Towards an integrated theory of causal
scenarios and evidential arguments
Floris BEX
Department of Information and Computing Sciences, Utrecht University

Abstract. The process of proof is one of inference to the best explanation, in
which alternative scenarios are supported and attacked by arguments. This
combination of scenarios and arguments was previously presented as a formal
hybrid theory. In this paper, the aim is to further integrate scenarios and arguments
by defining a notion of attack between alternative explanations. Thus, scenarios
and arguments can be incorporated in the same dialectical framework.
Keywords. causal reasoning, evidential reasoning, scenarios, arguments

1. Introduction
The study of reasoning with evidence constitutes the study of the process of proof,
which involves reasoning with evidence and commonsense knowledge in order toestablish whether something is or was the case. Whether in law, medicine or science,
the reasoning in the process of proof can be viewed as inference to the best explanation(IBE): alternative scenarios that explain the evidence are constructed and ultimately thebest scenario is chosen.
The logical frameworks for IBE that have been proposed in the AI literature[7][10] use causal rules to model the scenarios, which are then compared on basic
criteria such as the minimum number of hypothetical assumptions. Because theseframeworks were originally intended for automated diagnosis within relatively small
and pre-defined domains, they do not incorporate aspects like reasoning with bothcausal rules (c causes e) and evidential rules (e causes c), reasoning about the causal
rules that comprise the scenarios and more fine-grained scenario comparison criteria.
Hence, these frameworks are less suited to modelling and supporting more complex
tasks such as analysing and making sense of the evidence in large criminal cases[4][13].
One way to expand the analytical possibilities of these classic frameworks for IBE
is to combine them with a general argumentation framework for evidential reasoning,
so that arguments based on evidence can be used to support and attack the scenarios.
This is the approach taken by the hybrid theory [4], in which causal, scenario-driven
reasoning is combined with argumentative, evidence-driven reasoning.
The expressive power of the hybrid theory, however, negatively influences its
formal rigour. Because of the combination of two separate frameworks, causal
scenarios are compared differently from evidential arguments. For example, twocontradictory arguments attack each other but two contradictory scenarios do not, andtwo alternative scenarios are in competition, whilst two arguments that present

alternative explanations are not. Furthermore, the status of arguments is determinedusing standard semantics of [8], whereas the status of scenarios is determined withmore absolute measures (e.g., the more observations a scenario explains the better it is[11][13]). Whether or not a certain claim is considered acceptable can therefore depend
on if it is modelled as part of a scenario or an argument. What is missing from thehybrid theory is a standard way to compare both alternative explanations andcontradictory claims irrespective of whether they are part of a scenario or an argument.
The present paper aims to further integrate arguments and scenarios by treatingthem as equals in the dialectical process. It will be shown how alternative scenarios canbe considered as counterarguments. Furthermore, it will be discussed how argumentsthat present alternatives can attack each other. Thus, the status of both scenarios andarguments can be determined using the semantics of [8]. Note that these amendments
do not lead to a loss of expressiveness: it is still possible to reason about the scenarios
and the original criteria for comparing scenarios are maintained.

2. A Hybrid Theory of Scenarios and Arguments
Two influential methods for the analysis of evidence are the argument-based approach[2], which focuses on arguments based on evidence, and the story-based approach [11],which uses hypothetical stories or scenarios to explain the evidence. In the argumentbased approach the reasoning goes from evidence to conclusions by the application ofevidential rules (e is evidence for c), and arguments for and against the conclusion are
considered and compared. In the scenario-based approach multiple scenarios consistingof causal rules (c is a cause for e) are constructed to explain the evidence, and thescenarios are considered and compared.
Both the evidential argument and the causal scenario approach have advantages.
An inference from a piece of evidence (e.g. a witness testimony) to a conclusion is best
captured using evidential rules [4][6], and arguments based on evidence can becompared in a solid but intuitive framework [8]. However, certain parts of a case (suchas the cause of death, or the behaviour of the suspect) are more easily captured using
causal rules [6]. Furthermore, scenarios help in providing an overview of the case as
they are understandable stories that explain the evidence in a natural way [11].
The combination of evidential arguments and causal scenarios seems to be an
intuitive and analytically useful perspective for looking at cases and evidence. Hence,in [4] a hybrid theory for reasoning with arguments and scenarios is proposed, acombination of a formal theory of causal-abductive reasoning CT based on classicalmodel-based approaches [7][10] and an evidential argumentation theory ET based on
the ASPIC+ framework [12]. The logical language of this theory contains a connective (defeasible implication), and the logic includes an inference rule for this connective(defeasible modus ponens). Rules can be either causal (p C q) or evidential (q E p).
A causal theory CT contains a set of evidence, a set of causal rules and a set of
hypotheticals (literals). We can then construct scenarios, sequences of hypotheticals,
rules and elements that are derived from some previous element by application of
defeasible modus ponens. For example, consider S1 [9].
S1: [rain, rain C grass_wet, grass_wet]
Here, rain is hypothesized to explain the evidence grass_wet. A scenario S explains anobservation e  E iff S ⊢C e, where ⊢C stands for logical consequence according to theset of common deductive inference rules extended with modus ponens for C. For
example, scenario S1 explains grass_wet. An important part of abductive reasoning is
the consideration of alternative scenarios that explain the evidence. For example, analternative explanation for grass_wet is that the sprinkler was on during the night.
S2: [sprinkler_on_, sprinkler_on C grass_wet, grass_wet]
We can also infer conclusions from the evidence using evidential rules, thus creatingarguments that can be used to support or attack scenarios.
In the evidential theory ET we have a set of evidence and a set of evidential rules.
By applying defeasible modus ponens, we can then build arguments based on evidence
that support or attack scenarios. Take, for example, argument A1.
A1: [grass_looks_wet, grass_looks_wet E grass_wet, grass_wet]
Arguments can rebut each other (opposite conclusion) and one argument can undercut
another argument (attacking the inference) [12]. Given a collection of arguments andtheir attack relations, the status of the arguments can be determined according to [8].
In the hybrid theory, the status of alternative scenarios depends on various criteria.
One example is evidential support, the number of pieces of evidence that support a
scenario (i.e., evidence that is part of the scenario or is the premise of an argument thathas as its conclusion an element in the scenario). The second is evidential contradiction,
the number of pieces of evidence that contradict a scenario (i.e., that are premises of anargument that contradicts the scenario). These criteria can be used to compare scenarios,and by contrasting them to the total number of pieces of evidence in a case we get anindication of the relevance and coverage of a scenario [11].
Causal and evidential reasoning are closely intertwined: if we have a causal rule ccauses e, and c is a normal (‘default’) cause for e, then we will usually also accept thate is evidence for c. For example, rain can cause the grass to be wet so the observationof wet grass can be seen as evidence for the fact that it rained. Exactly how we shouldmodel a relation (causally or evidentially) seems arbitrary. [6] found that while insome situations, people quite consistently chose either causal or evidential, there arealso many examples where people interpret the relation evidentially or causally in equalmeasure. Furthermore, some aspects of a case will more conveniently be modelled interms of evidential arguments, others in terms of causal scenarios.
Under the right assumptions, evidential and causal reasoning are formallyequivalent [10]: either we use causal rules and perform abductive reasoning or we use
evidential rules and perform modus ponens style reasoning. So, in a formal theory, we
should be able to change an evidential link into a causal link and still get the same
result from the scenario and argument comparison. For the hybrid theory, this is not thecase. Take, for example, the two alternative scenarios S1 and S2. If we change thecausal rules into the reverse evidential rules, we get two arguments, viz.
A2: [grass_wet, grass_wet E rain, rain]
A3: [grass_wet, grass_wet E sprinkler_on, sprinkler_on]
These two arguments, however, do not attack each other, while the two scenarios S 1and S2 were in competition. Similarly, say that we have the following argument.
A4: [trees_wet, trees_wet E rain, rain]This argument will contradict S2, adding to its evidential contradiction. A4 will also
attack A2, but since determining the ‘winning’ argument is done differently thandetermining the ‘winning’ scenario, it is not guaranteed that a case with S1 and A4 has
the same outcome as a case with A2 and A4. Also, if we turn A4 into a scenario:
S3: [rain, rain C trees_wet, trees_wet]
This scenario does not attack S1, even though the S1 and S2 are clearly incompatible.
Thus we cannot capture the idea of an alibi scenario, which is not an alternative
explanation (it does not explain the main questions in a case, e.g. why the victim died)but it does contradict the prosecution’s scenario (that the suspect killed the victim).

3. An integrated formal framework
The above discussion indicates that we need to reconsider the relation between
arguments and scenarios: we need to be able to reason with alternative explanations
even if they are modelled as arguments, using evidential rules, and we need to be able
to consider contradictory causal scenarios. The comparison of alternative scenarios andattacking arguments should be drawn level so that we have a fully integrated theory
instead of a hybrid of causal, scenario-based and evidential argumentative reasoning.
The logical language and logic of the integrated theory IT are the same as for the
hybrid theory: there are connectives E and C for evidential and causal rules, and a
defeasible modus ponens (DMP) inference rule for these connectives. Like in the
hybrid theory, there are sets of causal and evidential rules.
Definition [Rules] The set of rules is R = RE  RC, where RE is the set of evidential
rules and RC is the set of causal rules.
Following ASPIC+ [12], there is one knowledge base (with appropriate subsets).
Definition [Knowledge base] A knowledge base is a set K = Ke  Kh, where Ke is theset of evidence and Kh is the set of hypotheticals.
In the hybrid theory, arguments and scenarios are similar in structure: both arederivations in a defeasible logic. Scenarios can be considered as (hypothetical) causal
arguments, which puts them at the same level as evidential arguments.
Definition [Arguments] A causal/evidential argument is a finite sequence [φ1, …, φn],where n > 0, such that for all φi (1  i  n): φi  K  RC (causal argument/scenario), φi  K  RE (evidential argument); or φi follows from ψ1, …, ψn  {φ1, …,φi-1} by application of DMP.
We say that Conc(A) = {φ  A | φ is the last element in the sequence} is the conclusionof argument A, and argument B is a (proper) subargument of A iff B is a (proper)
subsequence of A that is also an argument.
In our example, A1 – A4 are evidential arguments and S1 – S3 are causal arguments. We
can now define when two arguments are alternatives.
Definition [Alternative Causal Arguments] Causal argument C1 is an alternative tocausal argument C2 if Conc(C1) = Conc(C2) and there exist C1', C2' such that Conc(C1')≠ Conc(C2') and C1', C2' are proper subarguments of C1,C2, respectively.

The idea here is that there are two possible causes, Conc(C1') and Conc(C2'), for theeffect denoted by the same conclusion of C1 and C2. In our example, S1 and S2 arealternatives: they both have the conclusion shoes_wet, but S1 has a subargument withconclusion rain and S2 has a subargument with conclusion sprinkler_on.
Evidential arguments can also be alternatives: an evidential rule is essentially an
explicit expression of an abductive reasoning step applied to a causal rule.
Definition [Alternative Evidential Arguments] Evidential argument E1 is an alternative to evidential argument E2 if Conc(E1) ≠ Conc(E2) and there exist E1', E2' suchthat Conc(E1') = Conc(E2') and E1', E2' are proper subarguments of E1, E2, respectively.
This definition, where the conclusion of an argument is different but some of its
premises are the same, is similar to [1], who discuss how arguments that represent
different actions that lead to the same goal can be seen as competing alternatives. The
idea is again that there are two possible causes, Conc(E1) and Conc(E2), for one effect(Conc(E1') and Conc(E2')). A2 and A3 are alternatives: A2 implies that grass_wet wascaused by rain, whilst A3 implies that grass_wet was caused by sprinkler_on.
Definition [Alternative Evidential/Causal Arguments] Evidential argument E is analternative to causal argument C if there exist C', E', C'', E'' such that Conc(C') =Conc(E') and Conc(C'') ≠ Conc(E''), where C', E' are subarguments of C, E, respecttively, and C'' is a subargument of C' and E' is a subargument of E''.
Again, the idea is that there are two alternative causes Conc(C'') and Conc(E'') for one
and the same effect, Conc(C') = Conc(E'). As a clarification of the above definition,
consider the different subarguments of the alternative arguments S1 and A3.
S1': [rain, rain C grass_wet, grass_wet]
A3 / A3'': [grass_wet, grass_wet E sprinkler_on, sprinkler_on]

S1'': [rain]
A3': [grass_wet]

Conc(S1') = Conc(A3') and Conc(S1'') ≠ Conc(A3''), where S1' and A3' are subargumentsof S1 and A3, S1'' is a subargument of S1' and A3' is a subargument of A3''.
Figure 1 visualizes the different alternative arguments. Arrows with open
arrowheads are causal rules and arrows with closed arrowheads are evidential rules.
rain

sprinkler_on

S1

S2
grass_wet

(a)

rain

sprinkler_on

A2

A3
grass_wet

(b)

rain

sprinkler_on

S1

A3
grass_wet(c)Figure 1: alternative causal (a), evidential (b) and causal/evidential (c) arguments

In explanatory reasoning from effect to cause the dialectical component lies in thealternatives: two explanations (whether scenarios or arguments) are in competition if
they are alternative causes for the same consequence. Thus, we can use thesedefinitions of alternatives in the definition of attack between arguments.
Definition [Attack] Argument A attacks argument B iff
 There are two subarguments A' and B' that have an opposite conclusion, andthe conclusion of B' is not in Ke (rebut); or There is a subargument A' which has conclusion (p E/C q) and there is anapplication of defeasible modus ponens to p E/C q in B (undercut); or A and B are alternatives (alternative attack).

Take the following examples of causal arguments (scenarios) and evidential arguments.
S1: [rain, rain C grass_wet, grass_wet]
S2: [sprinkler_on_, sprinkler_on C grass_wet, grass_wet]
S3: [rain, rain C trees_wet, trees_wet]
A1: [grass_looks_wet, grass_looks_wet E grass_wet, grass_wet]
A5: [trees_look_wet, trees_look_wet E trees_wet, trees_wet]
S1 and S3 mutually attack (rebut) and S1 and S2 also mutually attack (alternative attack).
Given arguments and attacks, it can be informative to consider particular
combinations arguments representing the positions in a case.
Definition [Position] Given an integrated theory IT, a position P is a conflict-free setof arguments such that for each evidential argument A in P, there is a subargument A'
which has as its conclusion an element in a causal argument S  P.
So a position represents a consistent outlook on a case consisting of causal arguments
and the evidential arguments supporting them. In Figure 2, the arguments from theexample are shown as an argumentation framework [8], where arrows stand for attack
relations, with the two positions P1 and P2 superimposed on the argumentation framework. It can be seen that S2, S3 and A4 form a coherent whole that is in conflict with S 1.
We could now define evidential support for a position: this would simply be thenumber of pieces of evidence that are the premise of an argument in that position.
Evidential contradiction can be defined similarly. This then allows us to indicate
preferences between positions (e.g. the position with the higher evidential support ispreferred) and thus give a rich way of defining preferences between sets of arguments
based on evidence (cf. [1]). In the example, P2 (supported by A1 and A5) is thepreferred position since it has a higher evidential support than P1 (supported by A1).
S2
P1

S1

A1

A4

P2

S3
Figure 2: Two positions in a case with attacking arguments/scenarios

In the integrated theory we want to be able to reverse the direction of the rules, fromcausal to evidential and vice versa, without influencing the outcome of the case.
However, causal and evidential rules – and more broadly, scenarios and arguments –behave as communicating vessels [5]: a change in an argument-oriented version of acase analysis requires a matching change in a story-oriented analysis. For example, if
we change a causal rule p C q into its corresponding evidential rule q E p and there
is no existing evidential argument with q in its conclusion, we have to add q to the set
K, as otherwise we cannot derive p using the rule q E p. Furthermore, any argumentswith conclusions (p C q) have to be changed into arguments with the conclusion
(q E p), so that they will still undercut any evidential argument corresponding to theoriginal causal argument that was undercut by (p C q).
If, however, we consistently apply these changes whenever we change a causal rule
into an evidential rule or vice versa, the outcome of a case will stay the same despite
the changes. In our example, say we change the causal rule rain C grass_wet into theevidential rule grass_wet E rain. This deletes the causal argument S1 and gives us a
new evidential argument A1', viz.
A1': [grass_looks_wet, grass_looks_wet E grass_wet, grass_wet, grass_wet E rain, rain]
Now rain and sprinkler_on will still be alternatives and S2 and the new A1' will attack
each other. Similarly, S3 will attack A1' and thus still argue against rain. Furthermore,
the positions in the case contain the same information: in P1, S1 is swapped for A1' but
this does not influence the conclusions we can draw from each of the positions or theirevidential support.

4. Conclusion
The hybrid theory [4] provides a way to reason with scenarios, arguments andevidence. The integrated theory presented in this paper expands the hybrid theory andfurther integrates the reasoning with scenarios and arguments. By treating scenarios as
causal-abductive arguments, they can be integrated into the dialectical framework of[12][8]. Furthermore, as long as we adhere to the principle of communicating vessels,
we can freely exchange causal rules for their corresponding evidential rules and viceversa without affecting which conclusions we can draw in the case and the makeup of
the positions in the case. Hence, the existing difficulties around whether or not tomodel a relation as causally or evidentially are alleviated. This approach is also
advocated by [13], whose explanatory coherence theory does not include directional
arrows but just links between ‘an explanation and what it explains’.
One distinction that has not been explicitly made in this paper is that between
argumentation, where the truth of some claim is disputed and we want to show that theclaim is true, and explanation, in which the claim is accepted but the question is why it
is true. In fact, it could be argued that by equating scenarios with causal arguments, wefail to model explanations at all. However, it is the context of the reasoning that
determines whether a particular reasoning structure (e.g. a causal sequence) is regarded
as argument or explanation. The moment you present an alternative explanation, you
are in fact arguing against the original explanation and thus the scenarios themselves
can be treated as arguments. So the integrated theory does not dismiss explanation, butrather shows how argument and explanation are related.
Causal and evidential reasoning has been studied in depth by other authors. The
most comprehensive work in AI is perhaps that in model-based diagnosis [7][10].
Because this work is originally intended to solve diagnosis problems with a pre-set
causal theory (ellicitated from, for example, a medical expert), it largely avoids the
problems associated with the sometimes vague distinctions between causal andevidential reasoning. Hence, these classic models are able to avoid complications
reasoning with both causal and evidential rules and reasoning about rules. However, it
has been argued [4][11][9][13] that one’s logical theory needs to be more expressive if
we want to capture the subtleties of more broad commonsense reasoning as takes placein, for example, legal evidential reasoning. Pearl [11] makes a first step in combining
causal and evidential reasoning in a principled way, but falls short of proposing a fullfledged logical theory. In future work we want to show that the integrated theory
correctly adheres to the constraints Pearl proposes in his C-E framework.
An approach which has gained influence (at the expense of logical models such as[10][9]) is the Bayesian approach. In this approach the scenarios and evidence is a caseare modelled as a Bayesian Network [16], a graphical representation of a joint
probability distribution, where the nodes are variables (e.g., events, evidence) and thelinks represent the (in)dependencies between the variables. The directed links in a
Bayesian network can be interpreted as causal rules, though a less strict criterion of‘correlation’ is often also used. While Bayesian Networks are a powerful tool inevidential reasoning, a major shortcoming is that, in order to calculate the probabilityof the claims represented in the network, many conditional and prior probabilities areneeded. More often than not these probabilities are simply not available. Furthermore,
Bayesian Networks struggle with the same knowledge representation issues regarding
causal and evidential reasoning as logical approaches do [15][16]. In my opinion, thework presented in this paper can therefore inform Bayesian approaches as much as
logical approaches to evidential reasoning.

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