Reinterpreting arguments in dialogue: anapplication to evidential reasoning
1

Floris Bex1 & Henry Prakken1,2
Institute of Information and Computing Sciences, Utrecht University, The Netherlands /2
Faculty of Law, University of Groningen, The Netherlands

Abstract. This paper presents a formalisation of two typical legal dialogue moves in aformal dialogue game for argumentation. The moves concern two ways of reinterpreting a general rule used in an argument, viz. by ‘unpacking’ and ‘refining’ the rule. The
moves can be made not only by the user but also by the attacker of the rule, in order toreveal new ways to attack it. The new dialogue game is illustrated with examples fromlegal evidential reasoning, in which these types of moves are particularly common.

1 Introduction
Generalisations play a key role in reasoning about evidence [17, 18, 16]. Fact finders andtriers of fact mostly leave them implicit in their arguments and decisions, but they are necessary to explain why certain evidence supports a certain conclusion. They are also dangerous,
however, since their quality is often dubious. Therefore, any model of rational evidential
reasoning should allow for the building of arguments with generalisations but also for theircritical testing. Such testing both has inferential and dialogical aspects.
As for inference with generalisations, we have argued in [15, 5] that this can be modelled
as defeasible argumentation, so that part of the critical testing can be modelled as the construction and comparison of arguments and counterarguments. However, our previous work
did not address the dynamic aspects of the critical testing process. As argued by e.g. [18],in evidential reasoning generalisations (which they call “anchors”) should be reshaped in a
critical inquiry. The aim of this paper, which is based on [4], therefore is to embed defeasible
inference with generalisations in a dialogue game where such reshaping can take place.
Dialogue games for argumentation are not new in AI & Law [8, 10, 3, 14]. The present
contribution is to add some new dialogue moves that are typical in legal dialogues with unwritten rules, such as commonsense generalisations. From the examples of [18] two patterns
emerge in particular: ‘unpacking’ and ‘refining’. Unpacking is when a single generalisationis replaced by a chain of generalisations with the same start and end point as the original one.
For instance, [16], the generalisation “if a witness testifies that P is the case then usually P
is the case” could be unpacked into “if a witness testifies that he observed P then usually
he believes that he observed P ”, “if a witness believes that he observed P then usually hissenses gave evidence of P ” and “if a witness’ senses gave evidence of P then usually P isthe case”. Refining is when a generalisation is given an extra condition. For example [18],“two witnesses who agree usually tell the truth” can be argued to have an extra condition thatthe witnesses did not confer. By making this explicit with “two witnesses who agree and did
not confer usually tell the truth”, the extra condition becomes open for challenge. Note thatrefining a generalisation is not the same as arguing for an exception to it. When refining ageneralisation of one’s opponent, one tries to achieve that the opponent must prove an additional condition before the generalisation can be applied. By arguing for an exception to a
generalisation one instead tries to prevent its application by proving that an exception holds.
Unpacking and refining are not matters of inference but of dialogue. They reinterpret anargument moved in a dialogue and thus they crucially differ from current dialogue models,
where an argument can only be modified by ‘backwards’ extending it with an argument for
one of its premises. Another difference with current models is that these two moves can bemade not only by the user but also by the attacker of the argument, who may thus reveal additional attacking points (for instance, if the attacker has evidence that the witnesses conferred).
In this paper we want to show how these two dialogue moves about generalisations canbe added to dialogue models of legal reasoning. A secondary aim is to propose a new way of
formulating dialogue games for argumentation, which is arguably better suited for evidential
reasoning (as will be explained in Section 3). We do not intend to model an actual legal
procedure but a rational theory of how two antagonists can debate the tenability of a claim.
[12] have previously formalised similar reasoning phenomena in the context of reasoning
with and about case rationales; see also [2] for informal discussions of relevant examples.
We draw inspiration from this work but also add to it, since [12] only focus on how moves
like unpacking change the set of arguments of a dispute; they do not embed such moves ina model with other speech acts. Also, of the two moves that we are interested in, they only
model ‘unpacking’ (although they model four other moves about case rationales).
The rest of this paper is organised as follows. In Section 2 we present two informal examples of dialogues with unpacking and refinement moves, after which in Section 3 we discuss
some formal preliminaries about logic and dialogue systems. In Section 4 we present ourformal dialogue game and we apply it in Section 5 to the example dialogues of Section 2.
2 Motivating examples
In this section we give two example dialogues that involve unpacking and refinement ofgeneralisations, both taken from [18] (the generalisations involved are displayed in boldface).
The dialogues are not intended to be complete; other ways of explaining or attacking thegeneralisations may be possible.
The first example concerns the Rijkbloem case, in which Danny Rijkbloem was prosecuted for shooting the father of his girlfriend, Nicole. The only other possible suspect was
Nicole’s mother, who was also present at the shooting. However, she was not prosecuted.
prosecution: Rijkbloem shot the father, because mother and daughter testified that Rijkbloem
shot the father, and we all know that if two people testify that something is true then it is true.
(making a claim and providing reasons for that claim)
If witness A testifies that ϕ and if witness B testifies that ϕ, then ϕ is true

defense: Your argument is based on the generalisation “if two people testify that somethingis true then it is true”. This only holds if the two witnesses do not profit from lying.(exposing
a possible “hidden condition” of the generalisation)
If witness A testifies that ϕ and if witness B testifies that ϕ and the witnesses do not profit from lying, thenϕ is true

defense: What makes you think that the mother and daughter in this case do not profit fromlying? (disputing the newly exposed premise)
prosecution: I see your problem. Nevertheless, if two witnesses profit from lying but madetheir statements separately then usually they still speak the truth (because two people rarelytell exactly the same lie) (refining the original generalisation in an alternative way).
If witness A testifies that ϕ and if witness B testifies that ϕ and the witnesses profit from lying and thewitnesses made their statements separately, then ϕ is true

The second example concerns the Caroll case, involving an IRA attack on Australian
tourists in Roermond, The Netherlands in 1990. A witness had seen a car speeding past withtwo people in it near the crime. This happened at dusk, so the witness may not have beenable to produce a reliable identification of the persons he saw in the car. However, the witness
later saw one of the suspects, a mr. Carroll, on the news and in the newspaper. So when thewitness was called to testify whether he saw Carroll, he had already seen Carroll in his roleas the suspect after the crime and this arguably influenced his positive identification.
prosecution: The witness saw Carroll at the crime, because the witness remembers he saw
Carroll at the time of the crime and if a person remembers something then it is true (making
a claim and providing reasons for that claim)
If witness A remembers that ϕ, then ϕ is true

defense: You say that “if a person remembers something then it is true”. What you actuallymean is that if a witness remembers something, then at the time of the crime the witness
believed that it is true, and that if the witness believed something at the time of the crime,then it was true at that time. (unpacking the original generalisation)
If witness A remembers that ϕ, then at the time of the crime the witness believed ϕ, if at the time of thecrime the witness believed ϕ, then ϕ is true

defense: In my opinion, your witness did not at the time of the crime believe he saw Carroll. I
say this because I heard that the witness did not have a good look at the person near the crime.
Furthermore, your witness saw Carroll on TV after the crime, and this may have influenced
his memories. (providing a counterargument to the unpacked generalisation)
3 Formal preliminaries
3.1 Dialogue games for argumentation
Dialogue games formulate principles for coherent dialogue, and coherence depends on thegoal of a dialogue. The goal of argumentation dialogues is fair and effective resolution ofa conflict of opinion. Formal dialogue games have a topic language L t with a logic L, anda communication language Lc with a protocol P . Dialogue games also have commitment
rules, which specify the effects of an utterance from Lc on the propositional commitmentsof the dialogue participants. For instance, claiming or conceding ϕ commits the speaker to ϕ
and a retracting ϕ removes ϕ from the speaker’s commitments. Commitments can be used toconstrain the allowed moves, for example, to disallow moves that make the speaker’s commitments inconsistent. They can also be used to define termination and outcome of a dialogue. In
the dialogues we are concerned with here a proponent and an opponent argue about a single
dialogue topic t ∈ Lt . Proponent aims to make opponent concede t while opponent aims to
make proponent give up t. So a plausible termination and outcome rule is that a dialogue
terminates with a win for proponent if opponent is committed to t, while it terminates with awin for opponent if proponent is not committed to t any more. In addition, for nonterminateddialogues a notion of the current winner can be defined to control turntaking and relevanceof moves.
In the literature several ways can be found to formulate a protocol for dialogue. Following[11, 8, 3, 6], our idea is that during a dialogue the participants implicitly build a ‘theory’ towhich the logic L can be applied to determine the current winner. More precisely, L will beapplied to all ‘current’ premises of either of the participants that are not challenged by theother party. Proponent is then the current winner if the joint theory implies the dialogue topic
t while opponent is the winner otherwise. Note that with a player being a current winner wemean no more than that the current state of the dialogue favours that party; the current winner
can change many times during a dialogue. The notion is used to control turntaking, with arule that a player is to move until he has succeeded in becoming the current winner (cf. [11]).
The precise form of our dialogue game is, to our knowledge, new. The idea is that thetheory constructed during a dialogue has the form of an inference graph and that the legalityof moves and the outcome of dialogues is defined in terms of this graph. The main reason
to take this approach is the resemblance of inference graphs to Wigmore charts, which have
proven useful in the semiformal analysis of evidential reasoning [1].
3.2 Defeasible inference
Since reasoning about evidence is defeasible, the logic L of our dialogue system must be anonmonotonic logic. Following our earlier work in [15, 5] we choose for an argument-based
logic, combining some elements from [13] with any suitable notion of skeptical inference.
Pollock augments the inference rules of classical logic with a set of defeasible inference
rules called ‘prima facie reasons’. Arguments can be constructed by chaining reasons into
trees, starting from given input information. Each defeasible reason comes with one or more
undercutters, which specify the circumstances under which the inference is not warranted.
Accordingly, a defeasible argument can be defeated in two ways. It can be rebut with anargument for the opposite conclusion, while it can be undercut with an argument why a prima
facie reason does not apply in the given circumstances. To be successful, an attack should beof a certain strength. In the present paper, we will not discuss issues of strength and therefore
implicitly assume a given measure of relative strength between arguments.
Since the present focus is on reasoning with empirical generalisations, we will consider
just one of Pollock’s prima facie reasons, viz. a qualitative version of the statistical syllogism(qualitative since in legal cases numerical probabilities are usually not available). Let ⇒ bea connective from Lt such that P ⇒ Q intuitively reads as“If P then usually Q”. The onlyinference rule that can be applied to formulas of this form is modus ponens. Furthermore, let
P1 , . . . , Pn  Q) be shorthand for the metalevel expression “{P1 , . . . , Pn } is a prima faciereason for Q”. Finally, let dϕe denote the translation of a metalevel expression ϕ in L t . Then
P, P ⇒ Q  Q (Statistical Syllogism)
There are two main undercutters for this reason, viz. weak and strong subproperty defeat,which say that if for a special case of the antecedent the conditional does not hold (weak) ora conflicting conditional holds (strong), then the original generalisation cannot be used:
weak s.d.: P ∧ R, ¬(P ∧ R ⇒ Q)  ¬dP ⇒ Q  Qestrong s.d.: P ∧ R, P ∧ R ⇒ ¬Q  ¬dP ⇒ Q  QePollock also defines other reasons relevant for evidential reasoning, such as perception, memory and temporal persistence. Moreover, in [5] we formulated reasons for expert and witness
testimonies. In this paper, however, we will regard all these reasons as empirical generalisations, so that they are applied with the statistical syllogism. The reason for this is thatinference rules are fixed and cannot be reinterpreted during a dialogue. In [5] we alreadybriefly discussed this issue, and [13] also mentions that his other reasons can be reduced tothe statistical syllogism. Finally, we deviate from Pollock’s logic in one respect. Pollock requires that all arguments are ultimately grounded in a fixed and undisputable set INPUT of
propositions. However, in order to allow for stepwise backwards construction of an argument
during a dialogue, we make this set relative to a dialogue stage.
4 The new dialogue game defined
In this section the dialogue game is defined. Because of space limitations the definitions willin some places be semiformal. Below ϕ is a well-formed formula of Lt and A is an argumentof L. Dialogues take place between two players, a proponent pro and an opponent con. The
variable p ranges over the players, so that if p is one player, then p̄ denotes the other player.
Also, the variable s denotes the speaker of a move. The topic language L t is that of first-orderlogic augmented with the defeasible connective introduced above, and the logic L is thevariant of Pollock’s system introduced above. As for notation, for any argument A, prem(A)is the set of leaves of A (its premises) and conc(A) is the root of A (its conclusion).
The communication language Lc consists of the following locutions:• argue A. The speaker states an argument, either in support of the dialogue topic, or to
support a premise of another argument, or by way of counterargument.
• why ϕ. The speaker challenges that a premise ϕ of an argument is the case and asks forreasons why it would be the case.
• concede ϕ. The speaker admits that a premise or conclusion ϕ of an argument is the case.
• retract ϕ. The speaker declares that he is not committed (any more) to ϕ.
• explain (ϕ ⇒ ψ = A). The speaker reinterprets a generalisation used in an argument“ϕ, ϕ ⇒ ψ, so ψ”. A is an argument in L without ϕ ⇒ ψ such that ϕ is among thepremises of A and ψ is the conclusion of A.
The first four locutions are well-known from the literature, but the explain locution is new. It
captures unpacking since it allows the argument A to be of arbitrary complexity, as long as
ψ is its conclusion and ϕ is among its premises. It also captures refining, since it allows A tohave an additional condition besides ϕ.
A dialogue is now a sequence of utterances of locutions from L c . Each utterance is calleda move and a maximal sequence of moves in a dialogue by the same player is a turn. The
speaker of a move m is denoted by s(m). For any dialogue d = m1 , . . . , mn , . . ., the sequence
m1 , . . . , mi is denoted by di , where d0 denotes the empty dialogue.
4.1 Commitments and disputations
In our setup, dialogue moves have two kinds of effects, viz. on the players’ propositional
attitudes and on the dialogue’s inference graph. As for the propositional attitudes, we keep
track both of the players’ commitments and their disputations, i.e., the propositions they havedisputed. At the start of a dialogue these sets are empty. Since each proposition occurs only
once in the inference graph, these sets in fact label the nodes in the graph. Below C p (d) andDp (d) stand for the commitments, respectively, disputations of player p after dialogue d.
We first specify the commitment rules (only changes will be specified).
• Cs (d, argue A) = Cs (d) ∪ prem(A) ∪ {conc(A)}• Cs (d, concede ϕ) = Cs (d) ∪ {ϕ}• Cs (d, retract ϕ) = Cs (d)/{ϕ}It remains to define how an explain move affects the commitments. It is the only speech act
that can also commit the hearer.
• Cs (d, explain ϕ = A) = Cs (d) ∪ prem(A) ∪ {conc(A)} if ϕ ∈ Cs (d)• Cs̄ (d, explain ϕ = A) = Cs̄ (d) ∪ prem(A) ∪ {conc(A)} if ϕ ∈ Cs̄ (d).
An explanation only commits its mover to the premises and conclusion of the new argument
if he was already committed to the original generalisation. This is to allow for explaining theother player’s generalisation in order to attack it. Such an attack is impossible if the player
automatically commits himself to the explanation, as then he attacks one of his own commitments. Note that the commitment rule for explain can commit the hearer to explanations withwhich he disagrees. This is not a problem, however, since the hearer can always retract his
commitment and/or move an alternative explanation (as in the Rijkbloem case).
We must also keep track of the proposition a player has disputed, since arguments withdisputed premises will not count in determining the current winner. We define the following
disputation rules (again only changes are indicated).
• Ds (d, why ϕ) = Ds (d) ∪ {ϕ}• Ds (d, concede ϕ) = Ds (d)/{ϕ}• Ds (d, explain ϕ = A) = Ds (d) ∪ {ϕ} if ϕ 6∈ Cd (s)The first two rules are obvious. The third rule says that if a player explains a generalisation
to which only his adversary is committed, he disputes the original generalisation ϕ. The
rationale of this rule is that otherwise there would be no point in explaining the generalisation
to attack the explanation, as there would still be the original undisputed argument.
4.2 The inference graph of a dialogue
Recall that in our setup the dialogue participants jointly build a theory during a dialogue,
to which the logic L can be applied to identify the ‘current winner’. We implement theseideas by formulating the jointly built theory as an inference graph [13] which, as remarked
above, has a close correspondence with Wigmore charts. The nodes of an inference graph arepropositions and the links are of two kinds. Support links instantiate a deductive inference
rule or prima facie reason and thus capture inferential dependencies between nodes. If a
support link operates on more than one proposition it is in fact an AND-link (depicted below
with an arc). Defeat links between nodes reflect relations of defeat between the correspondingarguments for the nodes. The set INPUTd of current premises of d is defined as the set of allleaf nodes in the inference graph of d.
Actually, our inference graph differs from Pollock’s in one respect. In Pollock’s graphs, if
a sentence appears in more than one argument, it also appears more than once in the inferencegraph. In our graph each sentence appears only once, to avoid duplication of sentences. Thisrequires that, unlike in [13], defeat links are labelled with the arguments that are in the defeat
relation. Thus there can be more than one defeat link between the same two nodes.
Now the idea is that each move in a dialogue d is regarded as an operation on the inferencegraph Gd associated with d: arguments add nodes, support links and/or defeat links to thegraph, while all moves can affect the commitments and disputations stored as labels of thenodes. Because of space limitations we omit the formal definitions, which should be obvious.
4.3 The current winner of a dialogue and turntaking
To apply a Dung-style inference notion to a graph Gd to define the current winner of dialogued, we define the set ArgsGd of arguments in Gd . Recall that arguments are trees of inferences,so this is the set of all maximal subgraphs of Gd in which no node has more than one incoming
support link. The set DefArgsGd of all defended arguments in Gd is the set of all arguments
in ArgsGd of which all premises (= leaf nodes) are in the commitments of at least one player
and in the disputations of neither player. Thus the defended arguments are all arguments withno disputed or retracted premises. As proven in [9], Pollock-style defeat can be equivalently
transformed to Dung-style defeat, so that DefArgsGd corresponds to a Dung-style argumentation framework AFd to which any of [7]’s dialectical inference notions can be applied. To
make sense in the present context, it must be a skeptical inference notion (in our examples
below the choice does not matter). We then say that proponent currently wins a dialogue d if
the dialogue topic t is skeptically implied (according to the chosen inference notion) by AF dand that opponent currently wins d otherwise. Note that the protocol implements a ‘burden ofquestioning’ principle [19], as the current winner of a dialogue is calculated on the basis of
all commitments of one party that are not challenged by the other party.
With the notion of a current winner, a turntaking rule can be defined as follows: T is afunction that for each dialogue returns the players-to-move, such that T (d 0 ) = pro and else
T (d) = p̄ iff p currently wins d. Thus the ‘resources’ are always allocated to the losing side,which according to [11] promotes a fair and effective inquiry.
4.4 The protocol
The protocol P specifies the allowed moves at each stage of a dialogue. Its formal definitionis as follows. Together the conditions below imply that each dialogue starts with an argument
by proponent; its conclusion is the topic t of the dialogue.
For all moves m and dialogues d it holds that m ∈ P (d) if and only if all of the following
conditions are satisfied:
1 T (d) = s(m)2 m was not already moved in d by the same player
3 Cs (d, m) 6` ⊥4 Cs (d, m) ∪ Ds (d, m) = ∅5 If M is an argue A move (where ϕ is A’s conclusion), then extending G d with A adds atleast one node or link to Gd , and• either d = d0
• or ϕ ∈ Gd and ϕ ∈ Dd (s̄)• or ϕ defeats a node in Gd .
6 If m is a why ϕ move, then• ϕ ∈ INPUTd• ϕ 6∈ Ds (d)• Cs (d) 6` ϕ
7 If m is a concede ϕ move, then ϕ ∈ Gd and ϕ 6∈ Cs (d)8 If m is a retract ϕ move, then ϕ ∈ Cs (d)9 If m is an explain (ϕ = A) move, then ϕ ∈ Gd and extending Gd with argument A adds
at least one node or link to the graph
10 t ∈ Cpro (d) and t 6∈ Ccon (d).
The first two conditions say that only the player-to-move can make allowed moves and thata player may not repeat his moves. Conditions (3) and (4) regulate the players’ logical anddialogical consistency. Conditions (5)-(9) specify specific conditions for each speech act type.
Condition (5) ensures that an argue move is only moved if it is the initial move, or if itprovides a reason for a disputed proposition, or if it attacks another argument in the graph.

Also, the argument moved has to extend the graph, so that arguments that are already in thegraph cannot be repeated. Condition (6) says that a proposition ϕ can be challenged only
if ϕ is a leaf node in the inference graph, is not already disputed by the speaker and does
not classically follow from the speaker’s commitments. Condition (7) ensures that a player
concedes a proposition only if it is in the graph and the player is not already committed toit. Condition (8) says that a player can only retract a proposition to which he is committed.
Condition (9) ensures that only generalisations that are already in the graph can be explained
and that an explanation extends the graph. Finally, condition (10) regulates termination of
dialogues. It implies that a dialogue terminates if either the proponent is no longer committed
to the dialogue topic or the opponent has conceded it: then no further move is allowed.
The protocol ensures relevance of moves as follows. To start with, only propositions thatwere moved as premise of an argument can be challenged, conceded or retracted. Furthermore, a new argument must either be a counterargument to an argument in the inferencegraph or a support for a challenged node in the graph, so players cannot just move any argument. Finally, players cannot repeat their moves.
5 The examples formalised
We now formalise the examples of Section 2 in our dialogue system. In displaying inferencegraphs, parts involving retracted nodes will not be shown and undefended arguments will
be depicted with dotted lines. The sets Ci (p) and Di (p) stand for player p’s commitments,
respectively, disputations after dialogue di .
5.1 The Rijkbloem case
The prosecution, who is the proponent in this example, claims that Rijkbloem shot the father(the dialogue topic) and brings forward the mother’s and the daughter’s witness testimonies.
- pro1 :argue (1): mother’s testimony that Rijkbloem shot the father,(2): daughter’s testimony that Rijkbloem shot the father,r1 : (witness A testifies that ϕ) ∧ (witness B testifies that ϕ) ⇒ ϕ,so(3): Rijkbloem shot the father

(1)

(3)

pro1: the initial argument is moved(2)

C1(pro) = {(1), (2), r 1, (3)}C1(con) = {}D1(pro) = {}D1(con) = {}r1

The only argument in Gd1 concludes to the dialogue topic, so the turn shifts to con, who now
first adds a hidden condition to r1 and then disputes it.
- con2 :explain r1 =(1): mother’s testimony that Rijkbloem shot the father,(2): daughter’s testimony that Rijkbloem shot the father,(4): mother and daughter do not profit from lying,r2 : (witness A testifies that ϕ ∧ witness B testifies that ϕ ∧ the witnesses do not profit from lying) ⇒ ϕso (3): Rijkbloem shot the father

The generalisation r1 is now disputed. Note that if a player provides an explanation, theoriginal generalisation remains in the graph, to allow for alternative explanations of the samegeneralisation (as pro is about to give). Also note that through moving the above explanation,
con has committed pro to the premises and conclusion of the new argument.
Next, con makes a why move, disputing the new premise:- con3 : why mother and daughter do not profit from lying(3)

after explaning r 1, con disputes (4)

(2)
r2

(4)

r1

(1)

C3(pro) = {(1), (2), r 1, (3), r 2, (4)}C3(con) = {}D3(pro) = {}D3(con) = {r1, (4)}The graph now contains two arguments for the dialogue topic. Although they have no counterarguments, both have a disputed premise, so con is the current winner and the turn shifts backto pro (note that before con3 the argument with (4) was defended, so pro was still winning).
pro disagrees with con’s way to explain r1 and now explains r1 in an alternative way bysaying that even if the witnesses profit from lying, if they make their statements separately,
then they still usually speak the truth. To make this move, pro first has to retract (4) and r 2 .
- pro4 : retract (4): the witnesses profit from lying- pro5 : retract r1 : (witness A testifies that ϕ ∧ witness B testifies that ϕ ∧ the witnesses do not profit fromlying) ⇒ ϕ
pro retracts (4) and r2...

(3)

(1)

(2)

C5(pro) = {(1), (2), r1, (3)}C5(con) = {}D5(pro) = {}D5(con) = {r1, (4)}r1

By retracting these propositions, pro indicates that he does not accept con’s explanation of
his original generalisation r1 . pro must now provide a new argument for (3), as his original
argument using r1 is still disputed. pro moves an alternative explanation to r1 :- pro6 :explain r1 =(1): mother’s testimony that Rijkbloem shot the father,(2): daughter’s testimony that Rijkbloem shot the father,¬(4): mother and daughter profit from lying,(6): mother and daughter made their statements separately,
r3 : (witness A testifies that ϕ ∧ witness B testifies that ϕ ∧ the witnesses profit fromlying ∧ the witnesses made their statements separately) ⇒ ϕso (3): Rijkbloem shot the father
(3)
r3
(5)

...and explains r 1

(2)
not
(4)

r1

C6(pro) = {(1), (2), r 1, (3), not(4), (5), r 3}
C6(con) = {}D6(pro) = {}D6(con) = {r 1, (4)}(1)

By incorporating ¬(4) as a premise, pro effectively concedes that mother and daughter had
reason to lie. Nevertheless, the graph now again contains a defended argument for the dialogue topic and since it has no counterarguments, pro is the current winner.
5.2 The Carroll case
The prosecution, who argues for the fact that Carroll was one of the two men in the car, gives
a simple argument for the fact that Carroll was near the crime.
- pro1 :argue (1): the witness remembers he saw Carroll at the time of the crime,
r1 : a witness remembers ϕ ⇒ ϕ,so(2): the witness saw Carroll

propro11:: thethe initial
initial argument
argument isis moved
moved

(2)

(1)

C
(pro)1(pro)= {(1),= (2),} r 1}
Comm{(1),r 1(2),1C(con)1(con)= {} = {}
Comm
1
D
(pro)
= {}= {}
Disp
(pro)
1
1
D
(con)
= {}= {}
Disp
(con)
1
1

r1

The opponent now reveals a new attacking point by by explaining r 1 .
- con2 :explain r1 =(1): the witness remembers he saw Carroll at the time of the crime,
r2 : a witness remembers ϕ ⇒ at the time of the crime the witness believed ϕ,so (3): at the time of the crime the witness believed that he saw Carroll,r3 : at the time of the crime the witness believed ϕ ⇒ ϕso (2): the witness saw Carroll
(2)

r1

(1)

con
con explains
explains pro's
pro's generalisation
generalization rr11.
C(pro)2(pro)= {(1),= (2),, r2,rr13,}r2, r3}
Comm{(1),(3),(2),r 1(3),2C(con)2(con)= {} = {}
Comm
2
Disp
(pro)
=
{
}
D
(pro)
=
{}
2
2
Disp
(con)
D
(con)
= {r=1} {r 1}
2
2

(3)

r2

(1)

r3

Although pro is still winning, con can now move a counterargument.
- con3 :argue (4): the witness did not have a good look at the person near the crime,(5): the witness saw Carroll as a suspect on the news after the crime,
r4 : (a witness did not have a good look at person P1 during the crime∧ witness saw person P2 as a suspect after the crime)⇒ at the time of the crime the witness did not believe that he saw person P 2
so¬(3): at the time of the crime the witness did not believe that he saw Carroll(2)
Theexplained
explainedargument
argumentisisdefeateddefeatedby
byaanodenodeinincon's
con'snewnewThe
argument.
argument

r1

r2

not
(3)

(3)

(1)
r3

(1)

r4

(4)

(5)

{(1),
(2),
CComm
(pro) 3=(pro){(1),=(2),(3),
r 1,(3),
r2, r31}, r2, r3}3(con){(4),not(3),(5), notCComm(con)3={(4),=(5),r4}(3), r4}3(pro) = { }
DDisp
(pro)
3 = {}
3(con) = {r 1}
DDisp
(con)
3 = {r 1}
3

The arguments for (3) and ¬(3) are both defended. Supposing they are equally strong, they
defeat each other (indicated by the grey shadings) so that the dialogue topic is not skeptically
implied by AFd3 . So con is the current winner and the turn shifts back to pro.
6 Conclusion
In this paper we have shown how two typical moves in legal dialogues can be regulated ina formal dialogue game. Both moves reinterpret rather than extend an argument moved in a
dialogue. Such reinterpreting moves have so far received little attention in AI & Law research.
For the theory of evidential reasoning we have aimed to provide a better understanding of two
aspects of the critical testing of evidential arguments. For AI & Law models of legal dialogue
we have provided a possible formalisation of two types of dialogue moves that are importantin legal reasoning. The relevance of our contributions is not confined to evidential reasoning
since, as shown by [12] and [2], these moves also occur about other types of unwritten rules,
such as precedent rationales.

Acknowledgements
This research was partially supported by the EU under IST-FP6-002307 (ASPIC). We thankBart Verheij for his useful comments on earlier versions of the ideas reported in this paper.
References[1] T.J. Anderson and W. Twining. Analysis of Evidence. How to Do Things with Facts Based on Wigmore’s
Science of Judicial Proof. Little, Brown and Company, Boston, MA, 1991.
[2] K.D. Ashley. An AI model of case-based argument from a jurisprudential viewpoint. Artificial Intelligenceand Law, 10:163–218, 2002.
[3] T.J.M. Bench-Capon, T. Geldard, and P.H. Leng. A method for the computational modelling of dialecticalargument with dialogue games. Artificial Intelligence and Law, 8:233–254, 2000.
[4] F.J. Bex. Explaining anchors: laying the foundations for an evidential sense-making system. Master’s
thesis, Cognitive Artificial Intelligence, Utrecht University, Utrecht, 2004.
[5] F.J. Bex, H. Prakken, C. Reed, and D.N. Walton. Towards a formal account of reasoning about evidence:argumentation schemes and generalisations. Artificial Intelligence and Law, 12, 2004. to appear.
[6] G. Brewka. Dynamic argument systems: a formal model of argumentation processes based on situation
calculus. Journal of Logic and Computation, 11:257–282, 2001.
[7] P.M. Dung. On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic
programming, and n–person games. Artificial Intelligence, 77:321–357, 1995.
[8] T.F. Gordon. The Pleadings Game. An Artificial Intelligence Model of Procedural Justice. Kluwer Academic Publishers, Dordrecht/Boston/London, 1995.
[9] H. Jakobovits. On the Theory of Argumentation Frameworks. Doctoral dissertation Free University Brussels, 2000.
[10] A.R. Lodder. DiaLaw. On Legal Justification and Dialogical Models of Argumentation. Law and Philosophy Library. Kluwer Academic Publishers, Dordrecht/Boston/London, 1999.
[11] R.P. Loui. Process and policy: resource-bounded non-demonstrative reasoning. Computational Intelligence, 14:1–38, 1998.
[12] R.P. Loui and J. Norman. Rationales and argument moves. Artificial Intelligence and Law, 3:159–189,
1995.
[13] J.L. Pollock. Cognitive Carpentry. A Blueprint for How to Build a Person. MIT Press, Cambridge, MA,
1995.
[14] H. Prakken. Modelling reasoning about evidence in legal procedure. In Proceedings of the Eighth International Conference on Artificial Intelligence and Law, pages 119–128, New York, 2001. ACM Press.
[15] H. Prakken. Analysing reasoning about evidence with formal models of argumentation. Law, Probability
and Risk, 3:33–50, 2004.
[16] D.A. Schum. Evidential Foundations of Probabilistic Reasoning.
Evanston. Illinois, 1994.

Northwestern University Press,[17] W. Twining. Necessary but dangerous? Generalisations and narrative in argumentation about “facts” incriminal process. In M. Malsch and F. Nijboer, editors, Complex Cases. Perspectives on the Netherlands
Criminal Justice System, pages 69–98. Thela Thesis, Amsterdam, 1999.
[18] W.A. Wagenaar, P.J. van Koppen, and H.F.M. Crombag. Anchored Narratives. The Psychology of Criminal
Evidence. St. Martins Press, New York, 1993.
[19] D.N. Walton. Is there a burden of questioning? Artificial Intelligence and Law, 11:1–43, 2003.