Deductive and Abductive Reasoning with
Causal and Evidential Information
Remi WIETEN a,1 , Floris BEX a,b , Henry PRAKKEN a,c and Silja RENOOIJ aa Information and Computing Sciences, Utrecht University, The Netherlands
b Institute for Law, Technology and Society, Tilburg University, The Netherlandsc Faculty of Law, University of Groningen, The Netherlands
Abstract. In this paper, we propose the information graph (IG) formalism, which
provides a precise account of the interplay between deductive and abductive inference and causal and evidential information. IGs formalise analyses performed bydomain experts using the informal reasoning tools they are familiar with, such as
mind maps used in crime analysis. Based on principles for reasoning with causal
and evidential information given the evidence, we impose constraints on the inferences that may be performed with IGs. Moreover, we propose an argumentation
formalism based on IGs that allows arguments to be formally evaluated.
Keywords. Deduction, abduction, causal and evidential information, argumentation

1. Introduction
In the legal and forensic domain, reasoning about evidence plays a central role in therational process of proof [1]. To aid in this process, various graph-based tools exist thatallow domain experts to make sense of a mass of evidence in a case, including mind
maps, argument diagrams and Wigmore charts [2]. Because of their informal nature,these tools typically do not directly allow for formal evaluation using AI techniques.
Hence, we wish to formalise analyses performed with such tools in a manner that allows
for formal evaluation and that adheres to principles from the literature on reasoning about
evidence [1,3,4] while allowing inference to be performed in a manner closely related to
the way in which inference is performed using such tools.
In reasoning about evidence, inference is often performed using domain-specificgeneralisations [1], also called defaults [4], which capture knowledge about the worldin conditional form. We distinguish between causal generalisations (e.g. fire typicallycauses smoke) and evidential generalisations (e.g. smoke is typically evidence for fire)[1,4]. 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 defeasibly inferred; abduction [3] can also be performed with causal generalisations, where by affirming the consequent (smoke) the antecedent (fire) is defeasiblyinferred. Pearl [4, p. 264] argued that people generally consider it difficult to express
knowledge using only causal generalisations, and in an empirical study, van den Braakand colleagues [5] found that while there are situations in which subjects consistently
choose either causal or evidential modelling techniques, there are also many examples inwhich people use both types of generalisations in their reasoning. For instance, subjects
often considered testimonies to be evidential, whereas a motive for committing an act
1 Corresponding author: g.m.wieten@uu.nl.

is considered a cause for committing that act. This illustrates that in formal accounts ofreasoning about evidence, it is important to allow for both types of generalisations [1].
When performing analyses using aforementioned tools such as mind maps, domain
experts naturally mix both causal and evidential generalisations and perform both deductive and abductive inferences, where the used generalisations and the inference type(deduction, abduction) are typically left implicit. Hence, in formalising such analyses weneed a precise account of the interplay between the different types of inferences and generalisations and the constraints on performing inference we need to impose. In this paper
we propose the information graph (IG) formalism, which provides such an account. IGs
are knowledge representations that formalise analyses performed by domain experts using the informal reasoning tools they are familiar with in a manner that makes the causal
and evidential generalisations used in performing inference explicit. Based on principles
for reasoning with causal and evidential generalisations, we then define how deduction
and abduction can be performed with IGs given a set of propositions labelled evidence.
Most existing formalisms that allow both inference types with causal and evidential information are logic-based (e.g. [1,6]); instead, we propose a graph-based formalism toremain closely related to analyses performed using aforementioned graph-based tools.
Our argumentation formalism generates an abstract argumentation framework as inDung [7], that is, a set of arguments with a binary attack relation, which thus allows arguments to be formally evaluated according to Dung’s classical semantics. Moreover, ourargumentation formalism adheres to the constraints imposed by Pearl’s C-E system [4],which say that, in performing inference, care should be taken that no cause for an effectis inferred in case an alternative cause for this effect was already previously inferred.
The paper is structured as follows. In Section 2, we provide principles for reasoning
about evidence. In Section 3, we present an example of an analysis performed using amind mapping tool, which illustrates that both deduction and abduction is performed bydomain experts, using both causal and evidential generalisations. Based on this example,
in Section 4 we motivate and define our IG-formalism. In Section 5, we then propose anargumentation formalism based on our IG-formalism. In Section 6, we discuss related
work. In Section 7, we discuss future work and conclude.
2. Reasoning about Evidence
In this section, we provide principles for and review the terminology used to describe reasoning about evidence. Inference is the process of inferring claims from theobserved evidence using generalisations [1]. We distinguish between causal and evidential generalisations [1,4]. Causal generalisations are of the form ‘c1 , . . . , cn usually/normally/typically causes e’, whereas evidential generalisations are of the form
‘e1 , . . . , en is usually/normally/typically evidence for c’. We denote generalisations as fire→ smoke, where fire is the generalisation’s antecedent and smoke its consequent. A generalisation may have multiple antecedents, in which case the generalisation expresses
that only the antecedents together allow us to infer the consequent. The notation →c and→e is used for causal and evidential generalisations, respectively.
Deductive Inference Inference can be performed in a deductive fashion, where froma causal or evidential generalisation and by affirming the antecedents, the consequent isinferred by modus ponens on the generalisation. Note that while deduction is typicallyequated with strict inference (cf. [8]) in which the consequent universally holds given
the antecedents, we use the term ‘deduction’ for defeasible ‘forward’ inference in whichthe consequent tentatively holds given the antecedents (cf. [9]). Hence, deduction is notnecessarily a stronger or more reliable form of inference than abduction.
Abductive Inference Abduction [3] can also be performed: from a causal generalisation and by affirming the consequent, the antecedents are inferred, since if the antecedents are true it would allow us to deductively infer the consequent modus-ponensstyle. In case causes c1 , . . . , cn and c01 , . . . , c0m are abductively inferred from common effect e using causal generalisations c1 , . . . , cn →c e and c01 , . . . , c0m →c e, then ci and c0j
for i ∈ {1, . . . , n}, j ∈ {1, . . . , m} are considered to be competing alternative explanations for e. We assume that causes ci (and c0j ) are not in competition among themselves.
For instance, consider the causal generalisations fire →c smoke and smoke machine →csmoke. By affirming the common consequent (smoke), fire and smoke machine are inferred, which are then competing causes for smoke.
Mixed and Ambiguous Inference Deduction and abduction can be iteratively performed, where mixed abductive-deductive inference is also possible. Suppose that fromthe causal generalisation fire →c smoke and by affirming the consequent (smoke), the antecedent (fire) is inferred. Now, if the additional causal generalisation fire →c heat is provided, then consequent heat can be deductively inferred (or predicted [9]) as antecedent
fire has been previously abductively inferred.
Mixed deduction, using both causal and evidential generalisations, can also be performed [6], but as noted by Pearl [4] deductively chaining a causal and an evidential
generalisation can lead to undesirable results. Consider the example in which a causal
generalisation smoke machine →c smoke and an evidential generalisation smoke →e fire
are provided. Deductively chaining these generalisations would make us infer there is afire 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 isinferred if an alternative cause for this effect was already previously inferred. Consider
the above example, where instead of an evidential generalisation smoke →e fire a causal
generalisation fire →c smoke is provided. Upon seeing a smoke machine, this would
make us infer there is a fire in case deduction and abduction are iteratively performed,
which is again undesirable. Accordingly, we wish to prohibit these types of inference
patterns, and refer to the constraint that no cause for an effect should be inferred if analternative cause for this effect was already previously inferred as Pearl’s constraint [4].
Finally, situations may arise in practice in which both deduction and abduction canbe performed with the same causal generalisation. For instance, consider the causal generalisation fire →c smoke and assume that both fire and smoke are affirmed but not observed, then both deduction and abduction can be performed to either infer smoke fromfire or fire from smoke, respectively. The inference type is, therefore, ambiguous.
3. Example of an Analysis Performed Using a Mind Mapping Tool
In this section, we present an example of an analysis performed using a mind mapping
tool [2], which is an example of a tool typically used by domain experts, for instancein crime analysis. Based on this example, we motivate and illustrate the design choices
for our IG-formalism in Section 4. A mind map usually takes the shape of a diagram inwhich hypotheses and claims are represented by boxes and underlined text, and undirected edges symbolise relations between these hypotheses and claims. The mind map
represents various scenario-elements and the crime analyst uses evidence to support or
oppose these elements, indicated by plus and minus symbols, respectively.

Testimony 1

In which way

Motive 1Why
Testimony 2
Testimony 4

Motive 2
Lie

When
Marjan murdered Leo

Who
Where

Testimony 3
Police report

What
With

MurderExampleCase

Figure 1. Example of a partially filled out mind map.

Example 1 An example of a partially filled out mind map is depicted in Figure 1, which
also serves as our running example. In this example, adapted from [1], the high-level
hypothesis ‘Murder’ is considered. The case concerns the murder of Leo de Jager. Leo’s
body was found on the property of Marjan van der E.; we are interested in her involvement in the murder. As a police report (Police report) indicates that Leo’s body was foundon Marjan’s property, claim Marjan murdered Leo is added as an answer to the ‘Who’question. By means of a plus symbol and an undirected edge connecting the evidence tothe claim, it is indicated that the police report supports the claim that Marjan murdered
Leo. Possible motives (Motive 1 and Motive 2) are provided as to why Marjan may havewanted to murder Leo, which are connected to the ‘Why’ question via undirected edges.
Testimony 1 and Testimony 2 support these two motives, indicated by the plus symbols
connected to these claims. In her testimony (Testimony 3), Marjan denied any involvement in the murder of Leo, which is indicated by a minus symbol. This opposes the claimthat Marjan murdered Leo. Further testimony (Testimony 4) indicates that Marjan had
reason to lie when giving her testimony (Lie). By means of a minus symbol and an undirected edge connecting Lie to Testimony 3, it is indicated that this claim weakens theinference from her testimony to the claim that she did not murder Leo.
As the edges in a mind map are undirected, it is unclear from this graphical representation alone which types of generalisations and inferences were used in constructing thismap. Establishing this with certainty would require directly consulting the domain experts involved in constructing the chart. We note, however, that the reasoning performedin constructing this mind map can be interpreted in at least two possible ways. One interpretation is that the domain expert first (preliminarily) inferred that Marjan murdered
Leo from the police report via deduction using the evidential generalisation Police report→e Marjan murdered Leo, and then abductively inferred the two possible motives using
the causal generalisations gi : Motive i →c Marjan murdered Leo; i = 1, 2. These twocauses are then competing alternative explanations as to why Marjan murdered Leo andare subsequently grounded in evidence, namely via deduction from the testimonial evidence using evidential generalisations g0j : Testimony j →e Motive j; j = 1, 2. An alternative interpretation is that the mind map was constructed iteratively from the observed
evidence, where from testimonial evidence the motives are inferred via deduction using
evidential generalisations g01 and g02 . The claim that Marjan murdered Leo is then inferred
modus-ponens style: from causal generalisations g1 and g2 and the previously inferredantecedents, the consequent is deductively inferred. In this way, the two motives are notin competition for the common effect that Marjan murdered Leo.
Lastly, note that in mind maps the exact manner in which claims and links conflict is
not precisely specified: a minus symbol can either indicate support for the opposing claim(e.g. Testimony 3 supports the negation of Marjan murdered Leo) or indicate an exception
to the performed inference step (e.g. Lie opposes the inference step from Testimony 3 tothe negation of Marjan murdered Leo).

murderg1

police

e

c
g3
mot1
g2 e
tes1

c
g5
mot2
g4 e
tes2

¬murder
g6 e

tes3

lie
g7 e
tes4

(a)

g1

policeemurder
g3
g5
c
c
mot1
g2 e

mot2
e g4

tes1

¬murder
g6 e

lie

tes3

e g7
tes4

tes2
(b)

Figure 2. An IG corresponding to a possible interpretation of the mind map of Figure 1 (a); the same IG,
where evidence set E (shaded) and resulting inference steps () are also indicated (b).

4. The Information Graph Formalism
The example from Section 3 makes it plausible that both deduction and abduction is
performed by domain experts when performing analyses using reasoning tools they arefamiliar 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 theseanalyses ambiguous to interpret. For our current purposes of providing a precise accountof the interplay between the different types of inferences and generalisations, we wish toformalise and disambiguate these analyses in a manner that makes the used generalisations explicit. Information graphs (IGs), which we define in Section 4.1, are knowledge
representations that explicitly describe causal and evidential generalisations in the graph.
In Section 4.2, we define how deductive and abductive inferences can be read from IGs,
based on the principles for reasoning about evidence discussed in Section 2.
4.1. Information Graphs
IGs are defined as follows.
Definition 1 (Information graph) An information graph (IG) is a directed graph G =(P, A), where P is a set of nodes representing propositions from a propositional literallanguage with ordinary negation symbol ¬. A = G ∪ X is a set of directed (hyper)arcs
with G∩X = 0,
/ where G and X are sets of generalisation arcs and exception arcs, definedin Definitions 2 and 3, respectively.
We write p = −q in case p = ¬q or q = ¬p. Note that an IG G does not have to be aconnected graph (see Figure 2a). In the remainder of this paper, let G = (P, A) be an IG.
Definition 2 (Generalisation arc) A generalisation arc g ∈ G ⊆ A is a directed (hyper)arc g : {p1 , . . . , pn } → p, indicating a generalisation with antecedents P1 = {p1 , . . . ,pn } ⊆ P and consequent p ∈ P \ P1 . Here, propositions in P1 are called the tails of g, denoted by Tails(g), and p is called the head of g, denoted by Head(g). G divides into two
disjoint subsets Gc and Ge of causal and evidential generalisation arcs, respectively.
Curly brackets are omitted in case |Tails(g)| = 1. In figures in this paper, generalisationarcs are denoted by solid (hyper)arcs, which are labelled ‘c’ for g ∈ Gc and ‘e’ for g ∈ Ge .
Example 2 In Figure 2a, an IG is depicted for a possible interpretation of the running
example. First, we consider the undirected edges connected to the testimonies and thepolice report in the mind map of Figure 1. As noted earlier, testimonies are often considered to be evidential [5], where generalisations are of the form ‘Testimony to fact x is
normally evidence for x’. Police reports can similarly be considered evidential. The IG
therefore includes generalisation arcs g1 , g2 , g4 , g7 ∈ Ge to denote these generalisations.

As tes3 concerns Marjan’s testimony to denying any involvement in the murder, ¬murder
is included in P and g6 : tes3 → ¬murder in Ge . A motive for committing an act canbe considered a cause for committing that act [5]. The IG therefore includes arcs g3 :mot1 → murder and g5 : mot2 → murder in Gc to denote these generalisations.

As generalisations hardly ever hold universally, exceptional circumstances can be provided under which a generalisation may not hold; hence, we allow exceptions to generalisations to be specified in IGs by means of exception arcs.
Definition 3 (Exception arc) An exception arc x ∈ X ⊆ A is a hyperarc x : p
p ∈ P is called an exception to generalisation g ∈ G.

g, whereAn exception arc directed from p to g indicates that p provides exceptional circumstances
under which g may not hold.
Example 3 Proposition lie, which states that Marjan had reason to lie when giving her
testimony, provides an exception to evidential generalisation g6 : tes3 → ¬murder in Ge .
In Figure 2a, this is indicated by a curved hyperarc x : lie g6 in X.

4.2. Reading Inferences from Information Graphs
We now define how deductive and abductive inferences can be read from IGs. By itself,
a generalisation arc only expresses that the tails together allow us to infer the head incase this generalisation is used in deductive inference, or that the tails together can be inferred from the head in case of abductive inference. Only when considering the available
evidence can directionality of inference actually be read from the graph.
Definition 4 (Evidence set) An evidence set is a subset E ⊆ P for which it holds that forevery p ∈ E, ¬p ∈/ E.
In the remainder of this paper, let E be an evidence set. The restriction that for every p ∈ E
it holds that ¬p ∈/ E ensures that not both a proposition and its negation are observed.
In figures in this paper, nodes in G corresponding to elements of E are shaded and all
shaded nodes correspond to elements of E. We emphasise that various sets E can be used
to establish inferences from the same IG.
Example 4 In the running example, the evidence consists of the testimonies and thepolice report. In Figure 2b, the IG of Figure 2a is again depicted, with nodes in E = {tes1 ,tes2 , tes3 , tes4 , police} shaded.

4.2.1. Deductive InferenceFirst, we specify under which conditions we consider a configuration of generalisationarcs and evidence to express deductive inference.
Definition 5 (Deductive inference) 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, denoted p1 , . . . , pn g q, iff ∀pi , i = 1, . . . , n:
1. pi ∈ E, or;
2. pi is deductively inferred from propositions r1 , . . . , rm ∈ P using a generalisation
g0 : {r1 , . . . , rm } → pi , where g0 ∈ Ge if g ∈ Ge , or;
3. pi is abductively inferred from a proposition r ∈ P using a generalisation g0 : {pi , r1 , . . . ,rm } → r in Gc , g 6= g0 , r1 , . . . , rm ∈ P (see Definition 6).

q
g1 e
p
(a)

q
g1

c

p2

p1
(b)

fire
e g2
smoke
c g1
smoke_machine
(c)

r
c

q
g2

e

c

r
(d)

g3

q

g1

g1

p

p1

g2
c
p2

c

r1

(e)

Figure 3. Examples of IGs illustrating the restrictions put on performing deduction within our IG-formalism(a-c); examples of IGs illustrating abductive inference (d-e).

In accordance with our assumptions stated in Section 2, deduction can be performed using generalisations in both Gc and Ge . The condition q ∈/ E ensures that deduction cannot
be performed to infer propositions that are already observed. Deduction can only be performed using a g ∈ G to infer Head(g) from Tails(g) in case every tail pi ∈ Tails(g) has
been affirmed in that either pi ∈ E, pi is itself deductively inferred, or pi is abductively
inferred. In correspondence with Pearl’s constraint (see Section 2), we assume in condition 2 that a proposition q ∈ P cannot be deductively inferred from p1 , . . . , pn ∈ P usinga g ∈ Ge if at least one of p1 , . . . , pn is deductively inferred using a g0 ∈ Gc . Condition 3of Definition 5 is further explained in Section 4.2.3, after abduction is defined.
Example 5 Consider the running example. In Figure 2b, mot1 and mot2 are deductively
inferred from tes1 and tes2 using generalisations g2 and g4 , respectively, as tes1 , tes2 ∈ E(condition 1 of Definition 5). Similarly, murder, ¬murder and lie are deductively inferredfrom police, tes3 and tes4 using generalisations g1 , g6 and g7 , respectively, as police,tes3 , tes4 ∈ E. Proposition murder is also deductively inferred from mot1 and mot2 using
causal generalisations g3 and g5 , as mot1 and mot2 are deductively inferred (condition
2 of Definition 5). This illustrates mixed deduction using both types of generalisations.
We now illustrate the restrictions put on performing deduction within our IG-formalism.
Example 6 Figure 3a depicts an example of an IG in which q cannot be deductively
inferred from p using g1 , as Head(g1 ) = q ∈ E. In Figure 3b, q cannot be deductively
inferred from p1 and p2 using g1 , as p2 ∈/ E and p2 is neither deductively nor abductively inferred. In Figure 3c, the example of Section 2 illustrating Pearl’s constraint
for deduction is modelled. As smoke machine ∈ E, smoke is deductively inferred fromsmoke machine using g1 by condition 1 of Definition 5. fire cannot in turn be inferredfrom smoke using g2 , as g2 ∈ Ge and smoke is deductively inferred using g1 ∈ Gc . 
4.2.2. Abductive Inference
Next, we specify under which conditions we consider a configuration of generalisationarcs and evidence to express abductive inference.
Definition 6 (Abductive inference) Let p1 , . . . , pn , q ∈ P, with {p1 , . . . , pn } ∩ E = 0.
/
Then given E, propositions p1 , . . . , pn are abductively inferred from q using a generalisation g : {p1 , . . . , pn } → q in Gc , 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
g0 : r1 , . . . , rm → q in G, g 6= g0 (see Definition 5), where g0 ∈ G \ Gc , or;
3. q is abductively inferred from a proposition r ∈ P using a generalisation g0 : {q, r1 , . . . ,rm } → r in Gc , r1 , . . . , rm ∈ P.

In accordance with our assumptions stated in Section 2, abduction is modelled using only
causal generalisations. The condition {p1 , . . . , pn } ∩ E = 0/ ensures that abduction cannot
be performed to infer propositions that are already observed. Furthermore, abduction can
only be performed using a g ∈ Gc to infer Tails(g) from Head(g) in case Head(g) has
been affirmed in that either Head(g) ∈ E, Head(g) is deductively inferred, or Head(g)is itself abductively inferred. In correspondence with Pearl’s constraint (see Section 2),
we assume in condition 2 that propositions p1 , . . . , pn ∈ P cannot be abductively inferredfrom a proposition q ∈ P using a g ∈ Gc if q is deductively inferred using a g0 6= g ∈ Gc .
Example 7 In Figure 3d, p is abductively inferred from q using generalisation g1 ∈ Gc
by condition 2 of Definition 6, as q has been deductively inferred from r using generalisation g2 ∈ Ge . In Figure 3e, q and r1 are abductively inferred from r using generalisation g3 : {q, r1 } → r in Gc by condition 1 of Definition 6, as r ∈ E. Then by condition 3 of Definition 6, p1 and p2 are abductively inferred from q using generalisations
g1 and g2 , respectively. Consider Figure 4b, which illustrates that Pearl’s constraint for
mixed deductive-abductive inference is adhered to (see Section 2). As smoke machine∈ E, smoke is deductively inferred from smoke machine using g1 ∈ Gc . fire cannot be
inferred from smoke, as g2 ∈ Gc (condition 2 of Definition 6).

4.2.3. Mixed Abductive-Deductive and Ambiguous Inference
As apparent from Definitions 5 and 6, mixed abductive-deductive inference can be performed within our IG-formalism.
Example 8 In Figure 4a, the example of Section 2 illustrating mixed abductiondeduction is modelled. From smoke ∈ E, fire is abductively inferred using g1 . Then heatis deductively inferred (or predicted) from fire using g2 (Definition 5, condition 3).

The conditions under which we consider a configuration of generalisation arcs and evidence to express deduction and abduction according to Definitions 5 and 6 are not mutually exclusive. Under specific conditions, both inference types can be established fromthe same g ∈ Gc in an IG given the provided evidence; the inference type is, therefore,
ambiguous (see Section 2). Examples of such inferences are provided in Figure 2b.
5. An Argumentation Formalism Based on Information Graphs
Based on our IG-formalism, we now propose an argumentation formalism that allows
for both deductive and abductive argumentation. Our approach generates an abstract argumentation framework as in Dung [7], that is, a set of arguments with a binary attack
relation, which thus allows arguments to be formally evaluated according to Dung’s classical semantics. In Section 5.1, we define arguments on the basis of a provided G andE, which capture sequences of inference steps as defined in Definitions 5 and 6 startingwith elements from E. We then formally prove that arguments constructed on the basisof IGs conform to Pearl’s constraint. In Section 5.2, we define several types of attacks
between arguments on the basis of IGs and instantiate Dung’s abstract approach.
5.1. Arguments
In defining arguments on the basis of a G and E, we take inspiration from the definition ofan argument as given in [8]. In what follows, for a given argument A, the function C ONC
returns its conclusion, S UB returns its sub-arguments (including itself), I MM S UB returns
its immediate sub-arguments, G EN returns all the generalisations used in constructing A,
and T OP G EN returns the last generalisation used in constructing A.

A2

B2¬murdermurder

¬heat

heat
g2

c

smoke
c g1
fire
(a)

smoke
g3
e
c
g2 c
c g1
A3
A1 g1
fire
smoke_machine police
mot1
(b)

g
c 5

A4 B1

mot2
(c)

g6 e

tes3

C2
lie
e g7 C1
tes4

Figure 4. IGs illustrating mixed abduction-deduction (a) and Pearl’s constraint for mixed deduction-abduction(b); adjustment to the IG of Figure 2b, where arguments and attacks (99K) are also indicated (c).

Definition 7 (Argument) An argument A on the basis of G and E is any structure obtainable by applying one or more of the following steps finitely many times:
1. p if p ∈ E, where: C ONC(A) = p; S UB(A) = {A}; I MM S UB(A) = 0;
/ G EN(A) = 0;/
T OP G EN(A) = undefined.
2. A1 , . . . , An g p if A1 , . . . , An are arguments such that p is deductively inferredfrom C ONC(A1 ), . . . , C ONC(An ) using a generalisation g ∈ G \ (G EN(A1 ) ∪ . . . ∪G EN(An )), g : {C ONC(A1 ), . . . , C ONC(An )} → p according to Definition 5, where:
C ONC(A) = p; S UB(A) = S UB(A1 ) ∪ . . . ∪ S UB(An ) ∪ {A}; I MM S UB(A) = {A1 , . . . ,An }; G EN(A) = G EN(A1 ) ∪ . . . ∪ G EN(An ) ∪ {g}; T OP G EN(A) = g.
3. A0 g p if A0 is an argument such that p is abductively inferred from C ONC(A0 ) using ageneralisation g ∈ G\ G EN(A0 ), g : {p, p1 , . . . , pn } → C ONC(A0 ) for some propositions
p1 , . . . , pn ∈ P according to Definition 6, where: C ONC(A) = p; S UB(A) = S UB(A0 ) ∪{A}; I MM S UB(A) = {A0 }; G EN(A) = G EN(A0 ) ∪ {g}; T OP G EN(A) = g.
In the remainder of this paper, let the set of all arguments on the basis of G and E
be denoted by A. An argument A ∈ A is called a premise argument if only step 1 ofDefinition 7 is applied, deductive if only steps 1 and 2 are applied, abductive if only
steps 1 and 3 are applied, and mixed otherwise. The restrictions in steps 2 and 3 thatg∈
/ (G EN(A1 ) ∪ . . . ∪ G EN(An )) and g ∈/ G EN(A0 ), respectively, ensure that cycles inwhich two propositions are iteratively deductively and abductively inferred from eachother using the same g are avoided in argument construction.
Example 9 Consider the adjustment to the IG of Figure 2b depicted in Figure 4c, inwhich arguments on the basis of this IG and E = {police, tes3 , tes4 } are also indicated.
According to step 1 of Definition 7, A1 : police is a premise argument. Based on A1 ,deductive argument A2 : A1 g1 murder is constructed by step 2 of Definition 7, as
murder is deductively inferred from police using g1 : police → murder. Then A3 : A2
g3 mot1 is a mixed argument by step 3 of Definition 7, as mot1 is abductively inferredfrom murder using g3 : mot1 → murder. Consider Figure 3e, which illustrates step 3 inmore detail. On the basis of this IG 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 7,
as q and r1 are abductively inferred from C ONC(A01 ) using g3 : {q, r1 } → r. Again by step3, A04 : A03 g1 p1 and A05 : A03 g2 p2 are constructed using g1 and g2 , respectively. In performing inference, care should be taken that no cause for an effect is inferred in casean alternative cause for this effect was already previously inferred (Pearl’s constraint, see
Section 2). In the context of IGs, for g ∈ Gc , propositions in Tails(g) express causes forthe common effect expressed by Head(g), and for g ∈ Ge , Head(g) expresses a cause
for propositions in Tails(g). Hence, in defining how inferences can be read from IGs,
restrictions are put in Definitions 5 and 6 such that Pearl’s constraint is adhered to. We
now formally prove that Pearl’s constraint is indeed never violated when constructing
arguments on the basis of an IG G and an evidence set E.

Proposition 1 (Adherence to Pearl’s constraint) Let c1 , c2 ∈ P be alternative causes ofe ∈ P in that either:
1. ∃g ∈ Ge , e ∈ Tails(g), Head(g) = c1 , and either:1a) ∃g0 6= g ∈ Ge , e ∈ Tails(g0 ), Head(g0 ) = c2 , or;1b) ∃g0 ∈ Gc , c2 ∈ Tails(g0 ), Head(g0 ) = e.
2. ∃g ∈ Gc , c1 ∈ Tails(g), Head(g) = e, and either:2a) ∃g0 6= g ∈ Gc , c2 ∈ Tails(g0 ), Head(g0 ) = e, or;2b) ∃g0 ∈ Ge , e ∈ Tails(g0 ), Head(g0 ) = c2 .
Assume arguments A and B exist in A with C ONC(B) = e, A ∈ I MM S UB(B), andC ONC(A) = c1 . Then no argument C exists in A with B ∈ I MM S UB(C), C ONC(C) = c2 .
Proof. In constructing B from A, a generalisation g ∈ Ge , e ∈ Tails(g), Head(g) = c1
could not have been used (case 1), as this would be an instance of abduction while per the
restrictions of Definition 6 abduction can only be performed using generalisations g ∈Gc . Thus, we only need to consider case 2, which is a deductive inference. First, consider
case 2a. Then by Definition 6 (condition 2), no argument C with C ONC(C) = c2 can beconstructed from B using g0 . Next, consider case 2b. Then by Definition 5 (condition 2),
no argument C with C ONC(C) = c2 can be constructed from B using g0 .

5.2. Attack
In this section, several types of attacks between arguments on the basis of IGs are defined. In argumentation, two types of attacks are typically distinguished, namely rebuttal
and undercutting attack [8]. We also distinguish a third type of attack, namely alternative attack, inspired by [6]. In our argumentation formalism, these three types of attacks
directly follow from the constructed arguments and the specified exception arcs in an IG.
Definition 8 (Attack) Let A, B ∈ A. Then A attacks B iff A rebuts B, A undercuts B, or A
alternative attacks B, as defined in Definitions 9, 10 and 11, respectively.
First, rebuttal attack is considered, which informally is an attack on a p ∈/ E.
Definition 9 (Rebuttal attack) Let A, B, B0 ∈ A with B0 ∈ S UB(B). Then A rebuts B (onB0 ) iff C ONC(B0 ) ∈/ E and C ONC(A) = −C ONC(B0 ).
Example 10 Consider the IG of Figure 4c. Let A1 , A2 be the arguments introduced in
Example 9. Let B1 : tes3 and let B2 : B1 g6 ¬murder. Then A2 rebuts B2 (on B2 ) andB2 rebuts A2 (on A2 ), as C ONC(A2 ) = murder, C ONC(B2 ) = ¬murder, and both murder,¬murder ∈/ E. This symmetric rebuttal is indicated in Figure 4c by means of a bidirectional dashed arc between these propositions. Consider again Example 8 and Figure 4a,in which heat is predicted from fire. Assume that contrary to this prediction we observethat there is no heat (¬heat ∈ E). Let A001 : smoke; A002 : A001 g1 fire; A003 : A002 g2 heat;
B001 : ¬heat. Then B001 rebuts A002 (on A002 ), but A002 does not rebut B001 as C ONC(B001 ) ∈ E. 
Next, undercutting attack is considered. Informally, an undercutter attacks an inference
by providing exceptional circumstances under which the inference may not be applicable.
Undercutting attacks between arguments follow from the specified exception arcs in G.
Specifically, as an exception arc directed from p ∈ P to g ∈ G specifies an exception to g,
an argument A ∈ A with C ONC(A) = p undercuts an argument B ∈ A with g ∈ G EN(B).
Definition 10 (Undercutting attack) Let A, B, B0 ∈ A with B0 ∈ S UB(B). Then A undercuts B (on B0 ) iff there exists an x ∈ X such that x : C ONC(A) g and T OP G EN(B0 ) = g.

Example 11 Consider the IG of Figure 4c. Let B1 , B2 be the arguments introduced in
Example 10. Let C1 : tes4 ; C2 : C1 g7 lie. Then C2 undercuts B2 (on B2 ), as x : lie
g6 in X and T OP G EN(B2 ) = g6 . This attack is indicated in Figure 4c by means of adashed arc directed from lie to inference tes3 g6 ¬murder.
Lastly, alternative attack is defined. Arguments are involved in alternative attack iff theirabductively inferred conclusions are in competition for a common effect (see Section 2).
Definition 11 (Alternative attack) Let A, B, B0 ∈ A with B0 ∈ S UB(B). Then A alternative attacks B (on B0 ) iff there exists an argument C ∈ I MM S UB(A) ∩ I MM S UB(B0 ) suchthat C ONC(A) and C ONC(B0 ) are abductively inferred from C ONC (C) using generalisations g and g0 in Gc , g 6= g0 , respectively.
Under the conditions set out in Definition 11, arguments Ai : C g pi for pi ∈ Tails(g)constructed from C via abduction are involved in alternative attack with A0j : C g0 p0j
for p0j ∈ Tails(g0 ) constructed from C via abduction. Arguments Ai (and A0j ) are not
involved in alternative attack among themselves, in accordance with our assumption thatthe antecedents of causal generalisations are not in competition (see Section 2).
Example 12 Consider the IG of Figure 4c. Let A1 , A2 , A3 be the arguments introduced in
Example 9, and let A4 : A2 g5 mot2 , where mot2 is abductively inferred from murder.
Then A3 and A4 are involved in alternative attack, as indicated in Figure 4c by means ofa bidirectional dashed arc between their conclusions.

Finally, we instantiate [7]’s abstract approach with arguments and attacks based on IGs.
Definition 12 (Argumentation framework) An argumentation framework defined by G
and E is a pair (A, C ), where (A, B) ∈ C iff A ∈ A attacks B ∈ A (see Definition 8).
Given an argumentation framework, we can use any semantics for argumentation frameworks as defined by [7] for determining the acceptability status of arguments (cf. [8]).
6. Related WorkIn this paper, we have introduced the graph-based IG-formalism for deductive and abductive inference with causal and evidential information. Most related formalisms for
inference with this type of information are logic-based. In the hybrid theory proposedby Bex [1], deduction and abduction are used in constructing evidential arguments andcausal 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 deductive and abductive arguments. Building on the hybrid theory, Bex proposed the integrated theory of causal and evidential arguments [6]. 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 chaining
such inferences. Actual abduction is thus not performed by constructing arguments.
Graph-based formalisms for reasoning with causality information have also been
proposed, notably Pearl’s causal diagrams [10]. Compared to IGs, causal diagrams do
not allow for capturing asymmetric conflicts such as exceptions in the graph.
7. Conclusion and Future WorkIn this paper, we have introduced the IG-formalism, which provides a principled way
for representing and reasoning with causal and evidential information. Based on our IGformalism, we have proposed an argumentation formalism that generates an abstract argumentation framework as in Dung [7], that is, a set of arguments with a binary attack

relation, which thus allows arguments to be formally evaluated according to Dung’s classical semantics. Moreover, our argumentation formalism adheres to the constraints imposed by Pearl’s C-E system [4]. The added value of our argumentation formalism is thatit allows both deductive and abductive argumentation, the latter of which has received
relatively little attention in argumentation. In defining our argumentation formalism, wewere inspired by the ASPIC+ argumentation framework [8]. Our argumentation formalism can be regarded as an adaptation of a special case of ASPIC+ , which would among
other things require introducing a new form of attack, namely alternative attack, and restricting the manner in which arguments are constructed within this framework. In future
work, we intend to investigate the relations between our argumentation formalism and
ASPIC+ and whether Caminada and Amgoud’s rationality postulates [11] are satisfied.
IGs formalise analyses performed by domain experts using the informal reasoning
tools they are familiar with, such as mind maps. In interpreting a performed analysis as
an IG, an additional knowledge elicitation step may be required, as the generalisations
used in performing inference are typically left implicit in tools domain experts use. IGs
may also be directly constructed by domain experts in case work. In our future work, we
intend to investigate possible applications of our IG-formalism as intermediate formalism between informal tools and formalisms that allow for formal evaluation other thanthose for argumentation, for instance by extending on our previous work on facilitating
Bayesian network (BN) construction from a preliminary form of IGs [12].
In our future work, we also intend to increase the expressivity of our IG-formalism
by allowing generalisations that are neither causal nor evidential. For instance, definitions, or abstractions [13], allow for reasoning at different levels of abstraction, such as
stating that guns can generally be considered deadly weapons.
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