are constructed by forward chaining such inferences. In contrast to [5] we put special emphasis on the constraints that need to be imposed on the types of inferences that may be performed with the different types of generalisations, where we formally prove that arguments based on IGs indeed adhere to the identified constraints. Finally, compared to the ASPIC+ framework [20], which only allows for deductive reasoning, we allow for both deductive and abductive reasoning and introduce a new type of conflict, namely conflict between competing alternative explanations [17], which is currently not accounted for in that framework. The relations to existing formalisms is further discussed in Section 6.

Our approach generates an abstract argumentation framework as in Dung [13], that is, a set of arguments with a binary attack relation, which thus allows arguments to be formally evaluated according to Dung’s argumentation semantics. Besides allowing for rebuttal and undercutting attack, which are among the types of attacks that are typically distinguished in structured argumentation [20,27], we also define the notion of alternative attack among arguments based on IGs, a concept based on the notion of competing alternative explanations that is inspired by [3,5]. Alternative attack captures a crucial aspect of abductive reasoning, namely that of conflict between abductively inferred conclusions [17].

Our argumentation formalism extends a preliminary version proposed in [38] that was based on a more restricted version of our IG-formalism [39] in which only causal and evidential generalisations without enablers were considered. Moreover, in comparison to our earlier work [38] we now also prove that key rationality postulates [9] are satisfied by instantiations of our formalism, which implies that anomalous results as identified by [9] are avoided.

To summarise the main contributions of this paper, we propose an argumentation formalism that allows for both deductive and abductive argumentation, the latter of which has received relatively little attention in argumentation. Our argumentation formalism is based on an extended version of our IG-formalism, where in addition to causal and evidential generalisations we now also allow for abstractions and other types of generalisations, as well as generalisations that include enabling conditions, where constraints are imposed on the types of inferences that may be performed with these new types of generalisations. A new notion of attack is defined, namely alternative attack. Our approach allows arguments to be evaluated using Dung’s semantics. We formally prove that instantiations of our argumentation formalism satisfy key rationality postulates [9].

The paper is structured as follows. In Section 2 we provide a conceptual analysis of reasoning about evidence. In Section 3 we present examples of analyses performed using informal reasoning tools typically used by domain experts, namely Wigmore charts and mind maps, which illustrates that both deductive and abductive inference is performed by domain experts using both causal and evidential generalisations, abstractions, and other types of generalisations. Based on these examples, in Section 4 we motivate and define our IG-formalism. In Section 5 we then define our argumentation formalism based on our IG-formalism and prove formal properties of our approach. In Section 6 we discuss related work. In Section 7 we summarise our findings and conclude.

2.Reasoning about evidence

In this section, we provide a conceptual analysis of reasoning about evidence, where we review the terminology used to describe it and introduce assumptions that demarcate the scope of the work presented in this paper. This analysis extends the analysis provided in our previous work [39] in which only causal and evidential generalisations without enablers were considered. More specifically, we now also consider abstractions and other types of generalisations, as well as generalisations that include enabling conditions. The concepts and assumptions introduced in this section are formalised in Sections 4 and 5.

Inference is the process of drawing conclusions from premises starting from the evidence, where evidence is that what has been established with certainty in the context under consideration. For instance, in the context of a legal trial, the evidence consists of that what is actually observed by a judge or jury, such as documents (e.g. police and autopsy reports) and other tangible evidence, as well as testimonial evidence [2]. Inference is often performed using domain-specific generalisations [2,4,6], also called defaults [25,32], which capture knowledge about the world in conditional form. Generalisations can either be strict or defeasible, where defeasible generalisations are of the form ‘If a1,…,an, then usually/normally/typically b’ and strict generalisations are of the form ‘If a1,…,an, then always b’. Here, claims a1,…,an are called the antecedents of the generalisation and b its consequent, where we assume that claims are literal propositions and that generalisations have one or more antecedents and exactly one consequent. In case a generalisation has multiple antecedents, it expresses that only the antecedents together allow us to infer the consequent. We semi-formally denote generalisations as a1,…,an→b, among others to ease the description of examples in this section and in Section 3. For defeasible generalisations, exceptional circumstances can be provided under which the generalisation may not hold, whereas strict generalisations hold without exception. An example of a (defeasible) generalisation is ‘If fire, then typically smoke’, where ‘fire’ is its antecedent and ‘smoke’ its consequent. An example of an exception to this generalisation is that sufficient oxygen is present for complete combustion to occur.

A distinction can be made between causal and evidential generalisations [4,25], where instead of writing these generalisations in the form ‘If … , then …’, causal generalisations are written as ‘c1,…,cn usually/normally/typically cause e’ (e.g. ‘fire typically causes smoke’) and evidential generalisations are written as ‘e1,…,en are evidence for c’ (e.g. ‘smoke is evidence for fire’). For a causal generalisation, its antecedents express causes for the consequent, and for an evidential generalisation, its consequent expresses the usual cause for its antecedents. In the context of commonsense reasoning about evidence, causal and evidential generalisations are often assumed to be defeasible (see e.g. [4,18]); in this paper, this assumption is also made. The examples considered throughout this paper illustrate that causal and evidential generalisations are typically not strict.11

In this paper, we also consider generalisations that are neither causal nor evidential. For instance, abstractions [4,12] allow for reasoning at different levels of abstraction. More precisely, abstractions are of the form ‘p1,…,pn can usually/normally/typically/always be considered a specialisation of q’ (e.g. guns can usually be considered deadly weapons), where antecedents p1,…,pn are considered to be more specific than the more abstract consequent q. As noted by Console and Dupré [12], abstractions are syntactically the same as causal generalisations but they are semantically different in that the antecedents of abstractions do not express causes for the consequent or vice versa. Abstractions may be defeasible (cf. [4]) but may also be strict (cf. [12]); an example of a strict abstraction is generalisation lung_cancer →a cancer, which states that lung cancer is a type of cancer. An example of defeasible abstraction is gun →a deadly_weapon, where an example of an exception to this generalisation is that the gun is a non-functional replica, or a water gun.

Another example of a different type of generalisation is a generalisation representing a mere statistical correlation, such as a correlation between homelessness and criminality. While there may be one or more confounding factors that cause both homelessness and criminality (e.g. unemployment), a domain expert may be unaware of these factors or may wish to refrain from expressing them explicitly. In this paper, we distinguish between generalisations that are causal, evidential, abstraction, or of another type, where generalisations of type ‘other’ may be defeasible or strict. Specifically, as this category contains all possible types of generalisations other than causal, evidential and abstraction, we allow for the option to distinguish between strict and defeasible generalisations among these generalisations. Table 1 provides an overview of the different generalisation types, where for each type it is indicated whether generalisations may be defeasible or strict. The notation →c, →e, →a and →o is used for the different types of generalisations, respectively.

Different types of inferences can be performed with generalisations depending on whether their antecendents or consequent are affirmed in that they are either observed or inferred; here, a claim is inferred iff it is either deductively or abductively inferred, where in deductive inference the consequent is inferred from the antecedents and in abductive inference the antecendents are inferred from the consequent. These two inference types are now considered in more detail.

3.Examples of analyses performed using informal reasoning tools

In this section, we present examples of analyses performed using two tools that are typically used by domain experts, namely Wigmore charts [40] and mind maps [23,35]. Based on these examples, we motivate and illustrate the design choices for our IG-formalism in Section 4.

3.1.Example of an analysis performed in a Wigmore chart

First, Wigmore charts are considered, which are diagrams familiar to legal experts in which symbols indicating hypotheses and claims are joined by lines representing relations between these hypotheses and claims. Wigmore charts were introduced by John Henry Wigmore [40] and were applied and further developed by Anderson, Schum, Twining and others (e.g. [2,18]), who provided a modernised, more user-friendly version of Wigmore’s charting method. In this section, we consider these modern versions of Wigmore charts, specifically the version adopted by Kadane and Schum [18]. In their charts, each symbol represents a unique claim. As noted by Kadane and Schum [18, p. 88], vertical arcs between nodes in the chart indicate inferences between corresponding claims, where the generalisations used in performing these inferences are not explicitly recorded in the chart. To be able to interpret whether inferences are deductive or abductive, and hence what the antecedents and consequents are of generalisations used in performing the inferences, the evidence in the chart also needs to be considered.

Fig. 1.

Wigmore chart concerning Sacco’s consciousness of guilt, along with the corresponding key list, adapted from Kadane and Schum [18, pp. 330–331].

Example 17.

An example of a modern Wigmore chart, adapted from Kadane and Schum [18, pp. 330–331], is depicted in Fig. 1, which also serves as our running example. The Wigmore chart concerns parts of an actual legal case, namely the well-known Sacco and Vanzetti case. The case concerns Sacco and Vanzetti, who were convicted for shooting and killing payroll guard Berardelli during a robbery. In this example, we only consider the part of the case concerning Sacco’s consciousness of guilt. During their arrest, Sacco and Vanzetti were armed. According to the two arresting officers, Connolly and Spear, Sacco and Vanzetti made suspicious hand movements, from which the prosecution concluded that they intended to draw their concealed weapons in order to escape their arrest. This suggests that they were conscious of having committed a criminal act.

On the right-hand side of Fig. 1 the corresponding key list is depicted, which indicates for every number in the chart to which claim it corresponds. Claims provided by the defence and prosecution are represented as diamonds and circles in the chart, respectively, where nodes corresponding to the evidence are shaded. Finally, horizontal lines in the Wigmore chart indicate that information needs to be combined to draw a conclusion.

As noted earlier, the generalisations used in performing the indicated inferences are left implicit in the chart. Instead, in their analysis of the case some of the used generalisations are indicated in the text (see e.g. [18, pp. 97–98]). For instance, generalisations used in the inferences from the testimonies are of the general form ‘If a person testifying under oath tells us that event E occurred, then this event (probably, usually, often, etc,) did occur.’ [18, p. 88]. As noted by Kadane and Schum [18, pp. 74–76], in constructing their charts abduction is in some instances performed to generate interim hypotheses between the evidence and the ultimate claim Π3. However, Kadane and Schum do not explicitly indicate which inferences in their charts are abductive and which are deductive.

Lastly, it is important to note that the manner in which claims and links conflict is not precisely specified in Kadane and Schum’s Wigmore charts, as also observed by Bex and colleagues [6] in formalising such Wigmore charts as Pollock-style arguments [27]. For instance, multiple interpretations of the conflicts between the defence’s claims 462 and 465 and the prosecution’s claims 152 and 153 are possible. One possible interpretation is that 462 and 465 indicate support for the negation of claim 153: as Sacco carried his weapon for an innocent reason (either 462 or 465), he intended to surrender his weapon and, therefore, did not intend to use it. Alternatively, 462 and 465 can be considered competing alternative explanations of 152, and hence be interpreted as exceptions to the performed inference step from 152 to 153. Specifically, as Sacco carried his weapon for an innocent reason (462 or 465), this caused him to draw his weapon (152) with the intention of surrendering it.

3.2.Example of an analysis performed using a mind mapping tool

Next, we present an example of an analysis performed using a mind mapping tool [23], which is an example of a tool typically used by domain experts, for instance in crime analysis [35]. A mind map usually takes the shape of a diagram in which hypotheses and claims are represented by boxes and underlined text, and undirected edges symbolise relations between these hypotheses and claims. An example is depicted in Fig. 2, which is based on a standard template used by the Dutch police for criminal cases involving the suspicious death of a person. The mind map represents various scenario-elements and the crime analyst uses evidence to support or oppose these elements, indicated in the mind map by plus and minus symbols, respectively. Compared to Wigmore charts, which offer a wide range of symbols and arcs to allow users to be expressive and precise in modelling legal reasoning, mind maps are less precise and are used to obtain an overview of different possible alternative scenarios. In the following example, only supporting evidence is considered, which allows us to focus on the manner in which competing alternative explanations are captured in mind maps.

Fig. 2.

Example of a partially filled in mind map.

Example 18.

An example of a partially filled in mind map is depicted in Fig. 2. In this example case, a body was found; we are interested in the cause of death of this person. First, high-level hypothesis ‘Murder’ is examined. According to witness testimony (Testimony 1), the person was hit with a hammer (Hammer); however, according to another testimony (Testimony 2), the person was hit with a stone (Stone). By means of plus symbols and undirected edges connecting the evidence to these claims, it is indicated that claims Testimony 1 and Testimony 2 support claims Hammer and Stone, respectively. Hammer and Stone are connected via undirected edges to Hit angular, which indicates that hammers and stones can generally be considered to be angular. In turn, claim Hit angular is connected to the ‘With’ question to indicate that it provides an answer to this question. As an answer to the ‘In which way’ question, it is indicated that the person died because of a head wound (Head wound), which is again supported by the claim that the person was hit with an angular object (Hit angular). An autopsy report (Autopsy) further supports claim Head wound.

Next, high-level hypothesis ‘Accident’ is examined, which provides a competing alternative explanation of Head wound. As an answer to the ‘In which way’ question, it is again indicated that the person died because of a head wound and that this claim is supported by Autopsy; however, in contrast to the answer to this question for high-level hypothesis ‘Murder’, it is indicated that the head wound was caused as the person fell on a table by accident (Fell on table), a claim supported by Testimony 3.

As the edges in a mind map are undirected, it is unclear from the graphical representation alone which types of generalisations and inferences were used in constructing this map. Establishing this with certainty would require directly consulting the domain experts involved in constructing the chart. We note, however, that the reasoning performed in constructing this mind map can be interpreted in at least two possible ways. One interpretation is that the domain expert first (preliminarily) inferred that the person died because of a head wound from the autopsy report via deduction using the evidential generalisation g1: Autopsy →e Head wound, and then abductively inferred Hit angular using the causal generalisation g2: Hit angular →c Head wound. In turn, Hammer and Stone are abductively inferred from Hit angular using the abstractions g3: Hammer →a Hit angular and g4: Stone →a Hit angular. These two claims are then competing alternative explanations of Hit angular and are subsequently grounded in evidence, namely via deduction from the testimonies using evidential generalisations g5: Testimony 1 →e Hammer and g6: Testimony 2 →e Stone. An alternative interpretation is that the mind map was constructed iteratively from the evidence, where from the testimonies the claims Hammer and Stone are inferred via deduction using generalisations g5 and g6. Claim Hit angular is then inferred modus-ponens style: from abstractions g3 and g4 and the previously inferred antecedents, the consequent is deductively inferred. In this way, Hammer and Stone are not in competition for Hit angular.

This example illustrates that the types of generalisations and inferences involved in the analysis of a case using a mind mapping tool are typically left implicit. Similarly, the manner in which claims and links conflict is not precisely specified in mind maps: in particular, conflicts between competing alternative explanations are not explicitly indicated in the graph.

4.The information graph formalism

The examples from Section 3 make it plausible that both deductive and abductive inference is performed by domain experts when performing analyses using reasoning tools they are familiar with. In performing such analyses, the used generalisation, as well as the inference type (deduction, abduction), are left implicit. Furthermore, the assumptions of domain experts underlying their analyses are typically not explicitly stated, making these analyses ambiguous to interpret. For current purposes, we wish to provide a precise account of the interplay between the different types of inferences and generalisations that formalises and disambiguates these analyses in a manner that makes the used generalisations explicit. Information graphs (IGs), which we define in Section 4.1, are knowledge representations that explicitly describe generalisations in the graph. In constructing an IG from an analysis performed using a tool, an interpretation step may be required; we provide examples of this interpretation step by discussing possible formalisations of the Wigmore chart of Section 3.1 and the mind map of Section 3.2. In Section 4.2, we define how deductive and abductive inferences can be read from IGs given the evidence, based on our conceptual analysis of reasoning about evidence (Section 2). Compared to our previously proposed IG-formalism [39] in which only causal and evidential generalisations were considered, abstractions and other types of generalisations are now also considered, as well as generalisations that include enabling conditions, where constraints are imposed on the types of inferences that may be performed with these new types of generalisations.

4.1.Information graphs

First, the syntax of IGs is defined. Throughout this paper, boldface is used to indicate sets used in the IG-formalism.

Definition 1

Definition 1(Information graph).

An information graph (IG) is a directed graph GI=(P,A), where P is a set of nodes representing propositions from a propositional language consisting of only literals and that is closed under classical negation, where the negation symbol is denoted by ¬. A is a set of (hyper)arcs that divides into three pairwise disjoint subsets G, N and X of generalisation arcs, negation arcs and exception arcs, defined in Definitions 2, 4, and 5, respectively.

For IGs, there is a one-to-one correspondence between nodes and propositions, generalisation arcs and generalisations, exception arcs and exceptions, and negation arcs and negations. Throughout this paper, in the context of IGs, the terms ‘node’ and ‘proposition’, ‘generalisation arc’ and ‘generalisation’, ‘exception arc’ and ‘exception’, and ‘negation arc’ and ‘negation’ are therefore used interchangeably. We write p=−q in case p=¬q or q=¬p. Finally, note that while we currently only consider classical negation, our IG-formalism may be extended in future work to allow for more general notions of conflicts such as contrariness (cf. [20]).

Definition 2

Definition 2(Generalisation arc).

Let GI=(P,A) be an IG. A generalisation arc g∈G⊆A is a directed (hyper)arc g:{p1,…,pn}→p, indicating a generalisation with antecedents P1={p1,…,pn}⊆P and consequent p∈P∖P1. Here, propositions in P1 are called the tails of g, denoted by Tails(g), and p is called the head of g, denoted by Head(g). G divides into four pairwise disjoint subsets Gc, Ge, Ga and Go of causal generalisation arcs, evidential generalisation arcs, abstraction arcs, and all other types of generalisation arcs, respectively. Generalisations in Gc and Ge are defeasible, Ga divides into disjoint subsets Gsa and Gda of strict and defeasible abstraction arcs, respectively, and Go divides into disjoint subsets Gso and Gdo of strict and defeasible other types of generalisation arcs, respectively. For g∈G, Tails(g) divides into disjoint subsets Enabler(g) and Ant(g) of propositions representing enabling conditions and actual antecedents of the generalisation, respectively, where for g∈Gc∪Ge it holds that Ant(g)≠∅ and possibly Enabler(g)=∅, and for g∈Ga∪Go it holds that Enabler(g)=∅ (i.e. Tails(g)=Ant(g)).

Curly brackets are omitted in case |Tails(g)|=1. In figures in this paper, generalisation arcs are denoted by solid (hyper)arcs, which are labelled ‘c’ for g∈Gc, ‘e’ for g∈Ge, and ‘a’ for g∈Ga, where ‘o’ labels for g∈Go are omitted.

In accordance with our assumptions stated in Section 2, causal and evidential generalisations are defeasible and can include enablers. Abstractions and other types of generalisations can either be strict or defeasible. A causal generalisation g:c→e may have an evidential counterpart of the form g′:e→c (see Section 2.3), but only if c is the usual cause of e. Definition 2 does not prohibit the coexistence of a causal generalisation g:c→e and its evidential counterpart g′:e→c in an IG, and inferences can be read from IGs including both generalisations without yielding anomalous results; hence, both generalisations may be included if considered desirable. However, it should be noted that g and g′ represent the same knowledge, and that care should be taken in for instance modelling exceptions to generalisations (see Definition 5), as an exception to g can also be considered an exception to g′. Ultimately, it is the responsibility of the knowledge engineer in consultation with the domain expert to decide which knowledge to include in the IG and to ensure this knowledge is correctly and consistently represented.

In the following example, the Wigmore chart of Section 3.1 is modelled as an IG.

Example 19.

In Fig. 3, an IG is depicted for a possible interpretation of the Wigmore chart of Fig. 1. This interpretation is based on a previous interpretation of this Wigmore chart as a preliminary version of an IG in which only causal and evidential information is considered and the roles of generalisation and inference are not separated [37]. For every claim p in the Wigmore chart, a proposition node p is included in P. As noted by Kadane and Schum [18, p. 88], the generalisations used in the inferences from the testimonies are evidential. As propositions 150, 151, 463, 464, 466 and 470 denote testimonies, the IG includes generalisation arcs g1:{150,151}→149; g10:{463,464}→462, g11:466→465 and g12:470→469 in Ge. Here, testimonies 150, 151 and 463, 464 are combined in the antecedents of generalisations g1 and g10, respectively, as these sets of propositions concern testimonies to the same claim. As 461 concerns Sacco’s testimony to denying 149, proposition ¬149 is included in P and generalisation arc g2:461→¬149 is included in Ge.

Kadane and Schum do not indicate which (types of) generalisations were used in performing the inferences between propositions 149 and Π3. We note that the inferences between 149 and 155 fit a so-called episode scheme for intentional actions [4, p. 64], a story scheme in which someone’s psychological state causes them to form certain goals, which in turn lead to actions that have consequences. In this case, Sacco intended to escape from his arrest (154; goal) as he was conscious of having committed a criminal act (155; psychological state); therefore, we consider 155 a cause of 154. Sacco’s intention to use his weapon (153) can then be considered a sub-goal of 154 and his intention to draw his concealed weapon (152) a further sub-goal of 153. Sacco’s intention to draw his weapon (152) caused Sacco to attempt to put his hand under his overcoat (149; action); therefore, we consider 152 a cause of 149. The IG therefore includes generalisation arcs g3:149→152; g4:152→153; g5:153→154 and g6:154→155 in Ge to denote these generalisations.

Proposition 155 can be considered an abstraction of 155a: being involved in a robbery and shooting can generally be considered committing a criminal act. The involved generalisation is defeasible: involvement in a robbery and shooting does not imply that this involvement is of a criminal nature, as it may also imply that the person under consideration is the victim. Proposition 155a can be considered a strict abstraction of 156, as at a higher level of abstraction being conscious of having been involved in the specific robbery and shooting that took place in South Braintree can be considered being conscious of having been involved in a robbery and shooting. Π3 can be considered a cause of 156: committing a specific robbery and shooting typically causes a person (in this case Sacco) to be conscious of having been involved in this act. Therefore, generalisation g7:155a→155 is included in Gda, g8:156→155a in Gsa, and g9:Π3→156 in Gc. Finally, from 469 (Sacco believed he was being arrested because of his political beliefs), we can conclude that Sacco was not conscious of having been involved in a robbery and shooting (¬155a). We consider the relation between 469 and ¬155a to be defeasible and neither causal nor evidential nor an abstraction, and therefore include g13:469→¬155a in Gdo.

In the following example, the mind map of Section 3.2 is modelled as an IG.

Fig. 3.

IG corresponding to an interpretation of the Wigmore chart of Fig. 1, where ‘e’ labels denote evidential generalisations, ‘c’ labels denote causal generalisations, ‘a’ labels denote abstractions, ↭ is a negation arc and ⇝ is an exception arc.

Fig. 4.

IG corresponding to a possible interpretation of the mind map of Fig. 2.

Example 20.

Consider Fig. 4, which depicts an IG for a possible interpretation of the mind map of Fig. 2. The generalisations used in the inferences from the testimonies, as well as from autopsy, are considered to be evidential; therefore, generalisation arcs g1, g2, g5 and g8 are included in Ge. The relation between hammer (stone) and hit_angular is neither causal nor evidential; instead, generalisation arcs g3 and g4 are included in Gda to express that, at a higher level of abstraction, both hammers and stones can generally be considered angular objects. These generalisations are defeasible as not all hammers and stones are angular. Finally, hit_angular and fell_on_table can both be considered causes of head_wound; therefore, generalisation arcs g6 and g7 are included in Gc.

The following example illustrates generalisation arcs that include enabling conditions.

Example 21.

Consider g7′:{fell_on_table,no_helmet} → head_wound in Gc, which is an adjustment to generalisation g7 of Example 20 which states that falling on a table causes a head wound in case you are not wearing a helmet. As in Example 20, proposition fell_on_table expresses a cause for head_wound and hence, fell_on_table is included in Ant(g7′). Proposition no_helmet does not express a cause for head_wound and can thus be considered an enabler of g7′; therefore, no_helmet is included in Enabler(g7′). It should be noted that, while no_helmet does not express a cause for the consequent, it still is a necessary condition of generalisation g7′.

Specific configurations of generalisations express that two propositions are alternative explanations of a common proposition, as captured by Definition 3. The terminology used is illustrated in Fig. 5.

Fig. 5.

Illustration of the terminology used in Definition 3.

Definition 3

Definition 3(Alternative explanations).

Let GI=(P,A) be an IG. Then p1,p2∈P are alternative explanations of q∈P, as indicated by generalisations g and g′ in G, iff one of the following holds:

• (1) g∈Ge, Head(g)=p1, q∈Ant(g), and either:

  • 1a) g′∈Ge, g′≠g, Head(g′)=p2, q∈Ant(g′), or;

  • 1b) g′∈Gc, Head(g′)=q, p2∈Ant(g′).

• (2) g∈Gc, Head(g)=q, p1∈Ant(g), and either:

  • 2a) g′∈Gc, g′≠g, Head(g′)=q, p2∈Ant(g′), or;

  • 2b) g′∈Ge, Head(g′)=p2, q∈Ant(g′).

• (3) g∈Ga, Head(g)=q, p1∈Tails(g) and g′∈Ga, g′≠g, Head(g′)=q, p2∈Tails(g′).

Note that cases 1b and 2b are symmetrical in terms of p1 and p2 and the used generalisations; we opt to keep the distinction between these two cases as they simplify the proof of Proposition 1. In case 1a, q is an actual antecedent and not an enabler of both g∈Ge and g′∈Ge; hence, both p1 and p2 are actual causes of q. Assuming that g and g′ both have multiple actual antecedents in case 1a, then p1 and p2 are alternative explanations of every proposition q∈Ant(g)∩Ant(g′). Hence, it is meaningful to define alternative explanations in the context in which generalisations have non-singleton sets of actual antecedents; this similarly holds for the other cases of Definition 3. In case 1b, p2 is an actual antecedent and not an enabler of g′∈Gc and thus a cause of q, and q is an actual antecedent and not an enabler of g∈Ge and thus p1 is a cause of q. In case 2a, p1 and p2 are actual antecedents of g∈Gc and g′∈Gc, respectively; hence, both p1 and p2 are actual causes of q. Finally, in case 3, p1 and p2 are antecedents of g∈Ga and g′∈Ga with the same consequent q, and hence are alternative explanations of q.

Example 22.

Consider the IG of Fig. 4. According to condition 2a of Definition 3, hit_angular and fell_on_table are alternative explanations of head_wound as indicated by generalisations g6 and g7. Similarly, according to condition 3 of Definition 3, hammer and stone are alternative explanations of hit_angular as indicated by generalisations g3 and g4.

A negation arc captures a conflict between a proposition and its negation expressed in an IG.

Definition 4

Definition 4(Negation arc).

Let GI=(P,A) be an IG. A negation arc n∈N⊆A is a bidirectional arc n:p↭q in GI that exists between a pair p,q∈P iff q=−p.

Example 23.

Consider the running example. As both 149 and ¬149 are included in the IG of Fig. 3, negation arc n1:149↭¬149 is also included in the graph. Similarly, the IG of Fig. 3 includes negation arc n2:155a↭¬155a. As noted in Section 3.1, one possible interpretation of the conflicts between propositions 462, 465 and 153 is that 462 and 465 indicate support for ¬153. Accordingly, generalisations g14:462→¬153 and g15:465→¬153 can be included, as depicted in the adjusted IG of Fig. 6. As these generalisations are defeasible and neither causal nor evidential nor an abstraction, g14 and g15 are included in Gdo. Negation arc n3:153↭¬153 is then included in N. An alternative interpretation of these conflicts is provided in Example 24.

As defeasible generalisations do not hold universally, exceptional circumstances can be provided under which such a generalisation may not hold; hence, we allow exceptions to defeasible generalisations to be specified in IGs by means of exception arcs.

Fig. 6.

Adjustment to part of the IG of Fig. 3, where 462 and 465 indicate support for ¬153.

Definition 5

Definition 5(Exception arc).

Let GI=(P,A) be an IG. An exception arc x∈X⊆A is a hyperarc x:p⇝g, where p∈P is called an exception to defeasible generalisation g∈Gc∪Ge∪Gda∪Gdo.

An exception arc directed from p to g indicates that p provides exceptional circumstances under which g may not hold.

Example 24.

Consider the running example. Instead of interpreting the conflicts between propositions 462, 465 and 153 as negations (see Example 23), an alternative interpretation is that 462 and 465 indicate exceptions to generalisation g4∈Ge. Specifically, 462 and 465 can be considered competing alternative explanations of 152: as Sacco carried his weapon for an innocent reason (462 or 465), this caused him to draw his weapon (152) with the intention of surrendering it. In Fig. 3, these exceptions are indicated by curved hyperarcs x1:462⇝g4 and x2:465⇝g4 in X.

4.2.Reading inferences from information graphs

We now define how deductive and abductive inferences can be performed with constructed IGs. By itself, a generalisation arc only expresses that the tails together allow us to infer the head in case this generalisation is used in deductive inference, or that the tails together can be inferred from the head in case of abductive inference. Only when considering the available evidence can directionality of inference actually be read from the graph.

Definition 6

Definition 6(Evidence set).

Let GI=(P,A) be an IG. An evidence set is a subset E⊆P for which it holds that for every p∈E, ¬p∉E.

The restriction that for every p∈E it holds that ¬p∉E ensures that not both a proposition and its negation are observed.

In figures in this paper, nodes in GI corresponding to elements of E are shaded and all shaded nodes correspond to elements of E. We emphasise that various evidence sets E can be used to establish (different) inferences from the same IG.

Example 25.

In Fig. 1, the evidence consists of the testimonies. In Figs 7 and 8, the IGs of Figs 3 and 6 are again depicted with nodes in E={150,151,461,463,464,466,470} shaded.

Fig. 7.

The IG of Fig. 3, where evidence set E (shaded) and resulting inference steps (↠) are also indicated.

Fig. 8.

The IG of Fig. 6, where evidence set E (shaded) and resulting inference steps (↠) are also indicated.

We now define when we consider configurations of generalisation arcs and evidence to express deductive and abductive inference.

4.2.1.Deductive inference

First, we specify under which conditions we consider a configuration of generalisation arcs and evidence to express deductive inference, where strict and defeasible deduction are distinguished.

Definition 7

Definition 7(Deductive inference).

Let GI=(P,A) be an IG, and let E⊆P be an evidence set. Let p1,…,pn,q∈P, with q∉E. Then given E, q is deductively inferred from propositions p1,…,pn using a generalisation g:{p1,…,pn}→q in G iff ∀pi, i=1,…,n:

• (1) pi∈E, or;

• (2) pi is deductively inferred from propositions r1,…,rm∈P using a generalisation g′:{r1,…,rm}→pi, where g′∉Gc if g∈Ge, pi∉Enabler(g), or;

• (3) pi is abductively inferred from a proposition r∈P using a generalisation g′:{pi,r1,…,rm}→r in Gc∪Ga, g≠g′, r1,…,rm∈P (see Definition 8).

Here, proposition q is defeasibly deductively inferred from p1,…,pn, denoted p1,…,pn↠gq, iff g∈Gc∪Ge∪Gda∪Gdo, and proposition q is strictly deductively inferred from p1,…,pn, denoted p1,…,pn⇀gq, iff g∈Gsa∪Gso.

For ease of reference, symbols ↠ and ⇀ are annotated with the name of the generalisation used in performing a defeasible or strict inference. In accordance with our assumptions stated in Section 2.1, deduction can be performed using all types of generalisations in G, where strict deduction can only be performed using strict abstractions and strict other types of generalisations. The condition q∉E ensures that deduction cannot be performed with a generalisation to infer its consequent in case its consequent is already observed. Deduction can only be performed using a generalisation g∈G to infer its consequent Head(g) from its antecedents Tails(g) in case every antecedent pi∈Tails(g) has been affirmed in that either pi is observed (i.e. pi∈E), pi itself is deductively inferred, or pi is abductively inferred. In correspondence with Pearl’s constraint (see Section 2.4.1), we assume in condition 2 that a proposition q∈P cannot be deductively inferred from p1,…,pn∈P using a generalisation g∈Ge if at least one of its actual antecedents pi∈Ant(g) is deductively inferred using a generalisation g′∈Gc. In this case, q and propositions ri∈Ant(g′) are considered alternative explanations of pi as indicated by g and g′ (Definition 3, case 1b or case 2b). Condition 3 of Definition 7 is explained in Section 4.2.3, after abductive inference is defined.

Example 26.

In the IG of Fig. 7, given E propositions 149, ¬149, 462, 465 and 469 are defeasibly deductively inferred from 150 and 151, 461, 463 and 464, 466, and 470 using generalisations g1, g2, g10, g11, and g12, respectively, as 150,151,461,463,464,466,470∈E (Definition 7, condition 1). Proposition 152 is then defeasibly deductively inferred from 149 using g3, as 149 is deductively inferred (Definition 7, condition 2). Propositions 153, 154 and 155 are then iteratively defeasibly deductively inferred using generalisations g4, g5 and g6, respectively. Finally, from 469, ¬155a is defeasibly deductively inferred using g13, as 469 is deductively inferred.

The following example illustrates strict deductive inference.

Example 27.

Consider Example 2 from Section 2.1. In this example, generalisation arc g: lung_cancer → cancer is included in Gsa. As lung_cancer ∈E, cancer is strictly deductively inferred from lung_cancer (Definition 7, condition 1).

The next example illustrates the restrictions put on performing deduction in our IG-formalism.

Example 28.

Figure 9a depicts an example of an IG in which q cannot be deductively inferred from p using g1, as Head(g1)=q∈E. In Fig. 9b, q cannot be deductively inferred from p1 and p2 using g1, as p2∉E and p2 is neither deductively nor abductively inferred.

In Fig. 9c, Example 8 illustrating Pearl’s constraint is modelled. As smoke_machine ∈E, smoke is deductively inferred from smoke_machine using g1 by condition 1 of Definition 7. fire cannot in turn be inferred from smoke using g2 by condition 2 of Definition 7, as g2∈Ge and smoke is deductively inferred using g1∈Gc.

Fig. 9.

Examples of IGs illustrating the restrictions put on performing deduction within our IG-formalism (a–c).

4.2.2.Abductive inference

Next, we specify under which conditions we consider a configuration of generalisation arcs and evidence to express abductive inference.

Definition 8

Definition 8(Abductive inference).

Let GI=(P,A) be an IG, and let E⊆P be an evidence set. Let p1,…,pn,q∈P, with {p1,…,pn}∩E=∅. Then given E, propositions p1,…,pn are abductively inferred from q using a g:{p1,…,pn}→q in Gc∪Ga, denoted q↠gp1;…;q↠gpn, iff:

• (1) q∈E, or;

• (2) q is deductively inferred from propositions r1,…,rm∈P using a generalisation g′:{r1,…,rm}→q in G, g≠g′ (see Definition 7), where g′∉Gc if g∈Gc and g′∉Ga if g∈Ga, or;

• (3) q is abductively inferred from a proposition r∈P using a generalisation g′:{q,r1,…,rm}→r in Gc∪Ga, r1,…,rm∈P.

In accordance with our assumptions stated in Section 2.2, abduction is defeasible and is modelled using only causal generalisations and abstractions. Following Console and Dupré [12] and Bex [4], we assume that abductive inference can be performed with both strict and defeasible abstractions, where such an inference is always defeasible as it concerns an inference from the more abstract consequent to a more specific antecedent (see Section 2.2). The condition {p1,…,pn}∩E=∅ ensures that abduction cannot be performed with a generalisation to infer its antecedents in case at least one of its antecedents is already observed. Furthermore, abductive inference can only be performed using a generalisation g∈Gc∪Ga to infer its antecedents Tails(g) from its consequent Head(g) in case Head(g) has been affirmed in that either Head(g) is observed (i.e. Head(g)∈E), Head(g) is deductively inferred, or Head(g) is itself abductively inferred.

In correspondence with Pearl’s constraint (see Section 2.4.1), we assume in condition 2 that propositions p1,…,pn∈P cannot be abductively inferred from a proposition q∈P using a generalisation g∈Gc if its consequent q is deductively inferred using a generalisation g′≠g, g′∈Gc. In enforcing this constraint, we do not need to consider whether or not the antecedents of g or g′ include enablers, as illustrated in Example 12 from Section 2.4.1. More specifically, in Definition 2 it is assumed that ∀g∈Gc∪Ge, Ant(g)≠∅; therefore, at least one proposition pi is an actual antecedent of g and at least one proposition rj is an actual antecedent of g′, which are then alternative explanations of q according to case 2a of Definition 3 which may not be inferred from each other by inferring q as an intermediary step. Similarly, we assume in condition 2 that g′∉Ga if g∈Ga to account for our constraints on performing deduction and abduction in that order with two abstractions (see Section 2.4.2). In this case, propositions p1,…,pn∈Tails(g) are alternative explanations of r1,…,rm∈Tails(g′) as indicated by g and g′ according to case 3 of Definition 3.

Example 29.

In the IG of Fig. 7, given E proposition 155a is abductively inferred from 155 using g7∈Gda, as 155 is deductively inferred (Definition 8, condition 2). In turn, propositions 156 and Π3 are iteratively abductively inferred using generalisations g8∈Gsa and g9∈Gc, respectively. Note that although g8 is a strict abstraction, the abductive inference from 155a to 156 is defeasible and not strict; specifically, that Sacco was conscious of having been involved in a robbery and shooting does not allow us to strictly infer that he was conscious of having been involved in the specific robbery and shooting that took place in South Braintree.

Fig. 10.

Example of an IG illustrating abductive inference with causal generalisations (a); example of an IG illustrating abductive inference with abstractions (b).

In the IG of Fig. 10a, q and r1 are abductively inferred from r using generalisation g3:{q,r1}→r in Gc by condition 1 of Definition 8, as r∈E. Then by condition 3 of Definition 8, p1 and p2 are abductively inferred from q using g1 and g2, respectively.

The following example further illustrates abductive inference with abstractions.

Example 30.

In Fig. 10b, Example 15 from Section 2.4.2 is modelled as an IG. As smoking ∈E, cancer is deductively inferred from smoking using g3′. Propositions lung_cancer and colon_cancer are then abductively inferred from cancer using strict abstractions g1′ and g2′, respectively (Definition 8, condition 2). Hence, in this example, a cause (smoking) for an event (cancer) is known, after which this event is inferred and is in turn further specified at a lower level of abstraction (lung_cancer or colon_cancer). As noted in Section 2.4.2, this type of mixed inference using a causal generalisation and abstractions does not lead to undesirable results.

The following examples illustrate that Pearl’s constraint for mixed deductive-abductive inference (see Section 2.4.1), as well as our proposed constraints on performing inference with abstractions (see Section 2.4.2), are adhered to.

Fig. 11.

An IG illustrating Pearl’s constraint for mixed deductive-abductive inference (a); an IG illustrating our inference constraints for abstractions (b); an IG illustrating mixed abductive-deductive inference (c).

Fig. 12.

The IG of Fig. 4, where evidence set E (shaded) and resulting inference steps (↠) are also indicated.

Example 31.

In Fig. 11a, Example 9 is modelled as an IG. As smoke_machine ∈E, smoke is deductively inferred from smoke_machine using g1. fire cannot be inferred from smoke, as g2∈Gc and smoke is deductively inferred using g1∈Gc (Definition 8, condition 2).

In Fig. 11b, Example 13 is modelled as an IG. As gun ∈E, deadly_weapon is deductively inferred from gun using g1. knife cannot in turn be inferred from deadly_weapon, as g2∈Ga and deadly_weapon is deductively inferred using g1∈Ga (Definition 8, condition 2).

The following example describes the inferences that can be made based on the IG of Fig. 4 corresponding to the mind map example of Section 3.2.

Example 32.

Consider the IG of Fig. 12. Given E={tes1,tes2,tes3,autopsy}, head_wound is deductively inferred from autopsy using g5. Then, hit_angular and fell_on_table are abductively inferred from head_wound using g6 and g7, respectively (Definition 8, condition 2). head_wound is also deductively inferred from fell_on_table using g7, as fell_on_table is deductively inferred from tes3 using g8; the inference type of g7 is, therefore, ambiguous (see Section 2.5). hammer and stone are abductively inferred from hit_angular using g3 and g4, respectively (Definition 8, condition 3). hit_angular is also deductively inferred from hammer and stone using g3 and g4, respectively, as hammer is deductively inferred from tes1 using g1 and stone is deductively inferred from tes2 using g2. Then, head_wound is deductively inferred from hit_angular using g6.

4.2.3.Mixed abductive-deductive inference

As apparent from Definitions 7 and 8, mixed abductive-deductive inference can be performed within our IG-formalism.

Example 33.

In Fig. 11c, Example 7 from Section 2.4 is modelled as an IG. From smoke, fire is abductively inferred using g1, as smoke ∈E. Then heat is deductively inferred (or predicted) from fire using g2 (Definition 7, condition 3).

5.An argumentation formalism based on information graphs

Based on our IG-formalism from Section 4, we now define an argumentation formalism that allows for both deductive and abductive argumentation. Note that the IG-formalism is not an argumentation formalism, and that no semantics for IGs were defined in Section 4. Instead, we defined how inference can be performed with IGs and we defined different notions of conflicts. In the current section, we define an argumentation formalism based on IGs which allows us to assign a semantics to argumentation frameworks constructed on the basis of IGs. More specifically, our approach generates an abstract argumentation framework as in Dung [13], that is, a set of arguments with a binary attack relation, which thus allows arguments based on IGs to be formally evaluated according to Dung’s semantics. We can then study properties of generated AFs; in particular, we prove that Caminada and Amgoud’s [9] postulates are satisfied by instantiations of our formalism, which warrants the sound definition of instantiations of our argumentation system and implies that anomalous results such as issues regarding inconsistency and non-closure as identified by [9] are avoided. Our argumentation formalism extends a preliminary version proposed in [38] that was based on a more restricted version of our IG-formalism [39] in which only causal and evidential generalisations without enablers were considered. Moreover, satisfaction of rationality postulates was not proven in that paper.

In Section 5.1, we define arguments on the basis of a provided IG and an evidence set E, which capture sequences of deductive and abductive inference applications as defined in Definitions 7 and 8 starting with elements from E. We then formally prove that arguments constructed on the basis of IGs conform to our inference constraints (Section 2.4). In Section 5.2, we define several types of attacks between arguments based on IGs, which are based on the different types of conflicts defined for our IG-formalism. In Section 5.3 we instantiate Dung’s abstract approach with arguments and attacks based on IGs and provide the definitions of Dung’s argumentation semantics. In Section 5.4, we then prove that rationality postulates [9] are satisfied by instantiations of our formalism.

5.1.Arguments

In this section, we define how arguments on the basis of an IG and an evidence set E are constructed. Here, we take inspiration from the definition of an argument as defined for the ASPIC+ framework [20]. By remaining close to the ASPIC+ framework, this allows us to straightforwardly show that rationality postulates are satisfied for our argumentation formalism based on IGs (see Section 5.4). In what follows, for a given argument, the operator PREM returns all propositions in E used to construct the argument, CONC returns its conclusion, SUB returns all its sub-arguments (including itself), IMMSUB returns its immediate sub-arguments, GEN returns all the generalisations used in constructing the argument, TOPGEN returns the last generalisation used in constructing the argument, DEFINF and STINF return all the defeasible and strict inferences used in constructing the argument, respectively, and TOPINF returns the last inference used in constructing the argument. Definition 9 is explained and illustrated in Examples 34 and 35.

Definition 9

Definition 9(Argument).

Let GI=(P,A) be an IG, and let E⊆P be an evidence set. An argument A on the basis of GI and E is any structure obtainable by applying one or more of the following steps finitely many times, where steps 2 (i.e. step 2a or 2b) and 3 or vice versa are not subsequently applied using the same generalisation arc g∈G:

• 1. p if p∈E, where: PREM(A)={p}; CONC(A)=p; SUB(A)={A}; IMMSUB(A)=∅; GEN(A)=∅; TOPGEN(A)= undefined; DEFINF(A)=∅; STINF(A)=∅; TOPINF(A)= undefined.

• 2a. A1,…,An↠gp if A1,…,An are arguments such that p is defeasibly deductively inferred from CONC(A1),…,CONC(An) using a generalisation g:{CONC(A1),…,CONC(An)}→p according to Definition 7, where it holds that g∈Gc∪Ge∪Gda∪Gdo and if g is of the form g:c→e in Gc and its evidential counterpart g′:e→c is included in Ge, then g′∉GEN(A1)∪⋯∪GEN(An). For A, it holds that:

  PREM(A)=PREM(A1)∪⋯∪PREM(An); CONC(A)=p;

  SUB(A)=SUB(A1)∪⋯∪SUB(An)∪{A}; IMMSUB(A)={A1,…,An};

  GEN(A)=GEN(A1)∪⋯∪GEN(An)∪{g}; TOPGEN(A)=g;

  DEFINF(A)=DEFINF(A1)∪⋯DEFINF(An)∪{CONC(A1),…,CONC(An)↠gp};

  STINF(A)=STINF(A1)∪⋯STINF(An);

  TOPINF(A)=CONC(A1),…,CONC(An)↠gp.

• 2b. A1,…,An⇀gp if A1,…,An are arguments such that p is strictly deductively inferred from CONC(A1),…,CONC(An) using a generalisation g∈Gsa∪Gso, g:{CONC(A1),…,CONC(An)}→p according to Definition 7, where PREM(A), CONC(A), SUB(A), IMMSUB(A), GEN(A) and TOPGEN(A) are defined as in step 2a, and where:

  DEFINF(A)=DEFINF(A1)∪⋯DEFINF(An);

  STINF(A)=STINF(A1)∪⋯STINF(An)∪{CONC(A1),…,CONC(An)⇀gp};

  TOPINF(A)=CONC(A1),…,CONC(An)⇀gp.

• 3. A′↠gp if A′ is an argument such that p is abductively inferred from CONC(A′) using a generalisation g∈Gc∪Ga, g:{p,p1,…,pn}→CONC(A′) for some propositions p1,…,pn∈P according to Definition 8, where:

  PREM(A)=PREM(A′); CONC(A)=p; SUB(A)=SUB(A′)∪{A}; IMMSUB(A)={A′}; GEN(A)=GEN(A′)∪{g}; TOPGEN(A)=g; DEFINF(A)=DEFINF(A′)∪{CONC(A′)↠gp}; STINF(A)=STINF(A′); TOPINF(A)=CONC(A′)↠gp.

Note that we overload symbols ↠ and ⇀ to denote an argument while it also denotes a defeasible or strict inference. The set of all arguments on the basis of GI and E is denoted by A.

An argument A∈A is called strict if DEFINF(A)=∅; otherwise, A is called defeasible. An argument A∈A is called a premise argument if only step 1 of Definition 9 is applied, deductive if only steps 1, 2a and 2b are applied, abductive if only steps 1 and 3 are applied, and mixed otherwise. The restriction that steps 2 (i.e. step 2a or 2b) and 3 or vice versa are not subsequently applied using the same generalisation arc g∈G ensures that cycles in which two propositions are iteratively deductively and abductively inferred from each other using the same g are avoided in argument construction. Similarly, in case causal generalisation g:c→e has an evidential counterpart g′:e→c (see Sections 2.3 and 4.1), then the restriction in step 2a that g′∉GEN(A1)∪⋯∪GEN(An) ensures that cycles in which c and e are iteratively deductively inferred from each other using g′ and g are avoided. Note that cycles in which c and e are iteratively deductively inferred from each other using g and g′ in that order are already avoided due to the enforcement of Pearl’s constraint in condition 2 of Definition 7.

Fig. 13.

The IG of Fig. 7, where arguments and direct attacks (⇢) on the basis of the IG and E are also indicated.

Example 34.

Consider Fig. 13, in which arguments constructed on the basis of the IG of Fig. 7 are indicated. According to step 1 of Definition 9, A1:150 and A2:151 are premise arguments. Based on A1 and A2, defeasible deductive argument A3:A1,A2↠g1149 is constructed by step 2a of Definition 9, as 149 is defeasibly deductively inferred from 150 and 151 using g1∈Ge. Arguments A4:A3↠g3152; A5:A4↠g4153; A6:A5↠g5154 and A7:A6↠g6155 similarly are defeasible deductive arguments. Argument A8:A7↠g7155a is a defeasible mixed argument by step 3 of Definition 9, as 155a is abductively inferred from 155 using g7. Similarly, arguments A9:A8↠g8156 and A10:A9↠g9Π3 are defeasible mixed arguments. To illustrate the operators used in Definition 9, for A8, we have that PREM(A8)={150,151}; CONC(A8)=155a; SUB(A8)={A1,A2,A3,A4,A5,A6,A7,A8}; IMMSUB(A8)={A7}; GEN(A8)={g1,g3,g4,g5,g6,g7}; TOPGEN(A8)=g7; DEFINF(A8)={150,151↠g1149;149↠g3152;152↠g4153;153↠g5154;154↠g6155;155↠g7155a}; STINF(A8)=∅; TOPINF(A8)=155↠g7155a.

Step 3 of Definition 9 is now illustrated in more detail.

Example 35.

On the basis of the IG of Fig. 10a and E={r}, A1′:r is a premise argument. From A1′, arguments A2′:A1′↠g3r1 and A3′:A1′↠g3q are constructed by step 3 of Definition 9, as q and r1 are abductively inferred from CONC(A1′) using causal generalisation g3:{q,r1}→r. Then again by step 3, A4′:A3′↠g1p1 and A5′:A3′↠g2p2 are constructed using g1 and g2, respectively.

5.1.1.Properties of arguments based on IGs

We now prove a number of formal properties of arguments based on IGs. Lemma 1 states that the conclusions of deductive, abductive, and mixed arguments constructed in our argumentation formalism based on IGs are not observed.

Lemma 1.

Let A be a set of arguments on the basis of IG GI=(P,A) and evidence set E. Let A∈A be a deductive, abductive, or mixed argument. Then CONC(A)∉E.

Proof.

As A is not a premise argument, step 2a, step 2b or step 3 of Definition 9 is applied last in constructing A. In case step 2a or 2b of Definition 9 is applied last, then ∃g∈G such that Head(g)=CONC(A) is deductively inferred using TOPGEN(A)=g according to Definition 7. Hence, per the restrictions of Definition 7, Head(g)=CONC(A)∉E. In case step 3 of Definition 9 is applied last, then ∃g∈G such that CONC(A)∈Tails(g) is abductively inferred using TOPGEN(A)=g according to Definition 8. Hence, CONC(A)∉E per the restriction of Definition 8 that Tails(g)∩E=∅. □

In performing inference care should be taken that no cause for an effect is inferred in case an alternative cause for this effect was already previously inferred (see Section 2.4.1). Similarly, care should be taken that no version of an event at a lower level of abstraction is inferred if an alternative version of this event at a lower level of abstraction was already previously inferred (see Section 2.4.2). In the context of IGs, for g∈Gc, propositions in Ant(g) express causes for the common effect expressed by Head(g), for g∈Ge, Head(g) expresses the usual cause for propositions in Ant(g), and for g∈Ga, propositions in Tails(g) are at a lower level of abstraction than Head(g). Hence, in defining how inferences can be read from IGs, restrictions are put in Definitions 7 and 8 such that our inference constraints (see Section 2.4) are adhered to. We now formally prove that these inference constraints are never violated in constructing sequences of arguments on the basis of IGs.

First, we formally define the inference constraints of Section 2.4 in the context of arguments constructed on the basis of IGs.

Definition 10

Definition 10(Inference constraint).

Let A be a set of arguments on the basis of IG GI=(P,A) and evidence set E. Let p1,p2∈P be alternative explanations of q∈P as indicated by generalisations g1,g2∈G (Definition 3). If arguments A and B exist in A with CONC(B)=q, A∈IMMSUB(B), and CONC(A)=p1, then there does not exist an argument C∈A with B∈IMMSUB(C), CONC(C)=p2.

We now formally prove that this inference constraint is indeed adhered to.

Proposition 1

Proposition 1(Adherence to inference constraint).

Let A be a set of arguments on the basis of IG GI=(P,A) and evidence set E. Then A adheres to the inference constraint of Definition 10.

Proof.

Assume that p1,p2∈P are alternative explanations of q∈P as indicated by generalisations g1 and g2 in G, and assume that arguments A,B∈A exist with CONC(B)=q, A∈IMMSUB(B), CONC(A)=p1. Then we need to prove that no argument C exists in A with B∈IMMSUB(C) and CONC(C)=p2. In constructing argument B, either step 2a, step 2b or step 3 of Definition 9 is applied last, where generalisation g1 is used to infer CONC(B)=q. Here, g1 cannot be of the form g1∈Ge, q∈Ant(g1), Head(g1)=p1 (Definition 3, case 1) as in this case antecedent q of g1 is inferred from consequent p1 of g1, which would be an instance of abductive inference while per the restrictions of Definition 8 abductive inference can only be performed using generalisations in Gc∪Ga. More specifically, argument B cannot be constructed by applying step 2a, 2b and 3 of Definition 9 last if g1 is of that form. Thus, we only need to consider cases 2 and 3 of Definition 3, where a generalisation g1∈Gc, Head(g1)=q, p1∈Ant(g1) respectively a generalisation g1∈Ga, Head(g1)=q, p1∈Tails(g1) is used to construct B, namely by applying step 2a or 2b of Definition 9 last to deductively infer CONC(B)=q. We now show that for the given options for g1, no argument C with B∈IMMSUB(C), CONC(C)=p2 can be constructed using g2.

• First, consider case 2a of Definition 3 in which g2≠g1, g2∈Gc, Head(g2)=q, p2∈Ant(g2). Then no argument C with B∈IMMSUB(C), CONC(C)=p2 can be constructed using g2, as in this case abduction would be performed with g2 to infer p2 from q while per the restrictions in condition 2 of Definition 8 abduction cannot be performed with g2 as Head(g2) was previously deductively inferred using g1∈Gc. In particular, step 3 of Definition 9 cannot be applied in constructing C using g2. Furthermore, neither step 2a nor step 2b of Definition 9 can be applied in constructing C using g2, as these steps specify deductive and not abductive inferences.

• Next, consider case 2b of Definition 3 in which g2∈Ge, Head(g2)=p2, q∈Ant(g2). Then no argument C with B∈IMMSUB(C), CONC(C)=p2 can be constructed using g2, as in this case deductive inference would be performed with g2 to infer p2 while per the restrictions in condition 2 of Definition 7 deductive inference cannot be performed with g2 as q∈Ant(g2) was previously deductively inferred using g1∈Gc. In particular, step 2a of Definition 9 cannot be applied in constructing C using g2. Furthermore, step 2b of cannot be applied in constructing C using g2, as this step can only be applied using strict generalisations and g2∉Gsa∪Gso, and step 3 cannot be applied in constructing C using g2, as this step specifies an abductive and not a deductive inference.

• Finally, consider case 3 of Definition 3 in which g2≠g1, g2∈Ga, Head(g2)=q, p2∈Tails(g2). Then no argument C with B∈IMMSUB(C), CONC(C)=p2 can be constructed using g2, as in this case abduction would be performed with g2 to infer p2 from q while per the restrictions in condition 2 of Definition 8 abduction cannot be performed with g2 as Head(g2) was previously deductively inferred using g1∈Ga. In particular, step 3 of Definition 9 cannot be applied in constructing C using g2. Furthermore, neither step 2a nor step 2b of Definition 9 can be applied in constructing C using g2, as these steps specify deductive and not abductive inferences. □