Deductive and abductive argumentation
based on information graphs
Remi Wieten a,1 , Floris Bex a,b , Henry Prakken a,c and Silja Renooij aaDepartment of Information and Computing Sciences, Utrecht University,
Princetonplein 5, 3584 CC, Utrecht, The NetherlandsbInstitute for Law, Technology and Society, Tilburg University,
Warandelaan 2, 5037 AB, Tilburg, The NetherlandscFaculty of Law, University of Groningen,

Oude Kijk in ’t Jatstraat 26, 9712 EK, Groningen, The Netherlands
Abstract. In this paper, we propose an argumentation formalism thatallows for both deductive and abductive argumentation, where ‘deduction’ is used as an umbrella term for both defeasible and strict ‘forward’
inference. Our formalism is based on an extended version of our previously proposed information graph (IG) formalism, which provides aprecise account of the interplay between deductive and abductive inference and causal and evidential information. In the current version,
we consider additional types of information such as abstractions whichallow domain experts to be more expressive in stating their knowledge,
where we identify and impose constraints on the types of inferences that
may be performed with the different types of information. A new notionof attack is defined that captures a crucial aspect of abductive reasoning, namely that of competition between abductively inferred alternative explanations. Our argumentation formalism generates an abstract
argumentation framework and thus allows arguments to be formallyevaluated. We prove that instantiations of our argumentation formalism
satisfy key rationality postulates.
Keywords. Argumentation, deduction, abduction, causal and evidential
information, default reasoning

1 Corresponding author. Email address: g.m.wieten@uu.nl. Telephone number: +31 6 3956 8028.

11. Introduction
In the legal and forensic domains, reasoning about evidence plays a central role in the rational processof proof [2,5]. To aid in this process, various graph-based tools exist that allow domain experts tomake sense of a mass of evidence in a case, such as mind maps [24,36], argument diagrams [4,24] andWigmore charts [40]. Because of their informal nature, these tools typically do not directly allow for5formal evaluation using AI techniques such as computational argumentation [14]. Hence, we wish
to formalise and disambiguate analyses performed using such tools in a manner that (1) allows forformal evaluation and that (2) adheres to principles from the literature on reasoning about evidence[2,5,18,26], 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.

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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 [2,4,5], also called defaults [26,33], which capture knowledge about the world in conditional form. A distinction can bemade between causal generalisations (e.g. ‘fire typically causes smoke’ ) and evidential generalisations (e.g. ‘smoke is evidence for fire’ ) [5,26]. In the current paper, we also consider generalisations

15

that are neither causal nor evidential; examples are abstractions [5,13] 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 [13,18] or backward fashion, where from a causal generalisation or an abstraction and by affirming the consequent (smoke), the antecedent (fire) is defeasibly20

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. [21])or defeasible inference, in which the consequent tentatively holds given the antecedents (e.g. [34]).
To cover both meanings, in this paper ‘deduction’ is used as an umbrella term for both defeasible‘forward’ inference and strict ‘forward’ inference.

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Pearl [26, 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 [9] found thatwhile 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 theirreasoning. For instance, subjects often considered testimonies to be evidential, whereas a motive for

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committing an act is considered a cause for committing that act. This discussion illustrates thatin formal accounts of reasoning about evidence, it is important to allow for causal and evidential
generalisations [5]. 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 stating2

their knowledge. The need for including these types of generalisations will become apparent from35

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 typicallyleft implicit. Hence, in previous work [39] we set out to formalise analyses performed using these40

tools by providing a precise account of the interplay between the different types of inferences andgeneralisations 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 acategory of generalisations termed ‘other’, consisting of generalisations that are neither causal nor

45

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 thatno version of an event at a lower level of abstraction is inferred if an alternative version of this eventat a lower level of abstraction was already previously inferred. Hence, we extend on the constraints

50

imposed by Pearl’s C-E system [26] which say that, in performing inference, care should be takenthat 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[12] are provided under which a generalisation may be used in performing inference. Based on these55

constraints and our conceptional analysis of reasoning about evidence, we define how deductive andabductive 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 logicbased (e.g. [5,6,13,21]); instead, we opt for a graph-based formalism to remain closely related to theway analyses are visualised using aforementioned graph-based tools.

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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) [17], graphical models of joint probability distributions. In this paper, we instead focus onargumentation, where we propose an argumentation formalism based on IGs that allows for both65

deductive and abductive argumentation [38]. Previous work on abduction includes work on formal
logical models of abductive reasoning (e.g. [13,18]) and the work of Kakas and colleagues on abductive
logic programming [20]. However, to the best of our knowledge, our proposed formalism is one of3the first formalisms that models combined abductive and deductive reasoning in a formalism forstructured argumentation. The closest to the current paper is the integrated theory of causal and70

evidential arguments [6], which is based on the ASPIC+ framework [21]. In the 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. Actual abductionis thus not performed by constructing arguments. Moreover, in contrast with [6] we put special
emphasis on the constraints on performing inference that need to be imposed in the current paper,

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where we formally prove that arguments based on IGs indeed adhere to the identified constraints.
Finally, compared to the ASPIC+ framework [21], which only allows for deductive reasoning, weallow for both deductive and abductive reasoning and introduce a new type of conflict, namely
conflict between competing alternative explanations [18], which is currently not accounted for inthat framework. Other related formalisms are discussed in Section 6.

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Our argumentation formalism generates an abstract argumentation framework as in Dung [14],that is, a set of arguments with a binary attack relation, which thus allows arguments to be formallyevaluated 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 [21,28], we also define the notion of alternative attack among arguments based on

85IGs, a concept based on the notion of competing alternative explanations that is inspired by [3,6].
Alternative attack captures a crucial aspect of abductive reasoning, namely that of conflict between
abductively inferred conclusions [18].
Our argumentation formalism extends a preliminary version proposed in [38] that was based ona more restricted version of our IG-formalism [39] in which only causal and evidential generalisations

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without enablers were considered. Moreover, in comparison to our earlier work [38] we now also provethat key rationality postulates [10] are satisfied by instantiations of our formalism, which implies
that anomalous results as identified by [10] are avoided.
To summarise the main contributions of this paper, we propose an argumentation formalism thatallows for both deductive and abductive argumentation, the latter of which has received relatively

95

little attention in argumentation. Our argumentation formalism is based on an extended versionof 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 including enabling
conditions, where constraints are imposed on the types of inferences that may be performed withthese new types of generalisations. A new notion of attack is defined, namely alternative attack.

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Our approach allows arguments to be evaluated using Dung’s semantics. We formally prove that
instantiations of our argumentation formalism satisfy key rationality postulates [10].

4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
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that both deductive and abductive inference is performed by domain experts using both causal andevidential 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 wediscuss related work. In Section 7 we summarise our findings and conclude.

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2. Reasoning about evidence

In this section, we provide a conceptual analysis of reasoning about evidence, where we review theterminology 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, we115

now also consider abstractions and other types of generalisations, as well as generalisations including
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

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instance, in the context of a legal trial, the evidence consists of that what is actually observed bya judge or jury, such as documents and other tangible evidence, as well as testimonial evidence[2]. Inference is often performed using domain-specific generalisations [2,4,5], also called defaults[26,33], 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 usu-

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ally/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 generalisations130

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’5

its consequent. An example of an exception to this generalisation is that sufficient oxygen is present
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for complete combustion to occur.
A distinction can be made between causal and evidential generalisations [5,26], where insteadof 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

140

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. [5,19]); in this paper, this assumption is also made. The examples considered throughout
this paper illustrate that causal and evidential generalisations are typically not strict2 .

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In this paper, we also consider generalisations that are neither causal nor evidential. For instance,
abstractions [5,13] 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 [13],150

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. [5]) but may also be strict (cf. [13]); 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

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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 explic-160

itly. In this paper, we distinguish between generalisations that are causal, evidential, abstractions, orof another type, where generalisations of type ‘other’ may be defeasible or strict. Specifically, as thiscategory 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 thesegeneralisations. Table 1 provides an overview of the different generalisation types, where for each

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type it is indicated whether generalisations may be defeasible or strict. The notation →c , →e , →a
2 Note that strict generalisations such as strict rules from classical logic and definitions can be expressed using strict

generalisations of type ‘other’ and strict abstractions.
6

Table 1. Table indicating for each generalisation type whether generalisations may be defeasibleor strict.

Defeasible
StrictCausalEvidential
Abstractions
generalisations generalisations

V

V

V

Other
generalisations

X

X

V

V

V

and →o is used for the different types of generalisations, respectively.
Different types of inferences can be performed with generalisations depending on whether theirantecendents 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 conse170

quent is inferred from the antecedents and in abductive inference the antecendents are inferred fromthe consequent. These two inference types are now considered in more detail.
2.1. Deductive inference
Inference can be performed in a deductive fashion, where from a generalisation and by affirming
the antecedents, the consequent is inferred by modus ponens on the generalisation. As noted in175

the introduction, the term ‘deduction’ is used for both defeasible and strict ‘forward’ inference;
hence, deduction is not necessarily a stronger or more reliable form of inference than abduction,
which is a type of defeasible inference. Defeasible deduction can only be performed using defeasible
generalisations (of any type) and not using strict generalisations (see Table 2). Strict deductive
inference can only be performed using strict abstractions and strict generalisations of type ‘other’.

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For a given instance of deductive inference, it will be explicitly specified whether it concerns strict
or defeasible deductive inference.
Example 1. Consider causal generalisation g : fire →c smoke. By affirming g’s antecedent fire, itsconsequent smoke is defeasibly deductively inferred.



The following example illustrates strict deductive inference.
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Example 2. Consider strict abstraction g : lung cancer →a cancer. Upon observing that a person
has lung cancer, we can strictly deductively infer that the person has cancer using g.

Prediction [34] is a specific type of deductive inference in which the consequent of a causal
generalisation is deductively inferred by affirming its antecedents. Specifically, as the antecedents
of a causal generalisation express causes for the consequent, the consequent is said to be predicted
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from the antecedents in this case. Example 1 provides an example of prediction.
7

Table 2. Table indicating for defeasible and strict generalisations of every generalisation type which types of inferences
may be performed.

Defeasible
deduction
Strictdeduction
Abduction

Causalgeneralisations

Evidential
generalisations

Defeasible
abstractions

Strictabstractions

Defeasible
othergeneralisations

Strictothergeneralisations

V

V

V

X

X

X

X

V

X

V

V

X

V

V

X

X

V

X

2.2. Abductive inference
Abduction [13,18], a type of defeasible inference, can be performed using causal generalisations
and abstractions: from a causal generalisation or an abstraction and by affirming the consequent,
the antecedents are inferred, since if the antecedents are true it would allow us to deductively
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infer the consequent modus-ponens-style. Following [18], in case causes c1 , . . . , cn and c01 , . . . , c0m
are abductively inferred from common effect e using causal generalisations g1 : c1 , . . . , cn →c e andg2 : c01 , . . . , c0m →c e, then ci and c0j for i ∈ {1, . . . , n}, j ∈ {1, . . . , m}, ci 6= c0j are considered to becompeting alternative explanations for e. We assume that causes ci (and c0j ) are not in competition
among themselves.

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Example 3. Consider the following causal generalisations:g1 : fire →c smoke;g2 : smoke machine →c smoke.

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By affirming the common consequent ( smoke), fire and smoke machine are abductively inferred,
which are then competing alternative explanations of smoke.

Abductive inference can also be performed using abstractions [5,13], where the used abstraction
can either be defeasible (cf. [5]) or strict (cf. [13]). An example of a model including strict abstractions is that of Console and Dupré [13], in which both explanatory axioms (comparable to causal
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generalisations) and abstraction axioms are used to explain observations. Multiple explanations thatare inferred using abstraction axioms can then be considered competing alternative explanations.
Note that an abductive inference step with a strict abstraction is still defeasible, as it concerns aninference step from the more abstract consequent to a more specific antecedent. Following Console
and Dupré [13] and Bex [5], we allow for abductive inference using both strict and defeasible abstrac-

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tions, where in performing abduction with abstractions g1 : p1 , . . . , pn →a q and g2 : p01 , . . . , p0m →a q
8the antecedents pi and p0j for i ∈ {1, . . . , n}, j ∈ {1, . . . , m}, ci 6= c0j are considered to be competingalternative explanations of the common consequent q. We assume that antecedents pi (and p0j ) are
not in competition among themselves.
Example 4. Consider the following defeasible abstractions:220

g1 : gun →a deadly weapon;
g2 : knife →a deadly weapon.
By affirming the common consequent ( deadly weapon), gun and knife are abductively inferred using
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generalisations g1 and g2 , which are then competing alternative explanations of deadly weapon. The following example illustrates abductive inference with strict abstractions.
Example 5. Consider the following strict abstractions:g10 : lung cancer →a cancer;

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g20 : colon cancer →a cancer.
Upon observing that a person has cancer, lung cancer and colon cancer are abductively inferred,
which are then competing alternative explanations of cancer.



2.3. Representing causal knowledge
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Abductive inference with causal generalisations and deductive inference with evidential generalisations are related: in some cases, we will accept not only causal generalisation ‘c usually/normally/typically causes e’ but also evidential generalisation ‘e is evidence for c’ [6,26], which
we will call the evidential counterpart of the causal generalisation. However, it can be argued that
we only accept the evidential counterpart of a causal generalisation if c is the usual cause of e, where240

we assume that only one cause can be the usual cause of e.
Example 6. Fire can be considered the usual cause of smoke, so we will accept both causal generalisation g : fire →c smoke and its evidential counterpart g 0 : smoke →e fire. In this case, abduction
with generalisation g can be encoded as deduction with generalisation g 0 . Because a smoke machine
cannot be considered the usual cause of smoke, we will accept causal generalisation smoke machine245→c smoke but we will not accept evidential generalisation smoke →e smoke machine.

Note that a causal generalisation g can only have an evidential counterpart g 0 in case g has asingle antecedent, as we assume generalisations have a single consequent but multiple antecedents.
9

Furthermore, as we assume that only one cause can be the usual cause of e, only one of the causal
generalisations c1 →c e or c2 →c e can be replaced by an evidential generalisation. Hence, we do
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not consider c1 and c2 to be competing alternative explanations of e in case deductive inference isperformed using evidential generalisations e →e c1 and e →e c2 .
2.4. Mixed inference and inference constraints
Deductive and abductive inference can be iteratively performed, where mixed abductive-deductive
inference is also possible.

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Example 7. Suppose that from the causal generalisation g1 : fire →c smoke and by affirming its
consequent ( smoke), its antecedent ( fire) is inferred. Now, if the additional causal generalisation
g2 : fire →c heat is provided, then its consequent ( heat) can be deductively inferred (or predicted)
as the antecedent ( fire) has been previously abductively inferred.



2.4.1. Constraints on performing inference with causal and evidential generalisations
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Mixed deductive inference, using both causal and evidential generalisations, can also be performed[6], but as noted by Pearl [26] care should be taken in performing mixed inference that no cause for
an effect is inferred in case an alternative cause for this effect was already previously inferred.
Example 8. Consider the example in which a causal generalisation g1 : smoke machine →c smoke
and an evidential generalisation g2 : smoke →e fire are provided. Deductively chaining these gen-265

eralisations would make us infer that there is a fire when seeing a smoke machine, which is clearly
undesirable.



Similarly, in performing mixed deductive-abductive 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.
Example 9. Consider Example 8, where instead of an evidential generalisation g2 : smoke →e fire270

a causal generalisation g2 : fire →c smoke is provided. Upon seeing a smoke machine, this would
make us infer that there is a fire in case deduction and abduction are performed in sequence, whichis again undesirable.



Accordingly, we wish to prohibit these types of inference patterns, and refer to the constraintthat no cause for an effect should be inferred in case an alternative cause for this effect was already275

previously inferred as Pearl’s constraint [26].
The above discussion can be extended to generalisations with multiple antecedents.
10

Example 10. Suppose that the following generalisations are provided:

g1 : high body temperature →e fever;280

g2 : smoke →c coughing;g3 : fever, coughing →e pneumonia.

Upon observing that a person has high body temperature and that there is smoke, this would makeus infer that the person has a fever and is coughing using generalisations g1 and g2 , respectively.
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In turn, this would make us infer that the person has pneumonia using generalisation g3 , which isundesirable: as a cause for coughing was already previously inferred ( smoke), we should not be able
to infer a different cause for coughing ( pneumonia). Specifically, fever is in itself not a sufficientcondition for inferring pneumonia: coughing is also necessary. Only in case a separate evidential
generalisation g4 : fever →e pneumonia is provided should we be able to infer pneumonia.

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

Similar problems arise in performing inference using causal generalisations with multiple antecedents. Accordingly, we wish to extend Pearl’s constraint to generalisations with multiple antecedents. However, there are exceptions under which we do not wish to prohibit the aforementioned types of inference patterns, namely in case additional circumstances, also called enabling conditions [12], or enablers, are provided under which a causal or evidential generalisation295

may be used in performing inference. Generalisations including enablers are of the general form
e1 , . . . , em , a1 , . . . , an → b, where e1 , . . . , em are its enablers and a1 , . . . , an its actual antecedents.
For a causal generalisation, only its actual antecedents and not its enablers express causes for theconsequent. Similarly, for an evidential generalisation its consequent only expresses the usual cause
for its actual antecedents and not for its enablers. Causality is a contentious topic, and it is easy to300

disagree about whether an event is an actual cause or an enabler. Cheng and Novick [12] note thatan event is typically viewed as an actual cause if it describes a situation that deviates from ‘normal’circumstances. For instance, lighting a match is considered a cause of fire, but the presence of oxygen
is typically not consider a cause of fire as it is normal that oxygen is present. This is, however,also context-dependent, and oxygen can be considered a cause of fire in situations where oxygen

305

is typically not present (e.g. in space). We note that generalisations capture knowledge about theworld as perceived by the person stating the knowledge, and that the distinction between enablers
and actual causes allows domain experts to be more expressive in stating their knowledge.
The following example illustrates that deductively chaining a causal and an evidential generalisation does not lead to undesirable results for evidential generalisations including enablers.
11

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Example 11. Consider the example in which the following evidential generalisation is provided:

g1 : fire, dry wood →e lightning strike.

Generalisation g1 states that from the presence of dry wood and fire we can conclude that there may
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have been a lightning strike. In this case, dry wood is an enabler of the generalisation, and lightning strike cannot be considered a cause for antecedent dry wood. Only in case fire was previously
deductively inferred using a causal generalisation (e.g. g2 : torch →c fire) should the application ofevidential generalisation g1 be blocked. However, in case dry wood was previously inferred using a
causal generalisation (e.g. g3 : warm summer →c dry wood) and fire is not inferred using a causal

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generalisation, then we should be able to infer lightning strike using generalisation g1 .

Similarly, inference may be performed using causal generalisations including enablers, but Pearl’s
constraint does not need to be reconsidered in this case as illustrated by the following example.
Example 12. Consider the example in which the following causal generalisations are provided:

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g1 : torch →c fire;g2 : match, oxygen →c fire.

In this case, the presence of oxygen is an enabler of generalisation g2 , as it cannot be considered an
actual cause of fire. Upon striking a match in the presence of oxygen, we can deductively infer that330

there is a fire using generalisation g2 . Similar to Example 9, we should now not be able to abductively
infer torch using generalisation g1 . Similarly, performing deduction and abduction in sequence using
generalisations g1 and g2 is undesirable.

To summarise this section, we wish to prohibit (1) subsequent deductive inference using a causal
and an evidential generalisation in case the consequent of the causal generalisation is an actual335

antecedent of the evidential generalisation and not an enabler, and (2) subsequent deductive andabductive inference using two causal generalisations with the same consequent. Note that, whilethese constraints deviate from Pearl’s original constraints [26] as enabling conditions are now also
taken into account, we will refer to these constraints as Pearl’s constraint throughout this paper.
2.4.2. Constraints on performing inference with abstractions

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When performing inference with abstractions, care should be taken that no version of an event at alower level of abstraction is abductively inferred if an alternative version of this event at a lower level12

of abstraction was already previously inferred. In particular, performing deduction and abductionin that order with two abstractions with the same consequent leads to undesirable results.
Example 13. Consider generalisations g1 : gun →a deadly weapon and g2 : knife →a deadly weapon
345

from Example 4. Upon observing that a provided object is a gun, this would make us deductively
infer that this object is a deadly weapon using generalisation g1 . Upon performing abduction with
g2 , this would make us infer that the provided object is a knife, which is clearly undesirable.



Performing abduction and deduction in that order with two abstractions with the same consequent does not lead to undesirable results.
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Example 14. Consider abstractions g2 : knife →a deadly weapon and g3 : knife →a metal object.
Upon observing metal object, we can abductively infer knife using generalisation g3 . In turn, claim
deadly weapon is deductively inferred using generalisation g2 .

The following example illustrates that mixed inference, using either a causal generalisation andan abstraction or an evidential generalisation and an abstraction, does not lead to undesirable results.
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Hence, no additional inference constraints need to be imposed.
Example 15. Consider Example 5. Assume that in addition to strict abstractions g10 : lung cancer→a cancer and g20 : colon cancer →a cancer, causal generalisation g30 : smoking →c cancer is provided. Upon observing that a person smokes, we deductively infer that the person has cancer using
generalisation g30 . Using generalisations g10 and g20 , we can then in turn abductively infer that the360

person has either lung cancer or colon cancer, which are then competing alternative explanations ofcancer (see Example 5). Note that it is not undesirable to infer lung cancer or colon cancer fromcancer in this case, as smoking and lung cancer ( colon cancer) are not alternative explanations ofcancer; instead, smoking is a cause of cancer, while lung cancer ( colon cancer) is not a cause ofcancer but instead describes claim cancer at a lower level of abstraction. Similar observations can365

be made by replacing generalisation g30 by generalisation g40 : cancer →e smoking.

To summarise this section, we only wish to prohibit subsequent deduction and abduction using
two abstractions with the same consequent and not other inference patterns involving abstractions.
Finally, note that for generalisations of type ‘other’ no additional inference constraints are imposed.
2.5. Ambiguous inference
370

Situations may arise in practice in which both deduction and abduction can be performed with thesame causal generalisation or abstraction; the inference type is, therefore, ambiguous.
13

Example 16. Consider generalisation g1 : fire →c smoke. Suppose fire and smoke are not observed
but have been previously inferred, for instance via deduction using generalisations g2 : see fire →efire and g3 : see smoke →e smoke, where see fire and see smoke are provided as evidence. Then both375

deduction and abduction can be performed with g1 to infer smoke from fire and fire from smoke. 
Generally, we do not wish to prohibit this type of ambiguous inference patterns as we do notconsider them to be undesirable.

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
380

by domain experts, namely Wigmore charts [40] and mind maps [24,36]. 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

385

and claims. Wigmore charts were introduced by John Henry Wigmore [40] and were applied andfurther developed by Anderson, Schum, Twining and others (e.g. [2,19]), 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 [19]. In theircharts, each symbol represents a unique claim. As noted by Kadane and Schum [19, p. 88], arcs be-390

tween nodes in the chart indicate inferences between corresponding claims, where the generalisations
used in performing these inferences, as well as the inference type (deduction or abduction), are notexplicitly 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 theinferences, the evidence in the chart also needs to be considered.

395

Example 17. An example of a modern Wigmore chart, adapted from Kadane and Schum [19, pp. 330–
331], is depicted in Figure 1, which also serves as our running example. The Wigmore chart concerns
part 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

400

their arrest, Sacco and Vanzetti were armed. According to the two arresting officers, Connolly andSpear, Sacco and Vanzetti made suspicious hand movements, from which the prosecution concluded
14

PROSECUTION 149. Following his arrest, Sacco attempted to put

his hand under his overcoat.
Connolly’s testimony to 149.
Spear’s testimony to 149.
Sacco intended to draw his concealed weapon.
Sacco intended to use his weapon on
the arresting officers.
154. Sacco intended to escape from his arrest.
155. Sacco was conscious of having committeda criminal act.
155a. Sacco was conscious of having been involvedin a robbery and shooting.
156. Sacco was conscious of having been involved in therobbery and shooting that took place in South Braintree.
Π3. It was Sacco who, with the assistence of Vanzetti,
intentionally fired shots that took the life of Berardelli during
the robbery and shooting that took place in South Braintree.

Π3

150.
151.
152.
153.

156

155a

155

469

154

470

DEFENCE 461. Sacco’s testimony to denying 149.

153
462

152

463

149

150

151

465
464

466

462. Sacco carried a weapon because heintended to shoot rabbits with it.
463. Sacco’s testimony to 462.
464. Sacco’s wife’s testimony to 462.
465. Sacco carried a weapon because of his duties
as a night watchman.
466. Sacco’s testimony to 465.
469. Sacco believed he was being arrested becauseof his political beliefs.
470. Sacco’s testimony to 469.

461

Figure 1. Wigmore chart concerning Sacco’s consciousness of guilt, along with the corresponding key list, adaptedfrom Kadane and Schum [19, pp. 330–331].

that they intended to draw their concealed weapons in order to escape their arrest. This suggests thatthey were conscious of having committed a criminal act.
On the right-hand side of Figure 1 the corresponding key list is depicted, which indicates for every
405

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 theevidence 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
410

in the chart. Instead, in their analysis of the case some of the used generalisations are indicatedin the text (see e.g. [19, pp. 97–98]). For instance, generalisations used in the inferences from thetestimonies 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.’ [19, p. 88]. As noted by Kadane and Schum[19, pp. 74–76], in constructing their charts abduction is in some instances performed to generate

415interim hypotheses between the evidence and the ultimate claim Π3 . However, Kadane and Schum15

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 [4] informalising such Wigmore charts as Pollock-style arguments [28]. For instance, multiple interpreta420

tions of the conflicts between the defence’s claims 462 and 465 and the prosecution’s claims 152 and153 are possible. One possible interpretation is that 462 and 465 indicate support for the negationof 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 beconsidered competing alternative explanations of 152, and hence be interpreted as exceptions to the425

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 [24], which is an
example of a tool typically used by domain experts, for instance in crime analysis [36]. A mind map

430

usually takes the shape of a diagram in which hypotheses and claims are represented by boxes andunderlined text, and undirected edges symbolise relations between these hypotheses and claims. An
example is depicted in Figure 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, indicated435

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.

440

Example 18. An example of a partially filled in mind map is depicted in Figure 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 to445these claims, it is indicated that claims Testimony 1 and Testimony 2 support claims Hammer andStone, respectively. Hammer and Stone are connected via undirected edges to Hit angular, whichindicates 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.
16Fell on table

Testimony 3

Hit angular
Head wound

Who

Who

What

What

When

Testimony 1

HammerTestimony 2

Stone

In which way

In which way

Autopsy

Where
Hit angular

MurderExampleCase

Accident

Head wound

Autopsy

When
Where
With

With

Why

Why

Figure 2. Example of a partially filled in mind map.

As an answer to the ‘In which way’ question, it is indicated that the person died because of a head
450

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 thatthe person died because of a head wound and that this claim is supported by Autopsy; however, in

455

contrast to the answer to this question for the high-level hypothesis ‘Murder’, it is indicated thatthe head wound was caused because the person fell on a table by accident ( Fell on table), a claim
supported by further testimony ( 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 this460

with certainty would require directly consulting the domain experts involved in constructing thechart. 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 deductive inference using the evidential generalisation g1 : Autopsy →e Head wound, and then abductively inferred465

Hit angular using the causal generalisation g2 : Hit angular →c Head wound. In turn, Hammer andStone 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 explanationsof Hit angular and are subsequently grounded in evidence, namely via deductive inference fromthe testimonies using evidential generalisations g5 : Testimony 1 →e Hammer and g6 : Testimony

470

2 →e Stone. An alternative interpretation is that the mind map was constructed iteratively fromthe evidence, where from the testimonies the claims Hammer and Stone are inferred via deductive
inference 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.
17

475

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 competingalternative explanations are not explicitly indicated in the graph.

4. The information graph formalism

480

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,

485

we wish to provide a precise account of the interplay between the different types of inferences andgeneralisations that formalises and disambiguates these analyses in a manner that makes the usedgeneralisations 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 ananalysis performed using a tool, an interpretation step may be required; we provide examples of this490

interpretation step by discussing possible formalisations of the Wigmore chart of Section 3.1 andthe mind map of Section 3.2. In Section 4.2, we define how deductive and abductive inferences canbe 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 also495

considered, as well as generalisations including enabling conditions, where constraints are imposedon 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.

500

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 isa 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.
18

505

For IGs, there is a one-to-one correspondence between nodes and propositions, generalisationarcs 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 usedinterchangeably. We write p = −q in case p = ¬q or q = ¬p. Finally, note that while we currently

510

only consider classical negation, our IG-formalism may be extended in future work to allow for moregeneral notions of conflicts such as contrariness (cf. [21]).
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,

515

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 andGe are defeasible, Ga divides into disjoint subsets Gas and Gad of strict and defeasible abstraction
arcs, respectively, and Go divides into disjoint subsets Gos and God of strict and defeasible other types

520

of generalisation arcs, respectively. For g ∈ G, Tails(g) divides into disjoint subsets Enabler(g) andAnt(g) of propositions representing enabling conditions and actual antecedents of the generalisation,
respectively, where for g ∈ Gc ∪ Ge it holds that Ant(g) 6= ∅ and possibly Enabler(g) = ∅, and forg ∈ 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 are525

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 enabling conditions. Abstractions and other types of generalisations caneither be strict or defeasible. A causal generalisation g : c → e may have an evidential counterpart of530

the form g 0 : 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 0 : 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 this is considered desirable. However, it should
be noted that g and g 0 represent the same knowledge, and that care should be taken in for instance535

modelling exceptions to generalisations (see Definition 5), as an exception to g can also be considered
an exception to g 0 . Ultimately, it is the responsibility of the knowledge engineer in consultation withthe domain expert to decide which knowledge to include in the IG and to ensure this knowledge is19

correctly and consistently represented.
In the following example, the Wigmore chart of Section 3.1 is modelled as an IG.
540

Example 19. In Figure 3, an IG is depicted for a possible interpretation of the Wigmore chartof Figure 1. This interpretation is based on a previous interpretation of this Wigmore chart as apreliminary version of an IG in which only causal and evidential information is considered and theroles 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 [19, p. 88], the generalisations
Π3

PROSECUTION 149. Following his arrest, Sacco attempted to put

g9 c

his hand under his overcoat.
Connolly’s testimony to 149.
Spear’s testimony to 149.
Sacco intended to draw his concealed weapon.
Sacco intended to use his weapon on
the arresting officers.
154. Sacco intended to escape from his arrest.
155. Sacco was conscious of having committeda criminal act.
155a. Sacco was conscious of having been involvedin a robbery and shooting.
156. Sacco was conscious of having been involved in therobbery and shooting that took place in South Braintree.
Π3. It was Sacco who, with the assistence of Vanzetti,
intentionally fired shots that took the life of Berardelli during
the robbery and shooting that took place in South Braintree.
150.
151.
152.
153.

156

g8 a
155a

¬155a

g7 a

g13

155

469

g6 e

e g12

154

470

g5 e
153
465

g4 e

g11 e

152

466

g3 e

DEFENCE 461. Sacco’s testimony to denying 149.
e g10

463

149

g1
150

462. Sacco carried a weapon because heintended to shoot rabbits with it.
463. Sacco’s testimony to 462.
464. Sacco’s wife’s testimony to 462.
465. Sacco carried a weapon because of his duties
as a night watchman.
466. Sacco’s testimony to 465.
469. Sacco believed he was being arrested becauseof his political beliefs.
470. Sacco’s testimony to 469.

462

464
¬149
e g2

e

151

461

Figure 3. An IG corresponding to a possible interpretation of the Wigmore chart of Figure 1, where ‘e’ labels denoteevidential generalisations, ‘c’ labels denote causal generalisations, ‘a’ labels denote abstractions, ! is a negation arc
andis an exception arc.

¬153

153

g14
462
e

463

g15
465

g10

g11 e

464

466

Figure 4. Adjustment to part of the IG of Figure 3, where 462 and 465 indicate support for ¬153.
20

545

used in the inferences from the testimonies are evidential. As propositions 150, 151, 463, 464, 466 and470 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

550¬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 [5, p. 64], in which someone’s psychologicalstate causes them to form certain goals, which in turn lead to actions that have consequences. In this555

case, Sacco intended to escape from his arrest (154; goal) as he was conscious of having committed acriminal act (155; psychological state); therefore, we consider 155 a cause of 154. Sacco’s intention touse 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 toattempt to put his hand under his overcoat (149; action); therefore, we consider 152 a cause of 149.

560

The IG therefore includes generalisation arcs g3 : 149 → 152; g4 : 152 → 153; g5 : 153 → 154 andg6 : 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,565

as it may also imply that the person under consideration is the victim. Proposition 155a can beconsidered 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;
more specifically, committing a specific robbery and shooting typically causes a person (in this case570

Sacco) to be conscious of having been involved in this act. Therefore, generalisation g7 : 155a → 155
is included in Gad , g8 : 156 → 155a in Gas , 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 consciousof 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 include575

g13 : 469 → ¬155a in God .

In the following example, the mind map of Section 3.2 is modelled as an IG.
Example 20. Consider Figure 5, which depicts an IG for a possible interpretation of the mind map of21

g5

head_wound
g
g6 c
c 7

e

autopsy

hit_angular
g3
g4
a
a
hammer
g1 e

stone
e g2

tes1

tes2

fell_on_table
e g8

tes3

Figure 5. IG corresponding to a possible interpretation of the mind map of Figure 2.

Figure 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 .
580

The relation between hammer ( stone) and hit angular is neither causal nor evidential; instead,generalisation arcs g3 and g4 are included in Gad 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 bothbe considered causes of head wound; therefore, generalisation arcs g6 and g7 are included in Gc . 

585

The following example illustrates generalisation arcs including enabling conditions.
Example 21. Consider g70 : {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 forhead wound and hence, fell on table is included in Ant(g70 ). Proposition no helmet does not express590

a cause for head wound and can thus be considered an enabler of g70 ; therefore, no helmet is included
in Enabler(g70 ). It should be noted that, while no helmet does not express a cause for the consequent,
it still is a necessary condition of generalisation g70 .

Specific configurations of generalisation arcs express that two propositions are alternative explanations of a common proposition, as captured by Definition 3. The terminology used is illustrated
595

in Figure 6.
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 0 in G, iff one of the following holds:
1. g ∈ Ge , Head(g) = p1 , q ∈ Ant(g), and either:1a) g 0 ∈ Ge , g 0 6= g, Head(g 0 ) = p2 , q ∈ Ant(g 0 ), or;

6001b) g 0 ∈ Gc , Head(g 0 ) = q, p2 ∈ Ant(g 0 ).
22

p1
g e
...

e

q
(1a)

p1
e g
q

...

p2
g'

g'
...

...

c
p2

...

q
g
...

c

g

c g'

p1

(1b)

p2
e g'
q

p2

...

...

c
p1

q
g
...

(2b)

(2a)

a

a

p1

g'

p2

...

(3)

Figure 6. Illustration of the terminology used in Definition 3.

2. g ∈ Gc , Head(g) = q, p1 ∈ Ant(g), and either:2a) g 0 ∈ Gc , g 0 6= g, Head(g 0 ) = q, p2 ∈ Ant(g 0 ), or;2b) g 0 ∈ Ge , Head(g 0 ) = p2 , q ∈ Ant(g 0 ).
3. g ∈ Ga , Head(g) = q, p1 ∈ Tails(g) and g 0 ∈ Ga , g 0 6= g, Head(g 0 ) = q, p2 ∈ Tails(g 0 ).
605

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 0 ∈ Ge ; hence, both
p1 and p2 are actual causes of q. Assuming that g and g 0 both have multiple actual antecedents incase 1a, then p1 and p2 are alternative explanations of every proposition q ∈ Ant(g) ∩ Ant(g 0 ).

610

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 0 ∈ Gc and thus a cause of q, andq 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 0 ∈ Gc , respectively; hence, both p1 and p2 are actual

615

causes of q. Finally, in case 3, p1 and p2 are antecedents of abstractions g ∈ Ga and g 0 ∈ Ga withthe same consequent q, and hence are considered alternative explanations of q.
Example 22. Consider the IG of Figure 5. According to condition 2a of Definition 3, hit angular andfell 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

620

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 (Negation arc). Let GI = (P, A) be an IG. A negation arc n ∈ N ⊆ A is a bidirectionalarc 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 Figure625

3, negation arc n1 : 149 ! ¬149 is also included in the graph. Similarly, the IG of Figure 3 includes
negation arc n2 : 155a ! ¬155a. As noted in Section 3.1, one possible interpretation of the conflicts
23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 Figure 4. As these generalisations are defeasible and neither causal nor evidential nor an
630

abstraction, g14 and g15 are included in God . 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.
635

Definition 5 (Exception arc). Let GI = (P, A) be an IG. An exception arc x ∈ X ⊆ A is a hyperarc
x: pg, where p ∈ P is called an exception to defeasible generalisation g ∈ Gc ∪ Ge ∪ Gad ∪ God .
An exception arc directed from p to g indicates that p provides exceptional circumstances underwhich g may not hold.
Example 24. Consider the running example. Instead of interpreting the conflicts between propositions
640

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 competingalternative 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 Figure 3, these
exceptions are indicated by curved hyperarcs x1 : 462645

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 casethis generalisation is used in deductive inference, or that the tails together can be inferred from thehead in case of abductive inference. Only when considering the available evidence can directionality

650

of inference actually be read from the graph.
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.

655

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 toestablish (different) inferences from the same IG.
24

Example 25. In the Wigmore chart of Figure 1, the evidence consists of the testimonies.
In Figures 7 and 8, the IGs of Figures 3 and 4 are again depicted, with nodes in E =660{150, 151, 461, 463, 464, 466, 470} shaded.



We now define when we consider configurations of generalisation arcs and evidence to express
deductive and abductive inference.
4.2.1. Deductive inferenceFirst, we specify under which conditions we consider a configuration of generalisation arcs and665

evidence to express deductive inference, where strict and defeasible deduction are distinguished.
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;670

2. pi is deductively inferred from propositions r1 , . . . , rm ∈ P using a generalisation g 0 : {r1 , . . . , rm }→ pi , where g 0 ∈/ Gc if g ∈ Ge , pi ∈/ Enabler(g), or;
3. pi is abductively inferred from a proposition r ∈ P using a generalisation g 0 : {pi , r1 , . . . , rm } → r
in Gc ∪ Ga , g 6= g 0 , r1 , . . . , rm ∈ P (see Definition 8).
Here, proposition q is defeasibly deductively inferred from p1 , . . . , pn , denoted p1 , . . . , pn g q, iff675

g ∈ Gc ∪ Ge ∪ Gad ∪ God , and proposition q is strictly deductively inferred from p1 , . . . , pn , denoted
p1 , . . . , pn *g q, iff g ∈ Gas ∪ Gos .
For ease of reference, symbols  and * are annotated with the name of the generalisation usedin 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

680

only be performed using strict abstractions and strict other types of generalisations. The conditionq ∈/ E ensures that deduction cannot be performed with a generalisation to infer its consequentin case its consequent is already observed. Deduction can only be performed using a generalisationg ∈ G to infer its consequent Head(g) from its antecedents Tails(g) in case every antecedentpi ∈ Tails(g) has been affirmed in that either pi is observed (i.e. pi ∈ E), pi itself is deductively

685

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 ∈ Pusing a generalisation g ∈ Ge if at least one of its actual antecedents pi ∈ Ant(g) is deductively
inferred using a generalisation g 0 ∈ Gc . In this case, q and propositions ri ∈ Ant(g 0 ) are considered25

Π3

PROSECUTION 149. Following his arrest, Sacco attempted to put

g9 c

his hand under his overcoat.
Connolly’s testimony to 149.
Spear’s testimony to 149.
Sacco intended to draw his concealed weapon.
Sacco intended to use his weapon on
the arresting officers.
154. Sacco intended to escape from his arrest.
155. Sacco was conscious of having committeda criminal act.
155a. Sacco was conscious of having been involvedin a robbery and shooting.
156. Sacco was conscious of having been involved in therobbery and shooting that took place in South Braintree.
Π3. It was Sacco who, with the assistence of Vanzetti,
intentionally fired shots that took the life of Berardelli during
the robbery and shooting that took place in South Braintree.
150.
151.
152.
153.

156

g8 a
155a

¬155a

g a
7

g13

155

469

g6 e

e g12

154

470

g5 e
153

DEFENCE 461. Sacco’s testimony to denying 149.

465

g4 e

g11 e

e g10

152

466

g3 e

463

149

g1
150

462. Sacco carried a weapon because heintended to shoot rabbits with it.
463. Sacco’s testimony to 462.
464. Sacco’s wife’s testimony to 462.
465. Sacco carried a weapon because of his duties
as a night watchman.
466. Sacco’s testimony to 465.
469. Sacco believed he was being arrested becauseof his political beliefs.
470. Sacco’s testimony to 469.

462

464
¬149
e g2

e

151

461

Figure 7. The IG of Figure 3, where evidence set E (shaded) and resulting inference steps () are also indicated.

¬153

153

g14
462
e

463

g15
465

g10

g11 e

464

466

Figure 8. The IG of Figure 4, where evidence set E (shaded) and resulting inference steps () are also indicated.

alternative explanations of pi as indicated by g and g 0 (Definition 3, case 1b or case 2b). Condition690

3 of Definition 7 is explained in Section 4.2.3, after abduction is defined.
Example 26. In the IG of Figure 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, condition1). Proposition 152 is then defeasibly deductively inferred from 149 using g3 , as 149 is deductively

695

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 isdefeasibly deductively inferred using g13 , as 469 is deductively inferred.
26

The following example illustrates strict deductive inference.
Example 27. Consider Example 2 from Section 2.1. In this example, generalisation arc g :700lung cancer → cancer is included in Gas . As lung cancer ∈ E, cancer is strictly deductively inferredfrom 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 Figure 9b, q cannot be deductively inferred from p1 and p2 using705

g1 , as p2 ∈/ E and p2 is neither deductively nor abductively inferred.
In Figure 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 isdeductively inferred using g1 ∈ Gc .

710

4.2.2. Abductive inference
Next, we specify under which conditions we consider a configuration of generalisation arcs andevidence to express abductive inference.
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

715

inferred from q using a g : {p1 , . . . , pn } → q in Gc ∪ Ga , denoted q g p1 ; . . . ; q g pn , iff:
1. q ∈ E, or;
2. q is deductively inferred from propositions r1 , . . . , rm ∈ P using a generalisation g 0 : {r1 , . . . ,rm } → q in G, g 6= g 0 (see Definition 7), where g 0 ∈/ Gc if g ∈ Gc and g 0 ∈/ Ga if g ∈ Ga , or;
3. q is abductively inferred from a proposition r ∈ P using a generalisation g 0 : {q, r1 , . . . , rm } → r

720

in Gc ∪ Ga , r1 , . . . , rm ∈ P.

q

q
g1 e
p
(a)

g1

c

p2

p1
(b)

fire
e g2
smoke
c g1
smoke_machine(c)Figure 9. Examples of IGs illustrating the restrictions put on performing deduction within our IG-formalism (a-c).
27

In accordance with our assumptions stated in Section 2.2, abduction is defeasible and is modelledusing only causal generalisations and abstractions. Following Console and Dupré [13] and Bex [5],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 abstract725

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 caseat 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 consequentHead(g) in case Head(g) has been affirmed in that either Head(g) is observed (i.e. Head(g) ∈ E),

730

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 0 6= g, g 0 ∈ Gc .
In enforcing this constraint, we do not need to consider whether or not the antecedents of g or g 0

735

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) 6= ∅; therefore, at least one proposition pi is an actual
antecedent of g and at least one proposition rj is an actual antecedent of g 0 , which are then alternative explanations of q according to case 2a of Definition 3 which may not be inferred from eachother by inferring q as an intermediary step. Similarly, we assume in condition 2 that g 0 ∈/ Ga if740

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 0 ) as indicated by g and g 0 according to case 3 of Definition 3.
Example 29. In the IG of Figure 7, given E proposition 155a is abductively inferred from 155 using
g7 ∈ Gad , as 155 is deductively inferred (Definition 8, condition 2). In turn, propositions 156 and745

Π3 are iteratively abductively inferred using generalisations g8 ∈ Gas and g9 ∈ Gc , respectively. Notethat although g8 is a strict abstraction, the abductive inference from 155a to 156 is defeasible and notstrict; 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 robberyand shooting that took place in South Braintree.

750

In the IG of Figure 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.
28



r
c

q
g1

c

c

p1

colon_cancer
a g'2

g3

r1

g2

cancerc g'3
smokingp2

lung_cancer
g'1
a

(b)

(a)

Figure 10. Example of an IG illustrating abduction with causal generalisations (a); example of an IG illustrating
abduction with abstractions (b).

Example 30. In Figure 10b, Example 15 from Section 2.4.2 is modelled as an IG. As smoking ∈ E,
755

cancer is deductively inferred from smoking using g30 . Propositions lung cancer and colon cancer arethen abductively inferred from cancer using strict abstractions g10 and g20 , 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 orcolon cancer). As noted in Section 2.4.2, this type of mixed inference using a causal generalisation

760

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.
Example 31. In Figure 11a, Example 9 is modelled as an IG. As smoke machine ∈ E, smoke is765

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 Figure 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 anddeadly weapon is deductively inferred using g1 ∈ Ga (Definition 8, condition 2).

smoke
g2 c
c g1
fire
smoke_machine(a)deadly_weapon
g1
g2 a
a
knife
gun

¬heat

heat
g2



smoke
c
c g1
fire

(b)

(c)

Figure 11. An IG illustrating Pearl’s constraint for mixed deductive-abductive inference (a); an IG illustrating ourinference constraints for abstractions (b); an IG illustrating mixed abductive-deductive inference (c).
29

g5head_wound
g
g6
c
c 7

e

autopsy

hit_angular
g3
g4
a
a
hammer
g1 e

stone
e g2

tes1

tes2

fell_on_table
e g8

tes3

Figure 12. The IG of Figure 5, where evidence set E (shaded) and resulting inference steps () are also indicated.

770

The following example describes the inferences that can be made based on the IG of Figure 5
corresponding to the mind map example of Section 3.2.
Example 32. Consider the IG of Figure 12. Given E = {tes1 , tes2 , tes3 , autopsy}, head woundis 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 wound775

is also deductively inferred from fell on table using g7 , as fell on table is deductively inferred fromtes3 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

780

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 Figure 11c, Example 7 from Section 2.4 is modelled as an IG. From smoke, fire is785

abductively inferred using g1 , as smoke ∈ E. Then heat is deductively inferred (or predicted) fromfire using g2 (Definition 7, condition 3).

30

5. An argumentation formalism based on information graphs
Based on our IG-formalism from Section 4, we now define an argumentation formalism that allows forboth deductive and abductive argumentation. Note that the IG-formalism is not an argumentation
790

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 arguments
constructed on the basis of IGs. More specifically, our approach generates an abstract argumentation
framework as in Dung [14], that is, a set of arguments with a binary attack relation, which thus allows

795

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 [10] 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 [10] are avoided. Our argumentation formalism extends a preliminary

800

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, satisfactionof 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 and805

8 starting with elements from E. We then formally prove that arguments constructed on the basisof IGs conform to our inference constraints (Section 2.4). In Section 5.2, we define several types ofattacks between arguments based on IGs, which are based on the different types of conflicts definedfor our IG-formalism. In Section 5.3 we instantiate Dung’s abstract approach with arguments andattacks based on IGs and provide the definitions of Dung’s argumentation semantics. In Section 5.4,

810

we then prove that rationality postulates [10] 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[21]. By remaining close to the ASPIC+ framework, this allows us to straightforwardly show that815

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 toconstruct 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 usedin constructing the argument, TopGen returns the last generalisation used in constructing the
31

820

argument, DefInf and StInf return all the defeasible and strict inferences used in constructing theargument, 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 (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
825

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 g p if A1 , . . . , An are arguments such that p is defeasibly deductively inferred830

from Conc(A1 ), . . . , Conc(An ) using a generalisation g : {Conc(A1 ), . . . , Conc(An )} → paccording to Definition 7, where it holds that g ∈ Gc ∪ Ge ∪ Gad ∪ God and if g is of theform g : c → e in Gc and its evidential counterpart g 0 : e → c is included in Ge , theng0 ∈/ Gen(A1 ) ∪ . . . ∪ Gen(An ). For A, it holds that:
Prem(A) = Prem(A1 ) ∪ . . . ∪ Prem(An ); Conc(A) = p;

835

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 ) g p};
StInf(A) = StInf(A1 ) ∪ . . . StInf(An );TopInf(A) = Conc(A1 ), . . . , Conc(An ) g p.

840

2b. A1 , . . . , An *g p if A1 , . . . , An are arguments such that p is strictly deductively inferredfrom Conc(A1 ), . . . , Conc(An ) using a generalisation g ∈ Gas ∪ Gos , 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 );845

StInf(A) = StInf(A1 ) ∪ . . . StInf(An ) ∪ {Conc(A1 ), . . . , Conc(An ) *g p};
TopInf(A) = Conc(A1 ), . . . , Conc(An ) *g p.
3. A0 g p if A0 is an argument such that p is abductively inferred from Conc(A0 ) using a generalisation g ∈ Gc ∪ Ga , g : {p, p1 , . . . , pn } → Conc(A0 ) for some propositions p1 , . . . , pn ∈ P
according to Definition 8, where:

850

Prem(A) = Prem(A0 ); Conc(A) = p; Sub(A) = Sub(A0 ) ∪ {A}; ImmSub(A) = {A0 };Gen(A) = Gen(A0 ) ∪ {g}; TopGen(A) = g; DefInf(A) = DefInf(A0 ) ∪ {Conc(A0 ) g p};
StInf(A) = StInf(A0 ); TopInf(A) = Conc(A0 ) g p.
32

Note that we overload symbols  and * to denote an argument while it also denotes a defeasibleor strict inference. The set of all arguments on the basis of GI and E is denoted by A.
855

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

860

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 0 : e → c(see Sections 2.3 and 4.1), then the restriction in step 2a that g 0 ∈/ Gen(A1 ) ∪ . . . ∪ Gen(An ) ensuresthat cycles in which c and e are iteratively deductively inferred from each other using g 0 and g are
avoided. Note that cycles in which c and e are iteratively deductively inferred from each other using865

g and g 0 in that order are already avoided due to the enforcement of Pearl’s constraint in condition
2 of Definition 7.
Example 34. Consider Figure 13, in which arguments constructed on the basis of the IG of Figure 7
are indicated. According to step 1 of Definition 9, A1 : 150 and A2 : 151 are premise arguments. Basedon A1 and A2 , defeasible deductive argument A3 : A1 , A2 g1 149 is constructed by step 2a of Defini-870

tion 9, as 149 is defeasibly deductively inferred from 150 and 151 using g1 ∈ Ge . Arguments A4 : A3

g3 152; A5 : A4 g4 153; A6 : A5 g5 154 and A7 : A6 g6 155 similarly are defeasible deductive
arguments. Argument A8 : A7 g7 155a is a defeasible mixed argument by step 3 of Definition 9, as
155a is abductively inferred from 155 using g7 . Similarly, arguments A9 : A8 g8 156 and A10 : A9

g9 Π3 are defeasible mixed arguments. To illustrate the operators used in Definition 9, for A8 , we875

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g1 149; 149g3 152; 152g4 153; 153g5 154; 154g6 155; 155g7 155a}; StInf(A8 ) = ∅;TopInf(A8 ) = 155g7 155a.



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

880

Example 35. On the basis of the IG of Figure 10a and E = {r}, A01 : r is a premise argument. From
A01 , arguments A02 : A01 g3 r1 and A03 : A01 g3 q are constructed by step 3 of Definition 9, as q
and r1 are abductively inferred from Conc(A01 ) using causal generalisation g3 : {q, r1 } → r. Thenagain by step 3, A04 : A03 g1 p1 and A05 : A03 g2 p2 are constructed using g1 and g2 , respectively.

33

A10

A7
A6

C1

465

A4

g11 e

155a

¬155a

g7 a

g13

155

469

g6 e

e g12

154

470

B5
B4

B3

g5 e

DEFENCE 461. Sacco’s testimony to denying 149.

153

g4 e

466

A3
g1
150

e

463

g10

C3

149

464
¬149

e

151

A2

462. Sacco carried a weapon because heintended to shoot rabbits with it.
463. Sacco’s testimony to 462.
464. Sacco’s wife’s testimony to 462.
465. Sacco carried a weapon because of his duties
as a night watchman.
466. Sacco’s testimony to 465.
469. Sacco believed he was being arrested becauseof his political beliefs.
470. Sacco’s testimony to 469.

C5

462

152

g3 e

A1

150.
151.
152.
153.

156

g8 a

A8

C2

his hand under his overcoat.
Connolly’s testimony to 149.
Spear’s testimony to 149.
Sacco intended to draw his concealed weapon.
Sacco intended to use his weapon on
the arresting officers.
154. Sacco intended to escape from his arrest.
155. Sacco was conscious of having committeda criminal act.
155a. Sacco was conscious of having been involvedin a robbery and shooting.
156. Sacco was conscious of having been involved in therobbery and shooting that took place in South Braintree.
Π3. It was Sacco who, with the assistence of Vanzetti,
intentionally fired shots that took the life of Berardelli during
the robbery and shooting that took place in South Braintree.

g9 c

A9

A5

PROSECUTION 149. Following his arrest, Sacco attempted to put

Π3

e g2

461

C4

B2
B1

Figure 13. Annotation of the IG of Figure 7, where arguments and direct attacks (99K) on the basis of the IG and E
are also indicated.

5.1.1. Properties of arguments based on IGs885

We now prove a number of formal properties of arguments based on IGs. Lemma 1 states thatthe 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.

890

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 according895

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 analternative cause for this effect was already previously inferred (see Section 2.4.1). Similarly, careshould be taken that no version of an event at a lower level of abstraction is inferred if an alternative34

version of this event at a lower level of abstraction was already previously inferred (see Section 2.4.2).
900

In the context of IGs, for g ∈ Gc , propositions in Ant(g) express causes for the common effectexpressed 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 suchthat our inference constraints (see Section 2.4) are adhered to. We now formally prove that these905

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 (Inference constraint). Let A be a set of arguments on the basis of IG GI = (P, A) andevidence set E. Let p1 , p2 ∈ P be alternative explanations of q ∈ P as indicated by generalisations910

g1 and g2 in G (see 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)and Conc(C) = p2 .
We now formally prove that this inference constraint is indeed adhered to.
Proposition 1 (Adherence to inference constraint). Let A be a set of arguments on the basis of IG915

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 withB ∈ ImmSub(C) and Conc(C) = p2 . In constructing argument B, either step 2a, step 2b or step 3920

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 antecedentq 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 generalisationsin Gc ∪ Ga . More specifically, argument B cannot be constructed by applying step 2a, 2b and 3 of925

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 .
35

930

• First, consider case 2a of Definition 3 in which g2 6= 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 inthis 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

935

constructing C using g2 . Furthermore, neither step 2a nor step 2b of Definition 9 can be appliedin 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 ). Thenno argument C with B ∈ ImmSub(C), Conc(C) = p2 can be constructed using g2 , as in thiscase deductive inference would be performed with g2 to infer p2 while per the restrictions in

940

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 beapplied 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 ∈/ Gas ∪ Gos ,and step 3 cannot be applied in constructing C using g2 , as this step specifies an abductive

945

and not a deductive inference.
• Finally, consider case 3 of Definition 3 in which g2 6= 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 inthis 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

950

deductively inferred using g1 ∈ Ga . In particular, step 3 of Definition 9 cannot be applied inconstructing C using g2 . Furthermore, neither step 2a nor step 2b of Definition 9 can be appliedin constructing C using g2 , as these steps specify deductive and not abductive inferences. 

5.2. Attack

In this section, several types of attacks between arguments on the basis of IGs are defined. Among955

the types of attacks that are typically distinguished in structured argumentation (for instance in
ASPIC+ [21]) are rebuttal, undermining, and undercutting attack. Of these types of attacks, weonly consider rebuttal and undercutting attack and not undermining attacks (i.e. attack on anargument’s premises [21]), as in IGs we assume that all premises are certain and cannot be attacked(cf. ASPIC+ ’s axiom premises). We also distinguish a fourth type of attack, namely alternative960

attack, a concept based on the notion of competing alternative explanations (see Section 2.2) thatis inspired by [3,6]. In our argumentation formalism, attacks directly follow from the constructed36

arguments and the specified exception arcs in an IG. Hence, attacks between arguments do not need
to be separately specified by the user.
First, we define the general notion of attack, after which the different types of attacks are defined.
965

Definition 11 (Attack). Let A be a set of arguments on the basis of IG GI and evidence set E. Let
A, B ∈ A. Then A attacks B iff A rebuts B, A undercuts B, or A alternative attacks B, as definedin Definitions 12, 13 and 14, respectively.
5.2.1. Rebuttal attack
First, rebuttal attack is defined. Informally, a rebuttal is an attack on the conclusion of an argument970

for which it holds that the last inference used in constructing the argument is defeasible.
Definition 12 (Rebuttal attack). Let A be a set of arguments on the basis of IG GI = (P, A) andevidence set E. Let A, B, B 0 ∈ A with B 0 ∈ Sub(B). Then A rebuts B (on B 0 ) iff there exists a
negation arc n : Conc(A) ! Conc(B 0 ) in N and B 0 is of the form B100 , . . . , Bn00 g p for some
B100 , . . . , Bn00 ∈ A, p ∈ P.

975

Note that, as it is assumed that B 0 is of the form B100 , . . . , Bn00 g p (i.e. TopInf(B 0 ) is defeasible),it holds that B 0 is a deductive, abductive, or mixed argument; hence, by Lemma 1, Conc(B 0 ) ∈/ E.
Furthermore, while a negation arc expresses a symmetric conflict, our definition of rebuttal attack
allows for both symmetric or asymmetric rebuttal, as illustrated by the following example.
Example 36. Consider the IG of Figure 13. Let A1 , A2 , A3 be the arguments introduced in Exam-980

ple 34. Let B1 : 461 and let B2 : B1 g2 ¬149. Then B2 rebuts A3 (on A3 ) and A3 rebuts B2(on B2 ), as Conc(A3 ) = 149, Conc(B2 ) = ¬149 (and hence n : 149 ! ¬149 in N), where
TopInf(A3 ) = 150, 151g1 149 and TopInf(B2 ) = 461g2 ¬149 are defeasible. This symmetric
rebuttal is indicated in Figure 13 by means of a bidirectional dashed arc between these propositions.
Similarly, let A8 be as introduced in Example 34, and let B3 : 470; B4 : B3 g12 469; B5 : B4 g13985

¬155a. Then A8 rebuts B5 (on B5 ) and B5 rebuts A8 (on A8 ).
Consider again Example 33, in which heat is predicted from fire. Assume that contrary to thisprediction we observe that there is no heat (¬heat ∈ E). Let A001 : smoke; A002 : A001 g1 fire; A003 : A002

g2 heat; B100 : ¬heat. Then B100 rebuts A003 (on A003 ), but A003 does not rebut B100 as B100 is not of theform B1000 , . . . , Bn000 g p for some B1000 , . . . , Bn000 ∈ A, p ∈ P (i.e. B100 is a premise argument).
990



5.2.2. Undercutting attack
Next, undercutting attack is considered. Informally, an undercutter attacks a defeasible inference
by providing exceptional circumstances under which the inference may not be applicable. In our
37argumentation formalism based on IGs, undercutting attacks between arguments follow from thespecified exception arcs in GI . Specifically, as an exception arc directed from p ∈ P to g ∈ Gc ∪ Ge ∪995

Gad ∪God specifies an exception to defeasible generalisation g, an argument A ∈ A with Conc(A) = p
undercuts an argument B ∈ A with g ∈ Gen(B).
Definition 13 (Undercutting attack). Let A be a set of arguments on the basis of IG GI = (P, A)
and evidence set E. Let A, B, B 0 ∈ A with B 0 ∈ Sub(B). Then A undercuts B (on B 0 ) iff there exists
an exception arc x ∈ X such that x : Conc(A)1000

g and TopGen(B 0 ) = g ∈ Gc ∪ Ge ∪ Gad ∪ God .

Undercutting attack is illustrated by the following example.
Example 37. Consider the IG of Figure 13. Let A1 , A2 , A3 , A4 , A5 be the arguments introduced in
Example 34. Let C1 : 466; C2 : C1 g11 465. Then C2 undercuts A5 (on A5 ), as x : 465

g4 in

X and TopGen(A5 ) = g4 ∈ Ge . This direct attack is indicated in Figure 13 by means of a dashedarc directed from 465 to defeasible inference 152 g4 153. As undercutting attack is defined on
1005

subarguments, C2 also attacks Ai for i ≥ 6. Similarly, let C3 : 463; C4 : 464; C5 : C3 , C4 g10 462.
Then C5 undercuts A5 (on A5 ), as x : 462

g4 in X and TopGen(A5 ) = g4 . Argument C5 thenalso attacks Ai for i ≥ 6.



5.2.3. Alternative attack
Lastly, alternative attack is defined. Arguments are involved in alternative attack iff their abductively
1010

inferred conclusions are competing alternative explanations (see Section 2.2).
Definition 14 (Alternative attack). Let A be a set of arguments on the basis of IG GI = (P, A) andevidence set E. Let p1 , p2 ∈ P be alternative explanations of q ∈ P as indicated by generalisations
g and g 0 in G, where either g, g 0 ∈ Gc (Definition 3, case 2a) or g, g 0 ∈ Ga (Definition 3, case3). Let A, B, B 0 ∈ A with B 0 ∈ Sub(B). Then A alternative attacks B (on B 0 ) iff there exists

1015

an argument C ∈ ImmSub(A) ∩ ImmSub(B 0 ) such that Conc(A) = p1 and Conc(B 0 ) = p2 are
abductively inferred from Conc(C) = q using generalisations g and g 0 , respectively.
Note that A only alternative attacks B on B 0 iff TopInf(B 0 ) is an abductive inference andhence iff the last used inference in constructing B 0 is defeasible. Furthermore, unlike direct rebuttal
attack, which can either be symmetric or asymmetric, direct alternative attack is always symmetric1020

in that A alternative attacks B on B iff B alternative attacks A on A.
Under the conditions set out in Definition 14, arguments Ai : C g pi for pi ∈ Ant(g) constructed from C via abductive inference using g are involved in alternative attack with A0j : C g0 p0j
for p0j ∈ Ant(g 0 ) constructed from C via abductive inference using g 0 . We do not consider argu38

ments Ai : C g pi for pi ∈ Enabler(g) to be in competition with arguments A0j : C g0 p0j for1025

p0j ∈ Enabler(g 0 ), as enablers of causal generalisations do not express alternative causes for theconsequent. Arguments Ai (as well as A0j ) are not involved in alternative attack among themselves,in accordance with our assumption that the antecedents of a causal generalisation or abstraction
are not in competition. Finally, in case g ∈ Gc and g 0 ∈ Ga , then arguments Ai are not involved
in alternative attack with A0j , as the actual antecedents of g express causes for the effect expressed1030

by the consequent but the tails of g 0 are not alternative explanations of the consequent; instead,propositions in Tails(g 0 ) are at a lower level of abstraction than Head(g 0 ).
Example 38. Consider the IG of Figure 12. Given E, arguments D1 : autopsy; D2 : D1 g5head wound; D3 : D2 g6 hit angular; D4 : D2 g7 fell on table; D5 : D3 g3 hammer; andD6 : D3 g4 stone are constructed. Here, hit angular and fell on table are abductively inferred from1035

head wound using g6 and g7 , respectively, and hammer and stone are abductively inferred from
hit angular using g3 and g4 , respectively. Then D3 alternative attacks D4 (on D4 ) and D4 alternative attacks D3 (on D3 ), as Conc(D3 ) = hit angular and Conc(D4 ) = fell on table are alternative
explanations of Conc(D2 ) = head wound as indicated by g6 and g7 in Gc (Definition 3, case 2a).
As D3 ∈ Sub(D5 ) and D3 ∈ Sub(D6 ), D4 also alternative attacks D5 and D6 (on D3 ). Finally, D51040

alternative attacks D6 (on D6 ) and D6 alternative attacks D5 (on D5 ), as Conc(D5 ) = hammer
and Conc(D6 ) = stone are alternative explanations of Conc(D3 ) = hit angular as indicated by g3
and g4 in Ga (Definition 3, case 3).
Consider Example 12 from Section 2.4.1. Assume that in addition to generalisations g1 andg2 , evidential generalisation g3 : see fire → fire is provided. Given E = {see fire}, arguments E1 :1045

see fire; E2 : E1 g3 fire; E3 : E2 g1 torch; E4 : E2 g2 match; and E5 : E2 g2 oxygen are constructed. Then E3 and E4 are involved in alternative attack, as Conc(E3 ) = torch and Conc(E4 ) =match are alternative explanations of Conc(E2 ) = fire as indicated by g1 and g2 in Gc (Definition
3, case 2a), where torch and match are abductively inferred from fire using g1 and g2 , respectively.
E3 is not involved in alternative attack with E5 , as Conc(E5 ) = oxygen ∈ Enabler(g2 ).

1050

Consider Figure 10a. Let A01 , A02 , A03 be as defined in Example 35. Then A02 and A03 are not
involved in alternative attack, as r1 = Conc(A02 ) and q = Conc(A03 ) are abductively inferred fromr = Conc(A01 ) using the same generalisation g3 ; specifically, in case 2a of Definition 3 it is assumed
that g 6= g 0 , and hence r1 and q are not alternative explanations of r by that definition.

Finally, note that a causal generalisation g1 : c1 → e may be replaced by an evidential gener1055

alisation g10 : e → c1 if c1 is the usual cause of e, in which case abduction with g1 can be encoded
as deduction with g10 (see Section 2.3). Considering the case in which only g1 and not g10 is in39

cluded in IG GI and additional causal generalisation g2 : c2 → e is provided, then arguments A1 : e,
A2 : A1 g1 c1 , A3 : A1 g2 c2 are constructed upon observing e, where A2 and A3 are involved inalternative attack according to Definition 14. However, in case only g10 and g2 are included in GI and1060

not g1 , then arguments A1 : e, A02 : A1 g10 c1 , A3 : A1 g2 c2 are constructed, where A02 and A3 are
not involved in alternative attack as g10 ∈ Ge . Hence, if the knowledge engineer considers c1 and c2
to be competing alternative explanations of e, then the involved generalisations should be modelled
as causal generalisations in order to achieve alternative attack among constructed arguments. Alternatively, A3 can be interpreted as an undercutter of A02 as it provides an exception to the performed

1065

inference (see also [6, p. 15]). We reiterate that it is the responsibility of the knowledge engineer in
consultation with the domain expert to decide which knowledge (including conflicts) to represent in
an IG and to ensure this knowledge is modelled correctly (see also Section 4.1).
5.3. Argument evaluation
In this section, we provide Dung’s definitions for argumentation semantics [14] and illustrate these1070

definitions for our running example.
First, we instantiate Dung’s abstract approach with arguments and attacks based on IGs.
Definition 15 (Argumentation framework). Let GI = (P, A) be an IG, and let E ⊆ P be an evidence
set. An argumentation framework (AF) defined by GI and E is a pair (A, C ), where A is the setof all arguments on the basis of GI and E as defined by Definition 9, and where (A, B) ∈ C iff

1075

A, B ∈ A and A attacks B (see Definition 11).
An AF can be represented as a directed graph in which arguments are represented by circles
and attacks are indicated by solid arcs (→); an example of an AF is depicted in Figure 14.
Given an AF, we can use any semantics for AFs as defined in [14] for determining the dialecticalstatus of arguments (cf. [21]). The theory of AFs is built around the notion of an extension, which1080

is a set of arguments that is internally coherent and defends itself against attack.
Definition 16 (Dung extensions). Let (A, C ) be an AF defined by IG GI and evidence set E.
• A set of arguments S ⊆ A is conflict-free if there do not exist A, B ∈ S such that (A, B) ∈ C .
• An argument A ∈ A is acceptable with respect to some set of arguments S ⊆ A iff for allarguments B such that (B, A) ∈ C there exists an argument C ∈ S such that (C, B) ∈ C .

1085• A conflict-free set of arguments S ⊆ A is an admissible extension iff every argument A ∈ S is
acceptable with respect to S .
• An admissible extension S is a complete extension iff A ∈ S whenever A is acceptable withrespect to S ; S is the grounded extension iff S is the set inclusion minimal complete extension;40

S is a preferred extension iff S is a set inclusion maximal complete extension; and S is a stable
1090

extension iff it is preferred and ∀B ∈/ S , ∃A ∈ S such that (A, B) ∈ C .
The acceptability of arguments in abstract argumentation frameworks can then be evaluatedby establishing whether a given argument is a member of the various extensions. Arguments arethen assigned a dialectical status that can either be ‘justified’, ‘overruled’, or ‘defensible’, where
informally an argument is justified if it survived the competition, overruled if it did not survive the1095

competition, and defensible if it is involved in a tie.
Definition 17 (Justified, overruled and defensible arguments, adapted from [30]). Let (A, C ) be anargumentation framework.
• An argument is (i) justified under grounded semantics iff it is a member of the grounded extension,(ii) overruled under grounded semantics iff it is not justified under grounded semantics and it

1100

is attacked by an argument that is justified under grounded semantics, or (iii) defensible undergrounded semantics iff it is neither justified nor overruled under grounded semantics.
• Let T ∈ {complete, preferred, stable}. An argument is (i) justified under T semantics iff it is amember of all T extensions, (ii) overruled under T semantics iff it is not a member of any T
extension, or (iii) defensible under T semantics iff it is a member of some but not all T extensions.

1105

We now illustrate the evaluation of arguments based on IGs through our running example.
Example 39. Consider the IG of Figure 13. To prevent this example from becoming too involved, we
consider the following subset of arguments A = {A1 , A2 , A3 , A4 , A5 , B1 , B2 , C1 , C2 , C3 , C4 , C5 } andbinary attack relation C = {(A3 , B2 ), (B2 , A3 ), (B2 , A4 ), (B2 , A5 ), (C2 , A5 ), (C5 , A5 )} over A (seeExamples 34, 36 and 37). These arguments and attacks are depicted in Figure 14. The complete

1110

extensions of (A, C ) are:

A1

A2

A3

B2

B1

C1

A4

C3

C4

C2

A5

C5

Figure 14. AF corresponding to the IG of Figure 13.

41

S1 = {A1 , A2 , B1 , C1 , C2 , C3 , C4 , C5 };S2 = {A1 , A2 , B1 , B2 , C1 , C2 , C3 , C4 , C5 }S3 = {A1 , A2 , A3 , A4 , B1 , C1 , C2 , C3 , C4 , C5 }.
1115

Under complete semantics, A1 , A2 , B1 , C1 , C2 , C3 , C3 , C4 , C5 are justified as they are members of allcomplete extensions, A5 is overruled as it is attacked by a justified argument, and A3 , A4 and B2 are
defensible. For the other semantics, the same statuses are assigned; for grounded semantics, this is
the case as S1 is the set inclusion minimal complete extension. Furthermore, note that S2 and S31120

are set inclusion maximal complete extensions for which it holds that ∀B ∈/ Si , ∃A ∈ Si such that
(A, B) ∈ C for i = 2, 3; hence, S2 and S3 are preferred and stable extensions.

Dung’s abstract argumentation approach has been extended with new elements, for instance byadding support relations to abstract argumentation frameworks (e.g. [11]) or by adding preference
relations (e.g. so-called preference-based argumentation frameworks, or PAFs [1]), probabilities (see1125e.g. [16] for an overview), or weights [15] to AFs; a more complete overview is provided in [32]. We
opt for the approach introduced by Dung for the evaluation of arguments as it is a well-studied
and widely accepted approach in the field of computational argumentation. Moreover, the relations
between Dung’s fully abstract approach and formalisms for structured argumentation that are at anintermediate level of abstraction between concrete instantiating logics and Dung’s approach, such as

1130

ASPIC+ [21] and assumption-based argumentation (ABA) [7], have been previously investigated. In
our IG-formalism, we have currently opted not to account for preferences, as these are typically notindicated in tools domain experts use. As the components of our argumentation formalism based on
IGs are directly defined based on the elements that are accounted for in our IG-formalism, preferences
are currently not accounted for in our argumentation formalism. As shown in work on structured

1135

argumentation with preferences (e.g. [21]), the structure of arguments is crucial in determining how
preferences must be applied to attacks and one should be cautious in extending AFs with additional
elements without taking the structure of arguments into account. There is some work on the relations
between support relations in abstract argumentation frameworks and those at the inference level[31]. Relations between our proposed argumentation formalism and extended AFs such as [11] may

1140

be investigated in future research.
5.4. Satisfying rationality postulates
Caminada and Amgoud [10] studied rule-based argumentation systems and identified conditions
under which unintuitive and undesirable results are obtained upon performing inference. They thendefined principles, called rationality postulates, that can be used to judge the quality of a given rule42

1145

based argumentation system. More specifically, so-called consistency and closure postulates were
formulated for systems allowing for strict and defeasible inferences. Since these postulates are widely
accepted as important desiderata for structured argumentation formalisms, we prove in this sectionthat these postulates are satisfied by instantiations of our argumentation formalism based on IGs.

5.4.1. Comparison of our argumentation formalism based on IGs to the ASPIC+ framework
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In proving satisfaction of [10]’s rationality postulates, we follow Modgil and Prakken [21], who proved
satisfaction of these postulates for the ASPIC+ framework. As noted earlier, in defining our argumentation formalism based on IGs we were inspired by the definitions of argument and attack asgiven in [21]. In Definition 9, we defined how arguments on the basis of an IG and an evidence set E
are constructed. In step 2a of Definition 9, it is specified that an argument A with Conc(A) = p can

1155

be constructed from arguments A1 , . . . , An if p is defeasibly deductively inferred from Conc(A1 ),. . . , Conc(An ) according to Definition 7 using a generalisation g : {Conc(A1 ), . . . , Conc(An )} → p
in Gc ∪ Ge ∪ Gad ∪ God . Hence, in terms of the terminology used in the ASPIC+ framework, generalisations in Gc ∪ Ge ∪ Gad ∪ God can be interpreted as domain-specific defeasible inference rules 3 in
ASPIC+ ’s Rd that are applied when constructing arguments. Similarly, in step 2b of Definition 9 it1160

is specified that an argument A with Conc(A) = p can be constructed from A1 , . . . , An if p is strictlydeductively inferred from Conc(A1 ), . . . , Conc(An ) according to Definition 7 using a generalisationg : {Conc(A1 ), . . . , Conc(An )} → p in Gas ∪ Gos . Hence, generalisations in Gas ∪ Gos can be interpreted as domain-specific strict inference rules in ASPIC+ ’s Rs . Finally in step 3 it is specified thatan argument A with Conc(A) = p can be constructed from an argument A0 if p is abductively in-1165

ferred from Conc(A0 ) according to Definition 8 using a g ∈ Gc ∪Ga , g : {p, p1 , . . . , pn } → Conc(A0 )for some propositions p1 , . . . , pn ∈ P. Therefore, besides specifying aforementioned domain-specificdefeasible and strict deduction rules, generalisations g : {q1 , . . . , qn } → q in Gc ∪ Ga also specify
domain-specific abduction rules in ASPIC+ ’s Rd , namely for every i ∈ {1, . . . , n} a rule can be
specified in Rd that states that qi can be defeasibly inferred from q.

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Considering the different types of attacks that are defined in Section 5.2, rebuttal as defined in
Section 5.2.1 is identical to rebuttal as defined for a special case of ASPIC+ , namely one in whichconflict is based on the standard classical notion of negation. Undercutting as defined in Section
5.2.2 is a special case of undercutting as defined for ASPIC+ , as we only consider undercutters ofinferences in case an exception is provided to a defeasible generalisation used in an inference step.

3 For further details on using ASPIC+ to model domain-specific defeasible and strict inference rules, the reader isreferred to [22].
43

1175

Thus, of the types of attacks that are considered in our argumentation formalism, only alternative
attack is not accounted for in ASPIC+ . Furthermore, in comparison to our argumentation formalism,
Modgil and Prakken do not impose any additional restrictions on argument construction. Hence, toprove that instantiations of our argumentation formalism based on IGs satisfy rationality postulates,
in Section 5.4.3 we focus on showing how alternative attack and the additional restrictions that are1180

imposed on argument construction in our argumentation formalism can be taken account in the
results and proofs provided in [21].
5.4.2. Additional definitions and assumptions
Following Modgil and Prakken [21], we introduce the following definitions. We define what it means
for a set of propositions to be closed under strict generalisations.

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Definition 18 (Closure under strict generalisations). Let GI = (P, A) be an IG and let P0 ⊆ P.
Then the closure of P0 under strict generalisations, denoted Cl(P0 ), is the smallest set containing
P0 and the consequent Head(g) of any g ∈ Gas ∪ Gos whose antecedents Tails(g) are in Cl(P0 ).
Next, the terms directly consistent and indirectly consistent set are defined.
Definition 19 (Directly consistent set). Let GI = (P, A) be an IG and let P0 ⊆ P. Then P0 is1190

directly consistent iff @p, q ∈ P0 such that p = −q.
A set P0 is indirectly consistent if its closure under strict generalisations is directly consistent.
Definition 20 (Indirectly consistent set). Let GI = (P, A) be an IG and let P0 ⊆ P. Then P0 is
indirectly consistent iff Cl(P0 ) is directly consistent.
As noted by Caminada and Amgoud [10], one should search for ways to alter or constrain one’s

1195

argumentation formalism in such a way that rationality postulates are satisfied. Accordingly, following Modgil and Prakken [21] we assume that IGs and evidence sets satisfy a number of properties.
Similar to ASPIC+ , we leave the user free to make choices as to the strict and defeasible generalisations to include in G ⊆ A and the observations to include in E; however, some care needs to betaken in making these choices to ensure that the result of argumentation is guaranteed to be well-

1200

behaved. Specifically, to ensure rationality postulates are satisfied, we assume that evidence sets E
are indirectly consistent (referred to as the axiom consistency assumption), and we assume that G isclosed under transposition. Note that per definition every evidence set E ⊆ P is a directly consistentset, as it is assumed in Definition 6 that for every p ∈ E, ¬p ∈/ E. Furthermore, all examples of IGs
provided in this paper are axiom consistent, as they do not include generalisations g ∈ Gas ∪ Gos
44

1205

for which Tails(g) ⊆ E. Closure under transposition is one of the solutions proposed by Caminada
and Amgoud to ‘repair’ an argumentation system to ensure rationality postulates are satisfied [10,p. 16], as it can help generate rules needed to obtain an intuitive outcome.
Definition 21 (Closure under transposition). Let GI = (P, A) be an IG. A strict generalisation g 0 ∈ Gas ∪ Gos is a transposition of g : {p1 , . . . , pn } → p in Gas ∪ Gos iff g 0 is of the form

1210

{p1 , . . . , pi−1 , −p, pi+1 , . . . , pn } → −pi for some 1 ≤ i ≤ n. We say that G is closed under transposition iff for all strict generalisations g ∈ Gas ∪ Gos , the transpositions of g are also in Gas ∪ Gos .
An AF (A, C ) defined by an IG GI that is axiom consistent and for which G ⊆ A is closed
under transposition is said to be well defined. In the remainder of this section, we assume that any
given AF (A, C ) is well defined. Note that most examples of IGs provided in this paper only include1215

defeasible generalisations and not strict generalisations, and thus that AFs defined by these IGs arewell defined. The following example, adapted from Caminada and Amgoud [10], illustrates closure
under transposition and how ensuring it can help repair an argumentation system.
Example 40. In the IG depicted in Figure 15a, strict abstractions g2 : bachelor → ¬has wife and g4 :married → has wife are included. G is not closed under transposition, as generalisations has wife1220

→ ¬bachelor and ¬has wife → ¬married are not included. Arguments A5 and A6 constructed on thebasis of this IG have strict top inferences, as only step 2b of Definition 9 can be applied in constructing A5 from A3 and A6 from A4 using g2 and g4 in Gas , respectively. Note that, as TopInf(A5 ) and
TopInf(A6 ) are strict, A5 and A6 are not involved in rebuttal. In fact, C = ∅ for the AF corresponding to this IG, and hence under any semantics both A5 and A6 are justified. Thus, contradictory

1225

propositions has wife and ¬has wife are both justified at the same time, which is clearly undesirable
and among other things violates the direct consistency postulate (see Theorem 1). In the IG depicted

A7

A8¬bachelorg'2 aA6¬married
A6

A5

A3
A1

¬has_wife

has_wife

g2 a

g4 a

bachelor

marriedg1 ego_outg3
A2
wedding_ring

A4

A5 g'4 a¬has_wife
A3
A1

(a)

has_wife

g2 a

g4 a

bachelor

marriedg1 ego_outg3
A2
wedding_ring

A4

(b)

Figure 15. Example of an IG for which G is not closed under transposition (a); adjustment to this IG, in which
additional generalisations are included such that G is closed under transposition (b).
45

in Figure 15b, G is closed under transposition as additional generalisations has wife → ¬bachelorand ¬has wife → ¬married are now included. In the corresponding AF, A7 directly rebuts A4 andA8 directly rebuts A3 as TopInf(A3 ) and TopInf(A4 ) are defeasible. Then A7 indirectly rebuts A61230(on A4 ) and A8 indirectly rebuts A5 (on A3 ). Therefore, for this AF the more intuitive outcome isobtained that A5 and A6 cannot both be in the same extension at the same time.

Lastly, the following definitions introduce some terminology used in the below results. Following
Modgil and Prakken [23], we define strict continuations in a slightly different way than in [21], but
as noted by [23] this does not affect the proofs stated in [21].
1235Definition 22 (Strict continuations). Let (A, C ) be an AF defined by IG GI and evidence set E. The
set of strict continuations of a set of arguments from A is the smallest set satisfying the following
conditions:
1. Any argument A is a strict continuation of {A}.
2. If A1 , . . . , An are arguments and S1 , . . . , Sn are sets of arguments such that for every i ∈1240{1, . . . , n}, Ai is a strict continuation of Si and {Bn+1 , . . . , Bm } is a (possibly empty) set ofstrict arguments, and g : {Conc(A1 ), . . . , Conc(An ), Conc(Bn+1 ), . . . , Conc(Bm )} → p is astrict generalisation in Gas ∪ Gos , then argument A1 , . . . , An , Bn+1 , . . . , Bm *g p constructedfrom A1 , . . . , An , Bn+1 , . . . , Bm using g by applying step 2b of Definition 9 is a strict continuation of S1 ∪ . . . ∪ Sn .

1245

The maximal fallible sub-arguments of an argument B are those with the ‘last’ defeasible inferences in B. That is, they are the maximal sub-arguments of B on which B can be attacked.
Definition 23 (Maximal fallible sub-arguments). Let (A, C ) be an AF defined by IG GI and evidence
set E. The set M (B) of the maximal fallible sub-arguments of B is defined such that for any B 0 ∈Sub(B), it holds that B 0 ∈ M (B) iff:1250

1. TopInf(B 0 ) is defeasible, and;2. There is no B 00 ∈ Sub(B) such that B 00 6= B, B 0 ∈ Sub(B 00 ) and B 00 satisfies condition 1.
5.4.3. Proofs
We prove satisfaction of Caminada and Amgoud’s consistency and closure postulates for completesemantics, which implies satisfaction of these postulates for grounded, preferred, and stable seman-1255

tics. Caminada and Amgoud [10] also propose postulates for the intersection of extensions and theirconclusion sets, but since their satisfaction directly follows from satisfaction of the postulates forindividual extensions, these postulates will not be reconsidered.
46

First, a number of intermediate properties are proven. The intermediate result stated in Lemma
2 is identical to Lemma 37 of Modgil and Prakken [21], namely that any strict continuation B of a1260

set of arguments {A1 , . . . , An } is acceptable with respect to S if all Ai are acceptable with respect
to a set S . The proof follows similar to Lemma 37 of Modgil and Prakken [21], where alternative
attack is now also considered.
Lemma 2. Let (A, C ) be an AF defined by IG GI and evidence set E. Let B ∈ A be a strict
continuation of {A1 , . . . , An }, and for i = 1, . . . , n, let Ai be acceptable with respect to S ⊆ A. Then1265

B is acceptable with respect to S .
Proof. Let A be any argument such that (A, B) ∈ C . By Definition 11, A attacks B iff A rebuts
B (on B 0 ), A undercuts B (on B 0 ), or A alternative attacks B (on B 0 ) for some B 0 ∈ Sub(B) (seeDefinitions 12, 13, and 14). Here, it holds that TopInf(B 0 ) is defeasible; more specifically:
1. By Definition 12, A rebuts B (on B 0 ) iff B 0 is of the form B100 , . . . , Bn00 g p for some

1270

B100 , . . . , Bn00 ∈ A, p ∈ P and hence iff TopInf(B 0 ) is defeasible, and;2. By Definition 13, A undercuts B (on B 0 ) iff there exists an exception arc x ∈ X such thatx : Conc(A)g and TopGen(B 0 ) = g ∈ Gc ∪ Ge ∪ Gad ∪ God . Hence, in constructing B 0

step 2b cannot be applied last, as this step can only be applied with strict generalisationsg ∈ Gas ∪ Gos . Therefore, step 2a of step 3 of Definition 9 is applied last in constructing
1275

B 0 . Thus, the last used inference in constructing B 0 is a defeasible deductive inference using
TopGen(B 0 ) = g (step 2a of Definition 9) or an abductive inference using TopGen(B 0 ) = g(step 3 of Definition 9), and hence TopInf(B 0 ) is defeasible, and;3. By Definition 14, A alternative attacks B (on B 0 ) iff TopInf(B 0 ) is an abductive inference
and hence iff TopInf(B 0 ) is defeasible.

1280

Hence, by definition of strict continuations (Definition 22), it must be that (A, B) ∈ C iff (A, Ai ) ∈ C
for some (possibly more than one) Ai ∈ {A1 , . . . , An }. Specifically, if A does not undercut, rebutor alternative attack some Ai , then this contradicts that (A, B) ∈ C . Thus, we have shown that if(A, B) ∈ C , then (A, Ai ) ∈ C for some Ai ∈ {A1 , . . . , An }. By assumption, Ai is acceptable withrespect to S , thus ∃C ∈ S such that (C, A) ∈ C . Thus, B is acceptable with respect to S .

1285



The intermediate result stated in Lemma 3 is similar to Proposition 8 of Modgil and Prakken[21]. Compared to Proposition 8 of [21], in which no assumptions are made regarding A, we now
assume that A is defeasible with a strict top inference or that A is strict, as these are the only
cases needed in our proof of Theorem 1. As Modgil and Prakken do not impose any restrictions
on argument construction in their formalism, a result proven by Caminada and Amgoud [10] (i.e.
47

1290

Lemma 6 of [10]) can be directly used to complete their proof. Below, we show that the restrictions
that are imposed on argument construction in our argumentation formalism based on IGs do not
restrict the construction of strict continuations, and hence that the proof can similarly be completed.
Lemma 3. Let (A, C ) be an AF defined by IG GI and evidence set E. Let A, B ∈ A such that B
is defeasible, Conc(A) = −Conc(B). Let A be strict or let A be defeasible with TopInf(A) strict.

1295

Then for all B 0 ∈ M (B), there exists a strict continuation A+ of (M (B) \ {B 0 }) ∪ {A} such that
A+ rebuts B on B 0 .
Proof. Let A be strict or let A be defeasible with TopInf(A) strict. Let B be defeasible with
Conc(A) = −Conc(B). First, note that according to Definition 22 any strict continuation of a given
set of arguments from A is either (1) A if the set of arguments under consideration is {A} (Definition

1300

22, condition 1), or (2) is constructed by applying step 2b of Definition 9 one or more (but finitely
many) times (Definition 22, condition 2). As restrictions are imposed on argument construction in
our argumentation formalism based on IGs, we first show that in constructing any strict continuation
A+ of (M (B) \ {B 0 }) ∪ {A} step 2b of Definition 9 can be applied without restrictions.
Generally, in applying step 2b of Definition 9 an argument C with Conc(C) = p is1305

constructed from arguments C1 , . . . , Cn by strictly deductively inferring p from propositions
Conc(C1 ), . . . , Conc(Cn ) according to Definition 7 using a generalisation g : Conc(C1 ), . . . ,Conc(Cn ) → p in Gas ∪Gos . In Definition 7 no constraints are imposed on performing deduction withstrict generalisations g ∈ Gas ∪ Gos ; in particular, the only constraint that is imposed is in condition
2 of this definition, where constraints are imposed on performing deduction with defeasible generali-1310

sations in Ge (i.e. Pearl’s constraint). The only other case in which step 2b of Definition 9 cannot beapplied in constructing an argument C using a g ∈ Gas ∪ Gos is in case the same g was already usedin the previous construction step to construct an argument C 0 ∈ ImmSub(C), namely by applying
step 3 of Definition 9. Now again consider argument A. By assumption, A is strict or TopInf(A)strict, and therefore step 3 of Definition 9, which specifies a defeasible inference, could not have1315

been applied last in constructing A; therefore, no restrictions are imposed on constructing strict
continuations A+ of (M (B) \ {B 0 }) ∪ {A} in our argumentation formalism. By assumption, (A, C ) is
well defined and, therefore, closed under transposition; hence, by straightforward generalisation ofLemma 6 in [10] one can construct a strict continuation A+ that continues (M (B) \ {B 0 }) ∪ {A} withstrict inferences and that concludes −Conc(B 0 ). Since by construction of M (B), B 0 has a defeasible1320

top inference and therefore A+ rebuts B 0 . But then A+ also rebuts B.
The intermediate result stated in Lemma 4 is identical to Lemma 38 of [21].

48Lemma 4. Let (A, C ) be an AF defined by IG GI and evidence set E. Let A ∈ A be acceptable withrespect to admissible extension S ⊆ A. Let S 0 = S ∪ {A}. Then ∀B ∈ S 0 , neither (A, B) ∈ C nor(B, A) ∈ C .
1325

Proof. Suppose for contradiction that: (1) ∃B ∈ S 0 such that (A, B) ∈ C . As B ∈ S 0 , it follows
that B is acceptable with respect to S , as either B = A, which is acceptable with respect to S byassumption, or B is an element of admissible extension S . Hence, ∃C ∈ S such that (C, A) ∈ C .
Then, as A is acceptable with respect to S , ∃D ∈ S such that (D, C) ∈ C , contradicting S isconflict-free; (2) ∃B ∈ S 0 such that (B, A) ∈ C . As A is acceptable with respect to S , ∃C ∈ S such

1330

that (C, B) ∈ C , contradicting S is conflict-free.

The result stated in Lemma 5 is identical to Lemma 35-2 of Modgil and Prakken [21], namelythat an argument A attacks an argument B iff A attacks some sub-argument B 0 of B. Compared toLemma 35-2 of [21], alternative attack is now also considered in the proof.
Lemma 5. Let (A, C ) be an AF defined by IG GI and evidence set E. Let A, B ∈ A. Then (A, B) ∈ C
1335

iff (A, B 0 ) ∈ C for some B 0 ∈ Sub(B).
Proof. By Definition 11, (A, B) ∈ C iff A rebuts B (on B 0 ), A undercuts B (on B 0 ), or A
alternative attacks B (on B 0 ) for some B 0 ∈ Sub(B) (see Definitions 12, 13, and 14); hence, also
(A, B 0 ) ∈ C .



The intermediate result stated in Lemma 6 is identical to Proposition 10 of [21].
1340

Lemma 6. Let (A, C ) be an AF defined by IG GI and evidence set E. Let A ∈ A be acceptable withrespect to admissible extension S ⊆ A. Then S 0 = S ∪ {A} is conflict-free.
Proof. We need to show that there do not exist B, C ∈ S 0 such that (B, C) ∈ C . As S is anadmissible extension, S is conflict free: hence, there do not exist B, C ∈ S such that (B, C) ∈ C .
Thus, we need to show that (A, A) ∈/ C , and neither (A, B) ∈ C nor (B, A) ∈ C for all B ∈ S . As by1345

assumption A is acceptable with respect to S , this follows directly from Lemma 4.

Theorem 1, corresponding to the direct consistency postulate, states that the conclusions ofarguments in an admissible extension (and so by implication in a complete extension) are directly
consistent. The conclusions of arguments in an extension should not be contradictory, as this leads
to what Caminada and Amgoud call ‘absurdities’ [10, p. 15] in that two contradictory statements
1350

can then be justified at the same time.
Theorem 1 (Direct consistency). Let (A, C ) be an AF defined by IG GI and evidence set E. Thenfor all admissible extensions S of AF it holds that the set {Conc(A) | A ∈ S } is directly consistent.
49

Proof. Let S be an admissible extension of AF and let A, B ∈ S . We show that if Conc(A) = q,
Conc(B) = r with q = −r (i.e. {Conc(A) | A ∈ S } is not directly consistent), then this leads to a
1355

contradiction:
1. If A is a strict argument, and:
1.1 if B is also a strict argument, then this contradicts our axiom consistency assumption onevidence sets E;
1.2 if B is a defeasible argument, and:1360

1.2.1 if B has a defeasible top inference, then A rebuts B (on B) by Definition 12, as a negation arc n : Conc(A) ! Conc(B) exists in N (as q = −r). Hence, this contradicts

S is conflict-free.
1.2.2 if B has a strict top inference, then by Lemma 3 there exists a strict continuation A+of (M (B) \ {B 0 }) ∪ {A} for every B 0 ∈ M (B) such that A+ rebuts B on B 0 ; hence,
1365(A+ , B) ∈ C . By our Lemma 2, A+ is acceptable with respect to S , and by Lemma
6, S ∪ {A+ } is conflict-free, contradicting that (A+ , B) ∈ C .
2. If A is a defeasible argument and B is a strict argument, then the result follows similar to case
1.2 with the roles of arguments A and B reversed.
3. If A and B are defeasible arguments, and:1370

3.1 if TopInf(A) or TopInf(B) is defeasible, then the result follows similar to case 1.2.1 (eitherwith the roles of arguments A and B as they currently are or with their roles reversed).
3.2 if TopInf(A) and TopInf(B) are strict, then the result follows similar to case 1.2.2.

The result stated in Lemma 7 is identical to Lemma 35-3 of [21].
Lemma 7. Let (A, C ) be an AF defined by IG GI and evidence set E. Let S ⊆ A and let A ∈ S with
1375

A0 ∈ Sub(A). Then A0 is acceptable with respect to S if A is acceptable with respect to S .
Proof. Assume that A is acceptable with respect to S . We need to prove that for every argument
B such that (B, A0 ) ∈ C , ∃C ∈ S such that (C, B) ∈ C . Let B ∈ A and assume that (B, A0 ) ∈ C . By
Lemma 5, (B, A) ∈ C . Then, as A is acceptable with respect to S , ∃C ∈ S such that (C, B) ∈ C .
Hence, A0 is acceptable with respect to S .

1380



Below, Caminada and Amgoud’s [10] closure and indirect consistency postulates are stated.
Informally, the closure postulates state that the conclusions returned by an argumentation systemshould be ‘complete’ [10, p. 16]. The sub-argument closure postulate states that for any argument
A in a complete extension S , all sub-arguments of A are also in S .
Theorem 2 (Sub-argument closure). Let (A, C ) be an AF defined by IG GI and evidence set E. Then
50

1385

for all complete extensions S of AF it holds that if an argument A is in S then all sub-arguments
A0 ∈ Sub(A) of A are in S .
Proof. Let S be a complete extension of AF, let A ∈ S and let A0 ∈ Sub(A). Then A0 is
acceptable with respect to S by Lemma 7. Then S ∪ {A0 } is conflict-free by Lemma 6. Hence, since

S is complete, it holds that A0 ∈ S .
1390



Theorem 3, corresponding to the strict closure postulate, states that the conclusions of arguments
in a complete extension are closed under strict inference.
Theorem 3 (Closure under strict inferences). Let (A, C ) be an AF defined by IG GI and evidence
set E. Let S be a complete extension of AF. Then {Conc(A) | A ∈ S } = CL({Conc(A) | A ∈ S }).
Proof. It suffices to show that any strict continuation X of {A | A ∈ S } is in S . By Lemma 2,1395

any such X is acceptable w.r.t. S . By Lemma 6, S ∪ {X} is conflict-free. Hence, since S is complete,
it follows that X ∈ S .



Finally, Theorem 4, corresponding to the indirect consistency postulate, states the mutual consistency of the strict closure of conclusions of arguments in a complete extension.
Theorem 4 (Indirect consistency). Let (A, C ) be an AF defined by IG GI and evidence set E. Let S
1400

be a complete extension of AF. Then {Conc(A) | A ∈ S } is indirectly consistent.
Proof. The result follows from Theorems 1 and 3.



To conclude this section, we have shown that instantiations of our argumentation formalism
based on IGs satisfy Caminada and Amgoud’s [10] consistency and closure postulates. Satisfactionof these postulates warrants the sound definition of instantiations of our argumentation system and
1405

implies that anomalous results as identified by [10] are avoided.

6. Related work
In this paper, we have proposed an argumentation formalism based on IGs that allows for both deductive and abductive argumentation and which instantiates Dung’s [14] abstract approach. Earlier
work by Bex [5,6] is related, although only the integrated theory [6] is purely argumentation-based;1410

the relation to this work was discussed in the introduction. In the hybrid theory proposed by Bex [5],deductive and abductive inference are used in constructing evidential arguments and causal stories,
which are completely separate entities with their own definitions related to conflict and evaluation.
In comparison, our argumentation formalism based on IGs allows for the construction of both de51

ductive and abductive arguments. Bench-Capon and Prakken [3] offer a formalisation of Aristotle’s
1415

practical syllogism within a logic for defeasible argumentation. This approach allows for reasoning
about alternative goals and values to justify actions, which is of an abductive nature. In formalising
this syllogism, the authors only consider the abductive nature of reasoning about desires on the basisof beliefs and goals, whereas we offer a general account of abductive (and deductive) argumentation. Booth and colleagues [8] propose a top-down approach by developing a model of abduction in1420

abstract argumentation [14] and instantiating their approach with abductive logic programs [20]. In
comparison to our bottom-up approach, their approach does not allow for mixed abductive-deductive
inference with different types of information.
In defining our argumentation formalism based on IGs, we were inspired by ASPIC+ [21]. Our
argumentation formalism can be regarded as an adaptation of a special case of ASPIC+ , which

1425

would among other things require introducing a new form of attack, namely alternative attack, andrestricting the manner in which arguments are constructed within this framework.
The argumentation formalism presented in this paper is based on a version of the graph-based
IG-formalism that considers causal and evidential generalisations, abstractions and other types ofgeneralisations, as well as generalisations including enabling conditions. Most related formalisms for

1430

inference with these types of information are logic-based [5,6,13,18,25,29,34,35] and do not focus on
the constraints on performing inference that need to be imposed. The approaches of [18,29,34] only
allow for causal defaults and do not consider defaults including enabling conditions; complications
with reasoning using both causal and evidential defaults as identified by Pearl [26] are thus avoided.
Poole’s Theorist framework [29] allows for both deductive and abductive inference; however, only

1435

causal defaults are considered in this system. The approaches of Ortiz Jr. [25] and Shoham [35] similarly only allow for inference with causal rules, but in contrast also include enabling conditions. The
formalisms proposed by Console and Dupré [13] and Bex [5] allow for reasoning with abstractions.
Bex’s hybrid theory [5] does not allow for most types of mixed inference with causal and evidential
generalisations and abstractions, and largely avoids the problems associated with mixed inference as

1440

identified by Pearl [26] and as identified in the current paper. Console and Dupré [13] only consider
causal rules and abstractions and not evidential rules.
Graph-based formalisms for reasoning with causality information have also been proposed, notably Pearl’s causal diagrams [27]. Pearl provides a framework for causal inference in which diagrams
are queried to determine if the assumptions available are sufficient for identifying causal effects.

1445

Compared to our IG-formalism and our argumentation formalism based on IGs, this framework does
not allow for capturing asymmetric conflicts such as exceptions in the graph. Moreover, causal diagrams require probabilistic quantification to be queried, while IGs are qualitative. Pollock’s inference52graphs [28], in which arguments are represented as AND trees and sets of such trees are combined
into AND/OR graphs to which attack links are added, are also related to IGs. Compared to IGs,
1450

there is no emphasis on the distinction between the different types of generalisations and inferences.

7. Conclusion

In this paper, we have proposed an argumentation formalism that allows for both deductive andabductive argumentation, the latter of which has received relatively little attention in argumentation. Our argumentation formalism is based on an extended version of our previously proposed IG1455

formalism [39], where in addition to causal and evidential generalisations we now also allow for abstractions and other types of generalisations, thereby increasing the expressivity of our IG-formalism.
We have identified conditions under which performing inference with abstractions can lead to undesirable results, thereby extending the set of inference constraints imposed by Pearl’s C-E system
for reasoning with causal and evidential information [26]. Moreover, we have identified exceptional

1460

circumstances under which the constraints of Pearl’s C-E system should not be imposed, namely
in case enabling conditions 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 have defined how deductive and abductive inference may be performed with IGs. We have thenformally proven that arguments constructed in our argumentation formalism based on IGs indeed

1465

adhere to these constraints. In the paper, we have focused on the constraints that need to be imposed on performing inference with pairs of generalisations, which cover Pearl’s original constraints
and local constraints on performing inference with abstractions. In future work, additional inference
constraints may be imposed for longer chains of inferences involving more specific combinations ofgeneralisations, granted that the total set of constraints is consistent. Furthermore, as causality is1470

a contentious topic, our argumentation formalism may be extended in future work by allowing formeta-argumentation about labels of generalisations, as well as other elements of IGs.
Besides allowing for rebuttal and undercutting attack, which are among the types of attacks
that are typically distinguished in structured argumentation [21,28], we have also defined the notionof alternative attack among arguments based on IGs, a concept based on the notion of competing1475

alternative explanations that is inspired by [3,6]. Alternative attack captures a crucial aspect of
abductive reasoning, namely that of conflict between abductively inferred conclusions [18]. We havecontributed to the literature on computational argumentation by allowing for the formal evaluationof arguments involved in this type of conflict. Moreover, we have shown that instantiations of ourargumentation formalism satisfy key rationality postulates [10], which warrants the sound definition
53

1480

of instantiations of our argumentation system and implies that anomalous results such as issues
regarding inconsistency and non-closure as identified by [10] are avoided.
Our argumentation formalism generates an abstract AF as in Dung [14] and thus allows arguments to be formally evaluated according to Dung’s argumentation semantics. By formalising analyses performed by domain experts using the informal reasoning tools they are familiar with (e.g.

1485mind maps) as IGs as an intermediary step, this allows for the evaluation of IGs using computational
argumentation, as well as using other formal systems such as BNs [39].

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L. Amgoud and C. Cayrol, A model of reasoning based on the production of acceptable arguments. Annals of[2]

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