Artif Intell Law (2009) 17:79–99
DOI 10.1007/s10506-009-9074-z

Did he jump or was he pushed?
Abductive practical reasoning
Floris Bex Æ Trevor Bench-Capon Æ Katie Atkinson

Published online: 11 February 2009 Springer Science+Business Media B.V. 2009

Abstract In this paper, we present a particular role for abductive reasoning in law
by applying it in the context of an argumentation scheme for practical reasoning.
We present a particular scheme, based on an established scheme for practical reasoning, that can be used to reason abductively about how an agent might have acted
to reach a particular scenario, and the motivations for doing so. Plausibility here
depends on a satisfactory explanation of why this particular agent followed thesemotivations in the particular situation. The scheme is given a formal grounding interms of action-based alternating transition systems and we illustrate the approachwith a running legal example.
Keywords

Argumentation schemes  Abductive reasoning  Practical reasoning

1 Introduction
It is by now well-accepted that stories, or sequences of events, play an importantpart in theories of how people reason with the evidence in criminal cases. In legal
psychology, authors such as Pennington and Hastie (1993) and Wagenaar et al.
(1993) argue that judges, jurors, and police investigators construct and compare
stories about ‘what happened’ in a case using the available evidence. This approachhas been adopted by researchers in AI (& Law), who model stories as causal
F. Bex
Faculty of Law, University of Groningen, Groningen, The Netherlands
e-mail: f.j.bex@rug.nl
T. Bench-Capon (&)  K. Atkinson
Department of Computer Science, University of Liverpool, Liverpool, UKe-mail: tbc@liverpool.ac.uk
K. Atkinson
e-mail: katie@liverpool.ac.uk

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networks that explain the evidence; Thagard (2004) has applied his connectionist
model of inference to the best explanation to legal cases and Bex et al. (2007)
propose an approach that combines classical abductive inference to the best
explanation with defeasible argumentation.
A good story of a criminal case should not only be sufficiently supported byevidential data (e.g., testimonies, forensic data), but it should ideally also beplausible, that is, the story should conform to our beliefs about how things generally
happen in the world around us. This plausibility of a story partly depends on theplausibility of the causal links between the events in the story, which give a story its
coherence. For example, a story where one person died because he was shot byanother person is coherent because we believe that, in general, shooting someone
can cause that person to die. Many stories about crimes involve rational agents.
When rational agents are concerned we need to see events not simply as the result of
the operation of physical causal laws—what Dennet (1979) terms the physicalstance—but also as the result of choices made by the agents—what Dennet calls theintentional stance. For the intentional stance, plausibility comes from our view of
how likely it is that the agent would have made the required choice in the situation,and this in turn depends on the motivational preferences we believe the agent tohave. For example, an agent may act in a certain way only if he prefers fame tofortune. If we wish to say he is motivated by fame, we need to explain why webelieve this particular agent has this preference.1
The formal framework proposed by Bex et al. allows for a careful analysis andcritique of the causal links between the events. The validity of the causal rules canbe argued about and exceptions to these rules can be given. However, the model ofagent decision making as regarded from the intentional stance remains something ofa ‘black box’. In Bex et al. (2007), most causal links denote a physical causal
relation and while in both Bex et al. (2007) and Thagard (2004) explanations foractions can, in a sense, be given in terms of psychological states, the agents’motivations and the question of whether and how the agents act on thesemotivations remains implicit. For example, Thagard explains ‘Claus injected Sunnywith insulin’ with ‘Claus wanted to end his marriage to Sunny’ (Thagard 2004,
p. 238). Here Claus’ reasoning remains implicit and therefore, somewhat
unbelievable, as one can argue that there are less drastic ways of ending amarriage. For real plausibility we need a more elaborate explanation of why the
choice was made by Claus at the particular time.
In this paper, we will attempt to allow this elaborated conception of theintentional stance by giving agents’ motivations, and the priorities amongst theagents’ motivations, a clear place in evidential reasoning about actions. We do thisby using an argumentation scheme for abductive practical reasoning, based on thenormal (non-abductive2) practical reasoning scheme as proposed by Atkinson et al.
(2006). The combination of abduction and a scheme for practical reasoning has also
1

When speaking of persons we might call such a preference ‘character’ and may attempt to explain whatkind of character the person has, cf. Walton (2007).

2

We use ‘non-abductive’ instead of ‘deductive’ because deductive implies that normal practicalreasoning is not presumptive/defeasible, which, of course, it is.

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been used by Walton and Schafer in (2006). Although they make appeal to ideas
derived from computational agents built on the Belief-Desire-Intention (BDI)
paradigm, they do not provide any formal framework. In contrast, our abductive
scheme is formally grounded in terms of an Action-based Alternating Transition
System, or AATS (Wooldridge and van der Hoek 2005), and it allows us to explaina particular situation in terms of the choices made by the agents involved and theirmotivations. The resulting explanations, modeled as arguments in a value-based
argumentation framework (Bench-Capon 2003), can then be evaluated by considering the agents’ motivational priorities, or used to infer the agents’ motivational
priorities.
We recognize that, there are legal issues relating to the notion of motive,
discussed in work such as Walton and Schafer (2006) and Leonard (2001). In thispaper, we will use ‘‘motivation’’ rather than ‘‘motive’’, and present an accountwhich is intended to relate to the everyday notions that might be used by a juror or a
detective, and leave exploration of specifically legal notions of motive to morelegally qualified people. In particular we shall not address at all problems
concerning the legal admissibility of motive evidence, the central concern of
Leonard (2001). Nevertheless, we believe that, the account we give here is of
relevance to the legal notions, and could form a valuable basis for such exploration.
The rest of this paper is organised as follows. In Sect. 2 we will describe theargumentation scheme for abductive practical reasoning and its associated criticalquestions. In Sect. 3 we give the definitions of an AATS that we use to ground theapproach, then we show how the argument scheme and critical questions can be
defined in terms of an AATS. In Sect. 4 we will apply our model to an extended
example showing how explanations and objections can be constructed and how
conflicts between explanations may be resolved. In Sect. 5 we discuss related work,in particular (Walton and Schafer 2006), and we finish by making some concluding
remarks and identifying areas for future work in Sect. 6.

2 An argumentation scheme for abductive reasoning
In this section we will define an argumentation scheme for abductive practicalreasoning. This scheme is based on a well-known argumentation scheme forpractical reasoning defined by Atkinson et al. (2006). The original scheme and its
critical questions enables agents to propose, attack, and defend justifications foractions: presumptive or prima facie justifications of actions can be presented as
instantiations of the argument scheme, and then critical questions characteristic of
the scheme used can be posed to challenge these justifications. The original scheme
is stated as follows:
In the current circumstances R,
we should perform action A,
which will result in new circumstances S,
which will realise goal G,
which will promote some value V.

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This scheme is an extension of Walton’s sufficient condition scheme for practicalreasoning (Walton 1996). In the above scheme Walton’s notion of a goal isdisambiguated by separating it into three elements: the state of affairs brought about
by the action; the goal (the desired features in that state of affairs); and the value (thereason why those features are desirable). For example, I may diet to lose weight, withthe goal of not being overweight, to promote the value of health. The underlying idea
in making this distinction is that the agent performs an action to move from one stateof affairs to another. The new state of affairs may have many differences from thecurrent state of affairs, and it may be that only some of these differences are desired by
the agent. The significance of these differences is that they make the new state ofaffairs better with respect to some good valued by the agent.
An agent who does not accept a presumptive argument based on the above
scheme may challenge elements in the instantiation through the application ofcritical questions and an unfavourable answer to a critical question will identify apotential flaw in the argument. For example, one of the original critical questions(CQ8) is ‘Does doing the action have a side effect which demotes the value?’
Through the critical questions agents can attack the validity of the various elementsof the argument scheme and the connections between them, suggest alternative
possible actions, and draw attention to side effects of the proposed action.
The argumentation scheme for practical reasoning has roughly the same elements
as Pennington and Hastie’s episode scheme, a basic model about intentional actions(Pennington and Hastie 1993). In this scheme, some initiating states and events
cause the agent to have a set of goals, which give rise to actions that haveconsequences. In the argumentation scheme, we have the current circumstances (theinitiating states) in which the value acts as a motivation for some goal, which gives
rise to some action that results in the new circumstances (the consequences). An
important difference, however, between an episode and an instantiation of thepractical reasoning scheme is that in the argument based on the scheme the valuethat acts as a motivation is explicitly mentioned, whereas in an episode themotivation for the action remains implicit.
Now the argument scheme for abductive practical reasoning can be stated asfollows:
The current circumstances S,
are explained by the performance of action A,
in the previous circumstances R,
with motivation M.
By combining the normal and the abductive schemes for practical reasoning thiswill allow us to reason about intentional actions predictively as well as explanatorily.
Given this combination, two important questions need to be addressed. First, is theagent reasoning about past actions or is he reasoning about possible future actions, andsecond, is the agent reasoning about his own actions or about the actions of some otheragent? If he is reasoning about his own actions, then he can apply the abductive
practical reasoning scheme to justify with what motivations he took certain actions in
the past and apply the normal (non-abductive) practical reasoning scheme to guide his
future actions according to his values. If, on the other hand, he is reasoning about some123

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other agent, then he can apply the abductive practical reasoning scheme to explain
why and with what motivations the other agent took certain actions in the past andapply the normal (non-abductive) practical reasoning scheme to predict what actions
this other agent will take in the future to promote his values. The importance of the
distinction is stressed in Raz (1978). Agents can combine these different ways ofpractical reasoning. For example, a police investigator tracking a serial killer can be
guided in his actions by predicting what the killer will do next. Similarly, a judge
might be guided in his choice of action (i.e., acquit, convict) by accepting an
explanation of what happened and determining what could have motivated the suspectin this particular explanation.
Atkinson and Bench-Capon (2007), argued for the necessity of a well-founded
formal model underlying the generation of arguments and critical questions. They
provided the required grounding in terms of an Action-based Alternating Transition
System, or AATS (Wooldridge and van der Hoek 2005]. Essentially, an AATS
consists of a set of states and transitions between them, with the transitions labelledwith joint actions, that is, actions comprising an action of each of the agentsconcerned. To represent the fact that the outcome of actions is sometimes uncertain,in the scenario we use in this paper we will add a third ‘‘gent’’ which will determine
whether the actions had the desired or the undesired effect. The transitions will belabeled with motivations, corresponding to the values of Bench-Capon (2003),
encouraging or discouraging movement from one state to the next. Formal
definitions of the abductive argument scheme, and the associated critical questions
discussed below, are given in the next section. We use a transition system which is a
simplified version of the AATS used in Atkinson and Bench-Capon (2007) toground the practical reasoning argumentation scheme, but this will still allow us tohypothesise the reasoning concerning the events that may have taken place.
Given an AATS and a number of arguments generated from the AATS, a story (asequence of events) is a path through the AATS. An argument explains why that path
was followed, and so gives coherence and hence plausibility to the story. For example,‘John wrote a paper, John went to Florence’ is a story, but it has more coherence
expressed as ‘John went to Florence because he had to present the paper he had written.’Throughout this paper, we will use a simple example to illustrate our approach.
Picture two people on a bridge. The bridge is not a safe place: the footpath is narrow,the safety barriers are low, there is a long drop into a river, and a tramline withfrequent traffic passing quite close to the footpath. One of the persons, call him
Ishmael, is standing still, whereas the other, Ahab, is running. As Ahab reaches
Ishmael, Ishmael falls into the river. Did he jump or was he pushed? To answer this we
will need a story explaining either why Ahab chose to push Ishmael, or why Ishmael
chose to jump to his doom. If Ahab is on trial, the story we believe will be crucial: if
Ahab intended Ishmael’s death it will be murder, if there is a less damning explanation
for the push it may be manslaughter, and if Ishmael jumped, Ahab is completely
innocent. We illustrate the critical questions by reference to this example scenario.
Providing an explanation involves formulating the problem, generating candidate
explanations, and then choosing the best explanation. As in Atkinson and BenchCapon (2007) there are critical questions associated with both problem formulationand choice of explanation.

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We first present the critical questions relating to the choice of explanation for theabductive scheme. Below each critical question the answer to that question which
would attack the original argument is given, as well as an informal example of how
such an attack would be phrased in the example situation.
2.1 Critical questions for choice of explanation
CQ1

Are there alternative ways of explaining the current circumstances S?
(a)Could the preceding state R have been different?
Answer: action A was done in a different preceding state R‘You say that Ahab did not have a clear path in the previous state, but
actually Ahab already had clear progress’(b) Could the action A have been different?
Answer: a different action A0 was done in preceding state R‘You say that Ahab pushed Ishmael, but actually Ishmael jumped’
CQ2

CQ3

CQ4

CQ5

Assuming the explanation, is there something which takes away the
motivation?
Answer: doing action A in R to reach S demotivates M‘Ahab pushing Ishmael to the ground would not provide a clear path socannot be motivated by Ahab wanting clear progress’
Assuming the explanation, is there another motivation which is a deterrent
for doing the action?
Answer: some other motivation M0 deters from doing action A in R to reach S‘Ahab is deterred from pushing Ishmael off the bridge to get progressbecause he does not want to bring Ishmael into danger’
Can the current explanation be induced by some other motivation?
Answer: there is another motivation M0 which motivated doing A in R toreach S
‘Ahab pushing Ishmael of the bridge is not motivated by Ahab wanting clear
progress, but by Ahab wanting to revenge himself on Ishmael’
Assuming the previous circumstances R, was one of the participants in the
joint action trying to reach a different state?
Answer: in R, even though one agent performed his part of A with motivation
M, the joint action was actually A0 which led to S0 , where A0 = A and S0 = S‘Ahab wanted to push Ishmael out of the way of the tram to get him out ofdanger, but nature did not cooperate (and Ishmael fell off the bridge)’

In the above critical questions, ‘explanation’ stands for ‘the performance of joint
action A in previous circumstances R’. With this kind of explanation we mean
physical explanation, how performing an action in R caused the new state of affairs S,
as opposed to a mental explanation, what motivated an agent to do a particular action.
So answering CQ1 by giving an alternative cause of the current circumstances does
not require committing to a particular motivation for that alternative cause. Note that
it is possible to ask for an alternative mental explanation by posing CQ4. CQ2 and
CQ3 ask if there are any reasons for not doing the particular action.

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CQ5 is actually a critical question that does not apply to the abductive reasoning
step from the current state S to the previous state R, but rather to normal, nonabductive reasoning performed in R. It asks if it is at all possible that in the previous
state R, the agent wanted to perform a different joint action but was somehow
hindered by another agent (‘nature’ in the example) not cooperating. Typically, thiswill suggest that he was acting with a different, perhaps less culpable, motivation.
This reasoning with the abductive scheme and CQ5 actually combines multiple
reasoning steps into one. First, the previous circumstances are abduced using theabductive scheme. Then, assuming these previous circumstances were the case, we
try to justify the action by applying the normal scheme and finally we answer
critical question 17 from the original practical reasoning scheme ( Atkinson andBench-Capon 2007, p. 859), ‘are the other agents guaranteed to execute their part ofthe desired joint action?,’ unfavourably. For reasons of space, we have incorporated
these reasoning steps into one new critical question.
We now turn to the critical questions relating to problem formulation.
2.2 Critical questions for problem formulation
CQ6

CQ7

CQ8

CQ9

CQ10

CQ11

CQ12

CQ13

Are the current circumstances true?
Answer: the current state is not S‘Ishmael is not dead’
Could the action have had the stated preconditions?
Answer: A cannot be performed in R
‘Ahab is not strong enough to push Ishmael off the bridge’
Were the previous circumstances the same as the current circumstances?
Answer: for all propositions in S and R:
if a proposition p is true in S then pa was already true R
if a proposition p is false in S then pa was already false R‘Ishmael was already in the water when Ahab appeared’
Could the explanation for the current state provide the motivation?
Answer: doing action A cannot be motivated by M
‘Killing Ishmael would not give Ahab revenge’
Assuming the previous circumstances, would the action have the statedconsequences?
Answer: doing action A in in R does not bring about S‘Ishmael could not fall from the bridge (the safety barriers are too high)’
Assuming the previous circumstances, would the action have any
consequences?
Answer: doing action A in R does not get you to a new state‘Pushing Ishmael would have no effect’
Are the current circumstances S possible?
Answer: there is no state S (S is impossible)‘Suppose it was being claimed that Ishmael had flown to the moon’Is the joint action possible?
Answer: A is not a joint action‘Ahab pushed Ishmael and Ishmael jumped’

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CQ14

CQ15

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Are the previous circumstances R possible?
Answer: there is no state R (R is impossible)‘There is no bridge’
Is the motivation indeed a legitimate motivation?
Answer: M is not a motivation‘No civilised person acts out of revenge’3 Formal definitions
In this section we re-capitulate the definitions of an AATS as set out in Atkinson
and Bench-Capon (2007). We then go on to show how the argument scheme andcritical questions for abductive practical reasoning can be defined in terms of an
AATS.
3.1 AATS definitions
In order to be able to reason rigorously about actions and their effects, we need awell-defined structure in which we can represent how the actions of an agent willlead to transitions from one state to another. In particular we need to be able tocontextualise these transitions so that the effects of actions can be made dependent
on the action of other agents, and other events in the environment. One such
structure is provided by Alternating Transition Systems (ATS), originallydeveloped to underpin the Alternating-time Temporal Logic of Alur et al.
(2002). These structures have also been used by van der Hoek et al. (2005) to
explore the social laws paradigm for describing coordination in multi-agent
systems introduced largely through the work of Shoham, Tennenholtz and Moses(e.g., Moses and Tennenholtz (1995). Like Wooldridge and van der Hoek (2005)
we give the notions of actions and their pre-conditions a central role, so we adopt
their version of ATS in which actions and pre-conditions are first class entities.
This version is called an Action Based Alternating Transition Systems (AATS) in
Wooldridge and van der Hoek (2005), and it has been used in Atkinson andBench-Capon (2007) to provide formal definitions for an argument scheme andcritical questions for practical reasoning. In this report we also use this structure to
represent our argument scheme and critical questions for abductive practicalreasoning. We first provide the definition of an AATS, as given in Wooldridgeand van der Hoek (2005).
Assume first that the systems we wish to model may be in any of a finite set Q of
possible states, with some q0 2 Q designated as the initial state. Systems contain a
set Ag of agents and each agent i 2 Ag is associated with a set Aci of possible
actions. It is assumed that these sets of actions are pairwise disjoint (i.e., actions are
unique to agents).
A joint action jC for set of agents C (termed a coalition) is a tuple ha1 ,…, aki,where for each aj (where j B k) there is some i 2 C such that aj 2 Aci : Moreover,there are no two different actions aj and aj0 in jC that belong to the same Aci. The set

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Q
of all joint actions for coalition C is denoted by JC, so JC ¼ i2C Aci : Given anelement j of JC and an agent i 2 C; i’s action in j is denoted by ji.
An Action-based Alternating Transition System (AATS) is an (n ? 7)-tuple
S = hQ, q0, Ag, Ac1 ,…, Acn, q, s, U, pi, where:
–
–
–
–
–
–

–
–

Q is a finite, non-empty set of states;
q0 2 Q is the initial state;
Ag = {1 ,…, n} is a finite, non-empty set of agents;
Aci is a finite, non-empty set of actions, for each i 2 Ag where Aci \ Acj = ; forall i 6¼ j 2 Ag;
q : AcAg ? 2Q is an action pre-condition function, which for each action a 2
AcAg defines the set of states q(a) from which a may be executed;s : Q 9 JAg ? Q is a partial system transition function, which defines the states(q, j) that would result by the performance of j from state q—note that, as thisfunction is partial, not all joint actions are possible in all states (cf. the precondition function above);
U is a finite, non-empty set of atomic propositions; andp : Q ? 2U is an interpretation function, which gives the set of primitive
propositions satisfied in each state: if p 2 pðqÞ; then this means that the
propositional variable p is satisfied (equivalently, true) in state q.

In addition to the elements of an AATS given in Wooldridge and van der Hoek
(2005), we need to provide an extension to enable the representation of motivations
from the underlying argument scheme for abductive practical reasoning. First, wehave a set Am of motivations for each agent (which are a subset of a set M of
motivations). Every transition between two states from the set Q is either promoted,
demoted, or is neutral, with respect to each motivation. Note that motivations are
not unique to agents: individual agents may or may not have motivations incommon. Whether a motivations is promoted or demoted by a given action will be
determined by comparing the state reached with the state left. More formal
definitions of these elements are given below:––

Ami is a finite, non-empty set of motivations Ami  M; for each i 2 Ag:
d : Q  Q  AvAg ! fþ; ; ¼g is a valuation function which defines the status(promoted (?), demoted (–) or neutral (=)) of a motivation mu 2 AmAg ascribedby the agent to the transition between two states: d(qx, qy, mu) labels thetransition between qx and qy with one of {? , -, =} with respect to themotivation mu 2 AmAg :We can now extend the original specification of an AATS to accommodate thenotion of motivations and re-define an AATS as a (2n ? 8) tuple S ¼ hQ; q0 ;Ag; Ac1 ; . . .; Acn ; Am1 ; . . .; Amn ; q; s; U; p; di
3.2 Formal definitions for the argument scheme and CQs for abductive practicalreasoning
We now present the formal definitions of the argument scheme and criticalquestions in terms of an AATS. The critical questions can be grouped into two

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categories: those concerned with choice of explanation, and those concerned withproblem formulation. We present the formal definitions of the critical questions asgrouped into these categories. We begin by presenting the formal version of theargument scheme:
ABS1: The current circumstances q0 = qy
are explained by agent i participating in joint action jn where jin ¼ ai ;in the previous circumstances qx, where s(qx, jn) is qy
and 9pa 2 Usuch that either pa 2 pðqy Þ and pa 62 pðqx Þ; or pa 62 pðqy Þ and pa 2 pðqx Þsuch that for some mu 2 M; d(qx, qy, mu) is ?.
We now present the formal version of the critical questions that can be used tochallenge instantiations of the above argument scheme.
3.2.1 Critical questions for choice of explanation
CQ1a: The previous circumstances were not qx and were actually qz 2 Q; inwhich agent i 2 Ag could have participated in joint action jn 2 jAg ; agent suchthat s(qz, jn) is qy.
CQ1b: In the previous circumstances qx 2 Q; i 2 Ag; could have participated injoint action jm 2 JAg ; where jn = jm, such that s(qx, jm) is qy.
CQ2: There is a pb, where pa = pb, such that either pb 2 pðqy Þ; and pb 62 pðqx Þ;or pb 62 pðqy Þ and pb 2 pðqx Þ; such that d(qx, qy,mu) is –.
CQ3: There is a pb, where pa = pb, such that either pb 2 pðqy Þ; and pb 62 pðqx Þ;or pb 62 pðqy Þ and pb 2 pðqx Þ; such that d(qx, qy, mw) is –, where mu = mw.
CQ4: There is a pb, where pa = pb, such that either pb 2 pðqy Þand pb 62 pðqx Þ; orpb 62 pðqy Þ and pb 2 pðqx Þ; such that d(qx, qy, mw) is ?, where mu = mw.
CQ5: jin ¼ jim ; jn 6¼ jm and s(qx, jn) = s(qx, jm).
3.2.2 Critical questions for problem formulation
CQ6: q0 = qy.
CQ7: qx 62 qðai Þ:
CQ8: 8pj 2 U; pj 2 pðqy Þ; pðqx Þ; or pj 62 pðqy Þ and pj 62 pðqx Þ:CQ9: d(qx, qy, mu) is not ?.
CQ10: s(qx, jn) is not qy.
CQ11: s(qx, jn) is qx.
CQ12: qy 62 Q:
CQ13: jn 62 JAg :CQ14: qx 62 Q:
CQ15: mu 62 Ami :The above formalism can be used by agents in scenarios where reasoning takes
place to generate arguments to explain how an agent may have acted to find itself ina particular situation and its motivations for doing so. A worked example
demonstrating such a scenario is presented in the next section.

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4 Representation of the example
The first stage of our approach is to produce a transition diagram representing the
scenario. The objective is to include all that is relevant, but only what is relevant, so as
to avoid complicating the problem beyond what is necessary. Recall that the footpathis narrow, and that Ahab is running. He may be assumed to desire a clear path, and so
our first proposition is ‘‘Ahab’s path is clear’’ (C). Recall also that the bridge is adangerous place: a person on the bridge may be in danger by falling into the river (R),or by being hit by a tram (T). We also introduce a proposition ‘‘Ishmael’s life is indanger’’ (D), which is true whenever either R or T is true, since this will be usefulwhen we construct our arguments. In our example, we consider only the states which
are of interest to us, and for simplicity’s sake omit the other possible states from thediagram. For example, the state in which both T and C are true is of no interest to us,
since no one claims that it was the case, nor that anyone tried to reach it.
q1: CRTD = 0000—Ahab’s path is not clear and Ishmael is not in danger fromeither the river or the tram. This is the presumed situation before the incident.
q2: CRTD = 1101—Ahab’s path is clear and Ishmael is in danger from the river.
This is the situation immediately after the incident, which we wish to explain.
q3: CRTD = 1000—Ahab’s path is clear and Ishmael is not in danger from either
the river or the tram.
q4: CRTD = 0011—Ahab’s path is not clear and Ishmael is in danger from thetram. This situation would hold if Ishmael was too close to the tramline and a
tram was approaching.
Now consider the actions. Ishmael can jump or do nothing. Ahab can push
Ishmael or do nothing. Pushing and jumping are not simultaneously possible. Since,
however, the effect of a push is uncertain, we add a notional third agent (‘‘nature’’)to determine whether the push sends Ishmael into the river, or simply out of Ahab’s
way. Nature is irrelevant here when Ishmael jumps.
There are thus three joint actions which can be performed in states q1 and q4(see Fig. 1):
j1: Ishmael jumps and Ahab does nothing.
j2: Ishmael does nothing, Ahab pushes Ishmael and Ishmael falls off the bridge
into the river.
j3: Ishmael does nothing, Ahab pushed Ishmael and Ishmael is out of Ahab’s way
but still on the bridge.
We must next label the transitions with motivations. Moving from q1 to both q2
and q3 is motivated by Ahab’s progress: in q1 the path was not clear and in q2 andq3 it is clear. Equally the transitions from q1 to q2 demote Ishmael’s safety: in q1 heis in no danger and in q2 he is. But suppose also Ahab has a reason to seek revengeof Ishmael: this will apply only if q2 is reached from q1 by Ahab pushing Ishmael.
Finally suppose Ishmael wishes to kill himself, then, for him, oblivion will motivate
moving from q1 to q2. Turning to the transitions from q4, we can see that movingfrom q4 to q3 promotes both Ahab’s progress and Ishmael’s safety. Moving from q4
by j1, where Ishmael jumps, promotes Ahab’s progress, but does not affect

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q4

q3

j3
0011
Pa+

Pa+ Si+

j1

Pa+

1000
j3
Pa+

j2
Ra+

j2
0000

Pa+ Si− Ra+ Oi+

q1

j1

Pa+ Si− Oi+

1101
q2

Fig. 1 State transition diagram for the scenario

Ishmael’s safety or his chance of oblivion, since he dies in either case. Finally thetransition from q4 to q2 by j2 can be motivated either by Ahab’s progress or by
Ahab’s revenge, since it is important to satisfy this motivation that Ishmael is killed
by Ahab rather than by the tram.
Our final transition diagram is shown in Fig. 1. Of course, there are further transitions
such as those between q1 and q4, and those where both do nothing, which leave the state
unchanged, but those play no role in our considerations, and so are omitted.
This model could be challenged using the critical questions relating to problem
formulation given in the previous section. For example, someone might claim thatany push would send Ishmael into the river, so denying that j3 is possible, which
would remove the transitions into q3. For reasons of space, however, let us take ourformulation as accepted, and not consider these questions further here.
We now turn to the arguments that can be produced on the basis of Fig. 1. Recall
that we are trying to explain how we reached q2 and say that first we assume that theprevious state is q1. There are two transitions from q1 to q2, one promoting threevalues and one promoting two. We therefore have five possible instantiations of ourabductive argument scheme.
A1: q2 is explained by Ahab pushing Ishmael in q1 to reach q2 motivated byprogress.
A2: q2 is explained by Ahab pushing Ishmael in q1 to reach q2 motivated byrevenge.
A3: q2 is explained by Ahab pushing Ishmael in q1 to reach q2 motivated by
oblivion.
A4: q2 is explained by Ishmael jumping in q1 to reach q2 motivated by progress.
A5: q2 is explained by Ishmael jumping in q1 to reach q2 motivated by oblivion.
These arguments can be now be the subject of critical questioning. CQ1 applies,
attacking all five of the above arguments. Answering CQ1a, we can say that it ispossible that the preceding state was q4 rather than q1. Thus, we have:123Did he jump or was he pushed?

91

Obj1a: The preceding state was q4, not q1.
This objection can be met with the following rebuttal:
Reb1: The preceding state was indeed q1.
Notice that here, we label the answer to the critical question as an objection. It isalso possible to answer critical questions that ask for another explanation (CQs 1, 4
and 5) by providing an instantiation of the argument scheme, an alternative
explanation. If we take, for example, A1 as the current explanation, A4 and A5 areanswers to CQ1b and if we take A4 as our explanation, A1–A3 are answers to
CQ1b. So in virtue of CQ1b we can see that A1–A3 attack A4–A5 and vice versa.
CQ2 does not apply because none of the transitions has a motivation which both
encourages and discourages it at the same time.
CQ3 is important: the threat to Ishmael’s safety could be sufficient to deter Ahab
from pushing and Ishmael from jumping. Thus we have Obj3a attacking A1–A3 andObj3b attacking A4 and A5.
Obj3a: In q1 Ahab should not push Ishmael to reach q2 since it demotes Ishmael’s
safety.
Obj3b: In q1 Ishmael should not jump to reach q2 since it demotes Ishmael’s safety.
CQ4 applies in that if we take, for example, A1 as the current explanation, A2 andA3 are answers to CQ4. So in virtue of CQ4 we can see that A1–A3 all attack one
another and A4 and A5 mutually attack.
CQ5 also can be posed. If, for example, the critical question is targeted at A1, theanswer to this question could be an argument that Ahab did indeed push Ishmael
motivated by progress, but hoped he would not fall off the bridge.
Obj5a: Ahab pushed Ishmael in q1 to reach q3 motivated by progress, but nature did
not cooperate.
Obj5a attacks (and is attacked by) A1–A5 because, even though it does not directly
explain q2, it does provide us with a reason for believing things went differently
than is postulated in A1–A5.
We have one final objection to consider, which can be made if we accept Obj1a,
rather than taking it to be refuted by Reb1. Suppose that Ahab saw the situation as
q4 rather than q1, perhaps because he thought a tram was about to hit Ishmael. He
therefore, might have pushed Ishmael out of the way, hoping to reach q3, but
unfortunately Ishmael went off the bridge. Now Ahab’s claim would be:
Obj5b: Ahab pushed Ishmael in q4 to reach q3 motivated by safety, but nature did
not cooperate.
This objection is a combination of CQ1 and CQ5, where first CQ1 is answered
positively and then CQ5 is also answered positively. Similar to Obj5a, Obj5b
attacks A1–A5, and also attacks Obj5a.
We have identified a set of arguments and attacks between them, with eachargument associated with a motivating value. We now need to evaluate the status of

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A4

A5

Ishmael:Progress

Ishmael:Oblivion

Reb1
state =
q1

Obj5b

Obj5a

Ahab:
SafetyAhab:
Progress

A1

A2

Ahab:
Progress

Ahab:
RevengeAhabpushedIsmaelObj3b- Safety

Obj1astate =q4

Obj3a
- Safety

A3
Ahab:
OblivionFig. 2 VAF showing arguments, objections and rebuttals

the arguments. To do this we form the arguments into a Value-based Argumentation
Framework (VAF), introduced in Bench-Capon (2003). A VAF is an extension of
the argumentation frameworks (AFs) of Dung (1995). In an AF an argument is
admissible with respect to a set of arguments S if all of its attackers are attacked bysome argument in S, and no argument in S attacks an argument in S. In a VAF anargument succeeds in defeating an argument it attacks only if its value is ranked ashigh as, or higher than, the value of the argument attacked. In VAFs audiences arecharacterised by their ordering of the values. Arguments in a VAF are admissible
with respect to an audience A and a set of arguments S if they are admissible withrespect to S in the AF which results from removing all the attacks which do not
succeed with respect to the ordering on values associated with audience A. A
maximal admissible set of a VAF is known as a Preferred Extension (PE). The VAF
for our example is given below in Fig. 2.
Arrows between nodes denote attack relations. To improve readability, the‘jumping’ (A4–A5) and ‘pushing’ (A1–A3) explanations have been grouped
together. This is represented by a rounded box around the respective arguments; thearrow between, for example, Obj5a and the box surrounding A4–A5, means that
Obj5a attacks both A4 and A5 and vice versa.
Since we have a number of different orderings on values that yield different PEs,
we have a number of competing explanations which we must choose between. In
order to determine which of the arguments A1–A5 is in the preferred extension, wemust first provide an ordering on the motivations of Ahab and Ishmael which willallow one of our arguments to resist the others. In this case, only the most preferredvalue is important. We thus have the following possibilities:3
1.
2.
3.
4.
5.

ahab 2 fR [ fS; P; Ogg (murder)
ahab 2 fP [ fS; R; Ogg (manslaughter (arguably))
ahab 2 fS [ fP; R; Ogg (he did not push)ahab 2 fO [ fP; R; Sgg (mercy killing)ishmael 2 fO [ fS; P; gg (suicide)3

Determining the audience given a VAF and an admissible set is computable in polynomial time
(Bench-Copan 2007), so there are no complexity issues, even in large examples.

123Did he jump or was he pushed?

6.
7.

93

ishmael 2 fP [ fS; Ogg (sacrifice to let ahab pass)ishmael 2 fS [ fP; Ogg (he did not jump)

If we commit to such an ordering of motivating values, the arguments orobjections associated with the stronger motivation defeat the arguments andobjections that rely on one of the other motivations. This commitment tomotivational orderings is important in different ways, depending on the phase of
the case. In the investigation phase of a case, the assumed motivations of the actors
will direct the search for evidence; in the decision-making phase, the assumed
motivations can influence the decision of the judge or jury. Examples of both theseways of using motivational orderings will be given below.
We may well think that the normal priority is 3 and 7. Normally people would
not endanger the life of another to make progress, have no strong feelings ofrevenge and see oblivion as something to be avoided rather than sought. If weindeed assume that S is the most important motivation for both actors, the only
possible acceptable argument is Obj5b, because Obj3a and Obj3b would defeat A1–
A3 and A4–A5, respectively. For Obj5b to be in the preferred extension, however,
we have to suppose that the preceding state was q4 and that Obj1a defeats Reb1. Inother words, if there is evidence that Ishmael’s life was indeed threatened by a
rapidly approaching tram, we might believe that Ahab accidentally pushed him in
the river in an effort to save him. This would be a reasonable explanation, since itneeds no unusual preference, but it does require evidence for the tram, and it isunlikely that this would have been overlooked when the case was prepared. A
variant on Obj5b is saying that Ahab had a false belief: no tram was approaching but
Ahab thought that Ishmael’s life was threatened by a rapidly approaching tram. Thisrequires evidence that supplies reasons for why Ahab had the false belief, or very
convincing testimony from Ahab. It does not, however, require further explanationof Ahab’s ordering of values.
If, however, we assume the situation was indeed q1 and that Ahab was notjustified in believing the situation was any different from q1, we have to say thateither Ahab or Ishmael had an abnormal ordering of motivations.4 The question thenis which of the these abnormal orderings is the most plausible, and the role of thestory is to explain why the agent concerned can be thought to have this particularabnormal ordering.
In the example, it could have been the case that Ishmael preferred O to S or P to
S. The latter is implausible in the extreme: it is difficult to think of a story which
would explain why Ishmael would risk his own life in order to expedite Ahab’s
progress. But we are familiar with the fact that, in some exceptional circumstances,people are suicidal and this does indeed lead them to prefer O to S. So a possible
explanation is that Ishmael preferred O to S and therefore jumped to his death (A5).
While this is not an implausible explanation, we would also want the story to besupported by evidence. So we will investigate further. We might find witnesses whoheard Ishmael say that life was not worth living since his wife left him, or we mightfind other circumstances, for example, that Ishmael was bankrupt or terminally ill.
4

The point that crimes invariably involve an abnormal or deviant motivation is made in Walton andSchafer (2006), which we will discuss further in Sect. 5.

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Assuming that no such evidence can be found, however, this explanation must be
abandoned: we cannot accept A5 with no evidence for Ishmael’s suicidal state. So if
the motivational orderings for Ishmael are not 5 and 6 then he must have the normal
ordering 7. A4 and A5, the explanations in which Ishmael jumped, are now defeatedby Obj3b, which means that Ishmael did not jump, since he valued his own safety.
If Ishmael did not jump, one of the explanations that Ahab pushed Ishmael mustbe true. The question is now with which motivation Ahab did push Ishmael? One
explanation assumes that both Ahab and Ishmael preferred O to S; so Ishmael
wanted to die but could not bring himself to jump so Ahab pushed him to facilitate
his death. The problem with this explanation—apart from the intrinsic implausibility that anyone would agree to such a scheme—is that it would need
substantiation for Ishmael’s suicidal state. But this has already been looked forand not found in connection with A5.
We next consider the ordering for Ahab in which R is preferred to S, that Ahab
pushed Ishmael because he wanted revenge (A2). This explanation is not implausible:
feelings of revenge, although normally of little impact, have been known to increasesufficiently to dominate a person’s thinking. The explanation will, however, have tobe supported by evidence: for example, witness testimonies of people who heard
Ahab threaten Ishmael, or evidence that Ahab had been ruined by Ishmael. If no such
evidence can be found, we should reject A2 and thus also reject a verdict of murder,
which requires Ahab to have killed Ishmael with murderous intent.
Another option is that Ahab preferred P to S: Ahab cared only about his own
progress and so he pushed Ishmael to his death to clear his way (A1). Again,
however, we need to justify the context. For example, assume that Ahab was in arush to get to work; his boss told him he would be fired if he was late for work.
Ishmael was a co-worker of Ahab’s who for some reason had a score to settle with
Ahab. Ishmael then decided to hinder Ahab’s progress so that Ahab would be firedand Ahab pushed Ishmael to his death because he did not want to lose his job. Here,
Ahab’s culpability depends on the risk he took: if, as in A1, it was virtually certainthat Ishmael would die, Ahab could still be prosecuted for murder, even though he
did not explicitly wish for Ishmael to die.
It is also possible, however, that Ahab was merely reckless and that he pushed
Ishmael to clear his way, not realising that Ishmael would fall from the bridge (Obj5a).
While Obj3a should have been sufficient to defeat Obj5a, it is not entirely implausible
to suggest that Ahab miscalculated the risk, and so this attack failed in practice toinfluence his reasoning. There were, after all, safety barriers, although these proved too
low to be effective. Ahab is clearly at fault in that he did not take due account of the risk
to Ishmael. It could now be considered whether the charge should be one ofmanslaughter rather than murder, but we will not delve further into these legal niceties.

5 Related work
The most relevant related work is a paper by Walton and Schafer (2006). They
distinguish motive from intent on the one hand and character on the other, but argue
that identification of a motive can be important evidence of who committed a123

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particular crime. They give an example, taken from Leonard (2001), in which fivenurses had the opportunity to steal some Dermatol, but only one had the motive of
having been addicted to the drug. The motive was important evidence because
Dermatol is not something that anyone would normally want, and there was no
reason to think that it had been stolen for resale.
Walton and Schafer state that motives ‘‘are immediate goals to which an agent isstrongly committed.’’ They use a scheme for practical reasoning in which an action isjustified for an agent if that agent has a goal and the action will realise the goal. They also
give a scheme for value based practical reasoning in which the goal is required to besupported by the agent’s set of values, but even there the action is performed for the sakeof the goal rather than the value as in our scheme. The role of values in their scheme is toprovide support for the claim that the agent has the particular goal, rather than as themainspring of the action as in Atkinson et al. (2006) which is the scheme on which thework in this paper is based. Although Walton and Schafer claim to follow Wigmore(1931), who defined a motive as ‘‘a specific emotion or passion that is likely to lead to a
specific act’’, our notion of motivation used above seems closer to Wigmore’s definition
than their notion of an immediate goal. Wigmore does recognise that ‘‘motive‘‘ can be
ambiguous, but the ambiguity is between the emotion and the event that gave rise to theemotion, not the goal designed to satisfy the emotion. In our example, Ahab’s motivation
might be revenge or the event for which he wants revenge, such as Ishmael’s ruining
Ahab, but not Ishmael’s death, which is the immediate goal. In a second (hypothetical)
example, also taken from Leonard (2001), where a car thief murders someone to prevent
him informing the police, the motivating goal identified by Walton and Schafer is to
avoid punishment. Here, because the motivating goal is different from the event to be
explained, there is more plausibility in calling it the motive, but we would still prefer to
see the motivation as something like freedom which would be demoted by states inwhich the thief was punished. Committing murder is only one way to avoid
imprisonment: paying the potential informant or fleeing the country would bealternative actions with the same motivation.
Walton and Schafer’s account of the reasoning in (Walton and Schafer 2006) is
as follows. They proceed in two steps, a forward reasoning step based on theirpractical reasoning argument scheme, and then an abductive step to establish that
this provides the best explanation. The first step is based on Leonard (2001), which
gives the structure of the reasoning in the car theft example as:–––

EVIDENCE: D stole a car, V was aware of the fact, and V threatened to informthe police.
INFERENCE: D had a motive to prevent V from revealing the theft to thepolice.
CONCLUSION: D purposely killed V to prevent V from revealing the theft tothe police.

Looking at our example in this structure would give:–––

EVIDENCE: Ishmael’s financial dealings bankrupted Ahab
INFERENCE: Ahab had a motive to harm Ishmael
CONCLUSION: Ahab purposely killed Ishmael to get revenge

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Here the role of the evidence is to support the claim that D had V’s silence as agoal. It plays the same role as the value in Walton and Schafer’s value based
practical reasoning scheme. The problem is, however, that the conclusion does notfollow in any real sense from these premises: while we can say that D had anargument to kill V, it says nothing about why D should have found this argument tobe acceptable. At this point Walton and Schafer appeal to abductive reasoning toestablish that this is indeed the best explanation. Unfortunately their informal
presentation gives little clue as to how this is done: while they have a premise that itis the best explanation, no support for it being the best, as opposed to one plausible,
explanation is given unless there is no motive at all for any of the alternatives.
In contrast we reason backwards from the outset to identify the actions which
could have given rise to the current state, and the motivations that might have led anagent to perform them. The resulting argument is then subject to criticalquestioning: in the car theft example, having identified that V might have been
killed by a person wishing to avoid punishment we would consider critical questions
such as CQ10 (it appears that the police found out about the theft anyway), and CQ3(killing someone provides another possibility of punishment), and better ways ofachieving the goal such as paying V to keep silent, or leaving the country. We thenuse a VAF to determine what kind of audience would choose act on the original
argument. In this case D would have to be a person who could not afford to pay V,
who held life cheaply, accepted the risk of getting caught for murder, and who
preferred to kill V to leaving the country. At this point we will look for the evidence
which tells a story to explain why D is that sort of person. This where character andpast actions become relevant: we can infer that D has the required value order either
from past actions suggesting that he has used these preferences before, or from aparticular character manifest in such past actions.
It is this need to explain what is particular about the person that made him act
upon the motive that is lacking from the account presented in Walton and Schafer(2006), even though in their motivation the points showing the importance of this
aspect are well made. For example in discussing an example where a person kills his
daughter for financial gain they write ‘‘only a very few people kill their children, or
step children, for financial gain. Almost all of them can sleep soundly in their beds,
even if the financial burden on their parents is very heavy.’’ But the argument that X
needed money and would inherit on his daughter’s death, so that X had a motive forhis daughter’s death, so we can conclude that X killed his daughter for her money isof precisely the same form as that proposed for the car theft in Leonard (2001) andendorsed by Walton and Schafer. Another interesting example in Walton andSchafer (2006) is taken from the Sherlock Holmes story, The Mystery of the Noble
Bachelor, in which a bride disappears before her wedding reception. There theexplanation turns not on an unusual value ordering (the bride’s previous husband,presumed dead, appears and she leaves with him motivated by love, the desire to
avoid scandal and fear of a charge of bigamy, all of which seem quite normal). In
that case, before the discovery of the deus ex machina in the form of the long lost
husband there was simply no justification acceptable to any reasonable audience,
which led Holmes to look for some additional features in the situation which could
give a reasonable explanation. We believe that our account provides a much better

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way of recognising that crimes are, as stated in Walton and Schafer (2006), ‘‘bydefinition deviant behaviour, what people do not normally do.’’Our use of the AATS also allows us to clarify the notion of intent. Because theargument only motivates the particular agent’s component of a joint action, the statereached may not be the state that the agent wished to reach. For example the carthief may have hit V intending to intimidate him into silence, but through some
misfortune the blow proved fatal. Note that in Leonard’s formulation of the example
followed in Walton and Schafer (2006), the same evidence and inference would
equally license the conclusion D unintentionally killed V while attempting to
intimidate V to prevent V from revealing the theft to the police.
Finally, we should note that the primary concern of Leonard (2001), also touched
on in Walton and Schafer (2006), is the legal admissibility of motive evidence. We
do not want to go into these legal issues here: our aim is only to provide amechanism for rationally explaining actions in terms of their motivation.
Other relevant works in AI and Law are other accounts of justification through acausal story such as Bex et al. (2006, 2007) and Thagard (2004). In the next section
we will discuss how we see our account as extending and developing the work of
Bex et al. Similar remarks could also be made about Thagard (2004).

6 Concluding remarks
In this paper we have shown how we can use an argument scheme for practicalreasoning abductively in order to generate a set of explanations for a given state ofaffairs in terms of the motivations of the agents who brought it about. Typically wewill have competing explanations which we can resolve by considering the priorities
amongst motivations of the agents concerned. If the normal default priorities do not
explain the situation, we must find an ordering which does, and then justify this bymeans of a story which explains how the agent came to have this ordering. This willin turn guide search for further evidence to anchor this explanatory ‘‘back story’’.
Explanation in terms of motivations is important, both to make our chosen story
plausible and, in some cases, to determine the degree of culpability of the agents.
We have thus given motivations a clear and separate place in evidential reasoning
about actions. This distinction has been recognised in other work such as Modgil(2006), where a formal treatment is proposed. There Modgil defines a framework inwhich conflicts over value orderings are reasoned about at a separate meta-level.
This means that conflicts between arguments may be resolved through reasoning
about preference orderings at a different level, resulting in a hierarchicalargumentation framework. For a rule based treatment in the style of Bex et al.
(2007), the need for meta-level reasoning will require rules of an additional kindwhich allows us to conclude that an agent has a particular value preference, such as
if Ishmael ruined Ahab then Ahab prefers Revenge to Safety.
Our approach is firmly grounded on a formal structure provided by an AATS.
The AATS makes the underlying model of the stories less ad hoc than in previous
approaches which rely on a set of rules. In addition, when compared to Bex et al.
(2007) the problem formulation critical questions stated formally in Sect. 3 give us

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a better opportunity to reason about the causal model underlying the story. A
disadvantage is perhaps that the causal rules are not as explicit as in Bex et al.
(2007), since the causal relations are now implicit in the transition system. We
believe, however, that the possibilities for richer reasoning about motivation morethan compensate for this.
For future work we intend to apply the above analysis to provide a rule based
representation so as to facilitate computation. Taking as a starting point the rules
found in Bex et al. (2007), we will need three kinds of rules: rules describing
physical causation, such as ‘‘If q1 and Ahab pushes Ishmael then Ishmael is in theriver’’; rules to describe motivation ‘‘If q1 and Ahab pushes Ishmael revenge is
promoted’’; and rules to determine value preferences such as that mentioned above
to explain why Ahab prefers revenge to safety. The first kind can be straightforwardly derived from the AATS: each transition represents one such rule. The second
kind can also be extracted from the AATS: each label on a transition represents one
such rule. The third kind of rule is not present in the AATS, since these rules belong
to the meta-level reasoning associated with the VAF: these rules will require thekind of analysis we used in determining the appropriate audience to use for Ahab
and Ishmael. An excellent case to which these ideas can be applied is the Claus vonBülow case which is described by Thagard in (2004). There Thagard gives a fulldescription of the case as well as representations of the cases presented at both theinitial trial and the appeal. Using this example will allow direct comparisons with
the Explanatory Coherence and Bayesian Network approaches used by Thagard, as
well as the approach of Bex et al. (2007).

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