A common assumption for argumentation dialogues is that any argument exchanged is complete, i.e. its premises entail its claim. But in real world dialogues, agents commonly exchange enthymemes arguments with incomplete logical structure. This paper introduces a dialogue system that handles misunderstandings that may occur when there is a mismatch between what a participant excludes from an argument and what her interlocutor assumes to be the missing information. We also introduce a mechanism for determining the acceptability status of the dialogue moves so that, under certain conditions, the status of moves made during a dialogue conforming to our system, corresponds with the status of arguments in the Dung argument framework instantiated by the contents of the moves made at that stage in the dialogue.
In approaches to structured argumentation, arguments typically consist of a conclusion
deductively and/or defeasibly inferred from some premises [
Alice first asserts an argument (1), to which Bob responds with a counterargument (2). Alice then moves an enthymeme (3) for an argument that she believes counters Bob’s argument. Note, the enthymeme Alice presents does not explicitly contradict anything that Bob has said, and so Bob asks for clarification (4) on what he is meant to infer from this enthymeme, which Alice provides (5)1. However, Alice’s clarification still does not explicitly contradict anything Bob has said. Since Bob does not understand why Alice’s enthymeme is relevant to what he said, he asks Alice to explain what she assumed (6). Alice explains the assumption she had made (7), which Bob then corrects (8). Notice that Bob’s assertion in (2) is an enthymeme of his intended argument in (8).
This simple example illustrates the need for a dialogue system that allows agents to ask for and provide clarifications when enthymemes are involved (4 and 5, respectively). It also shows that agents need to be able to ask what another agent has assumed was intended by an enthymeme (6), to answer such a question (7), and to correct any erroneous assumptions (8). Additionally, observe that before Bob resolves the misunderstanding regarding what he intended and what Alice assumed (6-8), although it appears that Alice wins the dialogue (since she moves her enthymeme in 3 against Bob’s argument in 2), there is no formal defeat on Bob’s argument and so Bob’s argument (2) is determined acceptable in the argument framework AF constructed by the contents of the enthymemes revealed by the agents. In other words, a mismatch can exist between the pragmatic and the logical conclusions implied by a dialogue in which enthymemes are used.
Most works dealing with enthymemes focus on how an enthymeme can be constructed
from the intended argument and how the intended argument can be reconstructed from a
received enthymeme, based on assumptions that the sender and the receiver make about their
shared knowledge and context, e.g. [
Our primary contribution is the development of a dialogue system in which agents can
move any kind of enthymeme and seek clarification to elicit the intended arguments of these
enthymemes as well as resolve any misunderstanding that may occur between the participants
regarding what an agent assumes and what its counterparts intends, such that under certain
conditions, the dialogical status of the moves made during the dialogue—determined by what
we call the dialogue framework—corresponds to the acceptability of the arguments in the
Dung AF instantiated from the declarative contents of the moves made at that stage in the
dialogue. This correspondence—not considered previously for dialogue systems that support
1This example highlights requirements for dialogical ‘reification’ of relations between locutions, as proposed in [
The + framework [
An + argumentation theory AT is a tuple ⟨, ⟩ consisting of an argumentation system and a knowledge base . is a tuple ⟨, ( · ), , nom⟩ where is a logical language and ( · ) : → (2 − {∅} ) is a function that generalises the notion of negation, so as to declare that two formulae are in conflict (e.g., married = {single, unmarried}). Additionally, = ∪ def is a set of strict () and defeasible (def ) inference rules and nom : d*ef → (where d*ef is the class of all defeasible rules) is a naming function which assigns a name (or nominal) to each defeasible rule (so that rules can be referenced in the object language). Lastly, = ∪ def is the (disjoint) union of a set of strict rules of the form 1, . . . , → , and a set of defeasible rules def of the form 1, . . . , ⇒ , for , 1, . . . , ∈ . For each rule ∈ let antecedents() = {1, . . . , } denote the set of antecedents of and consequent() = denote the consequent of . Finally, a knowledge base ⊆ is a set of premises.
In our representation of an argument, a strict/defeasible inference rule is incorporated as a node, intermediating between the parent node (the rule’s conclusion) and the child node(s) (the rule’s antecedent(s)). This contrasts with the standard + notion of an argument in which the rules are represented by undirected edges (where solid lines represent strict rules and dotted lines represent defeasible rules) linking the conclusion to the rule’s antecedents (see Figures 1.(a) and 1.(b) for an example of an + argument and an argument as depicted in this paper, respectively).
Definition 1. Given an argumentation system AS = ⟨, ( · ), , nom⟩ and an argumentation theory AT = ⟨AS , ⟩, an argument is a labelled (downward directed) tree = ⟨Nodes(), Edges(), lab⟩ such that: 1. lab : Nodes() → ∪ is a node labelling; 2. Edges() ⊆ Nodes() × Nodes() such that if (, ) ∈ Edges(), then either: (a) lab() ∈ , lab( ) ∈ and consequent(lab( )) = lab(), or (b) lab() ∈ and antecedents(lab()) = {lab( ) | (, ) ∈ Edges()}, and if (, ), (, ) ∈ Edges() and ̸= then lab( ) ̸= lab(); 3. for every node ∈ Leaves(), we have lab() ∈ ; 4. |Roots()| = 1; 5. lab(Conc()) ∈ , where Conc() is the unique element in Roots(); where Leaves() = { ∈ Nodes() | ∄(, ) ∈ Edges()}, Roots() = { ∈ Nodes() | ∄( , ) ∈ Edges()}. Let A denote the set of arguments instantiated by , A* the class of all arguments, and Rules() = { ∈ Nodes() | lab() ∈ }, for all ∈ A* .
The attack and defeat relations over arguments are defined as for + arguments.
Argument attacks argument if ’s claim conflicts with an ordinary premise or the consequent or
name of a defeasible rule in , and may succeed as a defeat if the targeted sub-argument ′ of
( ′ ∈ Sub( )) is not strictly preferred to (for details see [
Enthymemes are incomplete arguments. Contrary to other approaches that handle
enthymemes [
Definition 2. Let AS = ⟨, ( · ), , nom⟩. An enthymeme is a forest of labelled (downward directed) trees = ⟨Nodes(), Edges(), lab ⟩ such that Nodes() ̸= ∅ and conditions 1 and 2 from Definition 1 hold.
We say that is an enthymeme of an argument (written ≤ ) if Nodes() ⊆ Nodes(), Edges() ⊆ Edges() and lab () = lab(), for all ∈ Nodes(). Then Roots() = { ∈ Nodes() | ∄( , ) ∈ Edges()} and Leaves() = { ∈ Nodes() | ∄(, ) ∈ Edges()}.
Note that an argument is an enthymeme of itself, where Roots() = {Conc()} and {lab() : ∈ Leaves()} ⊆ . Thus, we also write that ′ ≤ (where ′ and are enthymemes) to denote that ′ is an enthymeme of (or else extends ′, e.g. ′ ≤ and extends ′ in Figure 1.(b)). The class of all enthymemes is denoted E * and so includes the class of all arguments (A* ⊆ E * ). c p (a)
X
X ′ ¬a
An enthymeme dialogue is a sequence of moves. Each move is a 5-tuple comprising the move’s sender, locution, content, reply and target. The sender is either Prop or Op (the participants of the dialogue). Table 1 shows the locutions permitted in our system.
A move’s content depends on its locution: if a move’s locution is assert or w.d.y.a.b. its content is an enthymeme. If a move’s locution is intend, assumed, meant or agree its content is an argument, otherwise the move’s content is ∅. The w.d.y.i., intend, assumed, meant and agree moves explicit responses to a previous move. For any move with one of the aforementioned locutions, its reply is a natural number indexing the move to which it replies, otherwise its reply is ∅.
An asserted enthymeme is moved as a defeat against a previously moved enthymeme — the target of the assertion — whereas a w.d.y.a.b. move questions the defeat relation between the contents of two moves. So, if a move’s locution is assert, its target is the natural number indexing the move whose content is the enthymeme being defeated, whereas if a move’s locution is w.d.y.a.b., its target is the pair of natural numbers (, ) where targets (’s enthymeme is moved as a defeat on ’s enthymeme), otherwise its target is ∅.
When an agent asserts an enthymeme, reveals its intended argument, corrects or confirms its counterpart’s interpretation of an enthymeme it has in mind a complete argument and this is the intended argument of the move. Note that since we accommodate nefarious agents, we do not insist that a move’s intended argument necessarily extends the move’s enthymeme. Similarly, when an agent asserts an enthymeme so as to defeat some targeted enthymeme ′, the agent has in mind an argument that it believes was intended as the complete argument extending Locution assert w.d.y.i.
intend w.d.y.a.b. assumed meant agree stop
Meaning Assert an enthymeme.
Request the intended argument of an asserted enthymeme by asking “what do you intend”.
Provide the intended argument of an asserted enthymeme.
Check the other participant’s understanding of an enthymeme by asking “what did you assume by . . . ”.
Provide their own interpretation of an enthymeme.
Correct the other participant’s interpretation of an enthymeme.
Confirm the other participant’s interpretation of an enthymeme.
Indicate the participant’s intention to stop the dialogue. the targeted ′. is said to be the assert move’s intended target argument. See Table 2 for an example of an enthymeme dialogue.
Definition 3. An enthymeme dialogue between two participants and Op is a sequence of moves = [0, 1, . . . , ℓ], where each = ⟨s(), l(), c(), re(), t()⟩ is a 5-tuple comprising the following: Prop move 1. Sender: s() ∈ {Prop, Op}; 2. Locution: l() ∈ {assert, w.d.y.i., intend, w.d.y.a.b., assumed, meant, agree, stop}; 3. Content: if l() ∈ {assert, w.d.y.a.b.} then c() ∈ E * , else if l() ∈ {intend, assumed, meant, agree} then c() ∈ A* , otherwise c() = ∅; 4. Reply: if l() ∈ {w.d.y.i., intend, assumed, meant, agree}, re() ∈ {0, . . . , ( − 1)}, otherwise re() = ∅; 5. Target: if l() ∈ {assert}, t() ∈ {0, . . . , ( − 1)} ∪ {∅}, otherwise if l() = w.d.y.a.b., t() ∈ ({0, . . . , ( − 1)} × { 0, . . . , ( − 1)}), otherwise t() = ∅.
We denote the class of all moves as M * . If l() ∈ {assert, intend, meant, agree}, IntArg() ∈ (A* ∪ {∅}) is the intended argument of and if l() = assert, IntTarArg() ∈ (A* ∪ {∅}) is the intended target argument, else IntArg() = ∅ and IntTarArg() = ∅, respectively. We let = [0, . . . , ] be the enthymeme dialogue whose last move is and say that = [0, . . . , , . . . , ] expands where < < ℓ.
Abusing notation, we identify re() with the move re() to which replies and t() with the move t() (or the pair of moves ( , )) that targets, instead of the indices re() and t() (or (, )) of re() and t() (or ( , )), respectively.
We now define a well-formed enthymeme dialogue with two participants, Prop and Op. We assume that the agents share the same underlying argumentation system, ensuring they can understand each other, except that the defeasible rules they are aware of may difer. They may of course have diferent knowledge (belief) bases. The participants alternate turns and cannot repeat moves. Prop must start by asserting an argument (the conclusion of which we call the topic of the dialogue). Apart from this first assert move, which has no target, an assert move’s enthymeme targets (i.e., is moved as a defeat against) a previously asserted enthymeme.
Note that while targets , this does not necessarily imply that the content of is a valid defeat on the content ′ of , according to the + definition of defeat. Since enthymemes may only partially reveal the content of an argument, one may not be able to validate that a target relation corresponds to an + defeat; hence our definition of wellformed enthymeme dialogues does not enforce such a correspondence. This of course means that participants may be mistaken in the assumptions they have made regarding the other’s intended arguments (i.e., may not legitimately defeat the intended complete argument that extends ′), or, for strategic purposes, an agent may dishonestly target a move’s enthymeme ′, even though they know that their enthymeme does not defeat the intended target argument . A w.d.y.i. move is used to request the intended argument of a previously asserted enthymeme , while an intend move replies to a w.d.y.i. move and supplies the complete argument extending .
Recall that the receiver of an asserted enthymeme ′ might not be able to interpret correctly the intended argument ′ extending ′. Thus, if targets , the agent Ag who moved might not be able to understand why the content ′ of has been moved against the content of (i.e., Ag is unable to reconstruct an argument ′ from ′, such that ′ represents a valid defeat on Ag’s intended argument extending ). Moreover, the sender Ag′ of might be mistaken in her interpretation of the intended target argument extending in . So, Ag can check whether Ag′ correctly assumes that extends , by making a w.d.y.a.b. move (whose content repeats the content of ) and which essentially amounts to questioning the validity of the targeting relationship ( , ) (i.e., whether it corresponds to a valid defeat); thus, the target of is ( , ).
A participant of a well-formed enthymeme dialogue replies to a w.d.y.a.b. move only with an assumed move ℎ. We also impose that the content of ℎ is an argument that extends the content of since by using ℎ its sender (Ag′) reveals its understanding of the argument extending Ag’s enthymeme () in . A reply to the assumed move ℎ is a move with locution meant or agree, in which Ag reveals the intended argument extending the enthymeme in (and which is repeated in ). Now, if Ag′ (in the move ℎ) was mistaken in its assumption as to what was the intended argument, then ’s locution is meant and its content must not be the same as the mistaken content of ℎ. Otherwise, if Ag′’s interpretation was correct (i.e., Ag′ moved in ℎ), then ’s locution is agree and its content is ; that is, the argument in ℎ and is the intended argument that extends the enthymeme in Ag’s earlier move .
Finally, is terminated when two stop moves are made consecutively (indicating that neither agent wishes to continue the dialogue). For an example of a well-formed enthymeme dialogue see Table 2.
Definition 4. Let AS Ag = ⟨, ( · ), Ag, nom⟩ be an argumentation system for Ag ∈ {Prop, Op} such that Ag = ∪ dAegf , where dAegf is Ag’s defeasible rules. An enthymeme dialogue = [0, . . . , ℓ] between Prop and Op is said to be well-formed if ̸= for all ̸= , and for all ≤ ℓ: 1. s() = Prop if is even, otherwise s() = Op; 2. if = 0, then l() = assert, c() ∈ A* , and t() = ∅; 3. if > 0 and l() = assert, then c() ∈ E * and l(t()) = assert; 4. if l() = w.d.y.i., then l(re()) = assert; 5. if l() = intend, then l(re()) = w.d.y.i. and c(re( )) ≤ c(); 6. if l() = w.d.y.a.b., then t() = ( , t( )) such that l( ) = assert, and c() = c(t( )); 7. if l() = assumed, then l(re()) = w.d.y.a.b. and c() ∈ A* such that c(re()) ≤ c(); 8. if l() = meant, then l(re()) = assumed, c() ∈ A* and c() ̸= c(re()); 9. if l() = agree, then l(re()) = assumed, c() ∈ A* , and c() = c(re()).
The topic of (denoted Topic()) is the label of the conclusion of the argument moved in 0. If ∃, (0 ≤ < ≤ ℓ) such that = + 1 and l() = l( ) = stop, then = . We say that is terminated if l(ℓ) = l(ℓ− 1) = stop.
Henceforth, we use ‘dialogue’ as shorthand for a ‘well-formed enthymeme dialogue’. A dialogue ’s dialogue framework is a 4-tuple DF = ⟨︀ M , T , Re, S ⟩︀ where M = M ∪ {Yes, No}, M is the set of moves in (excluding stop moves), Yes and No are auxiliary elements used to respectively confirm and reject the other participant’s interpretation of an enthymeme, T is a defeat relation, Re a reply relation, and S a support relation. Recall that a dialogue framework is used to determine the dialogical status of the dialogue moves, similarly to a Dung AF which is used to determine the acceptability status of arguments.
When a move is made in , it is added to M (unless it is a stop). An agent Ag’s assert move moves the content of as a defeat on the content of ’s target, whereas Ag makes a w.d.y.a.b. move to question the defeat relation between two moves and (( , ) is the target of ). So, in both cases, we add a defeat relation between and its target; in all other cases T remains the same.
If is a w.d.y.i., or intend, or assumed that replies to , then this reply relation is added to Re, since these moves respectively request the intended argument of a move, provide the intended argument of a move and provide an interpretation of an enthymeme. If is a meant move, a reply relation between No and is added to Re. Essentially, has two efects on DF : firstly, the sender Ag of rejects its interlocutor’s Ag′’s understanding of Ag’s enthymeme , thus, we interpret as a negative reply to ; secondly, using , Ag reveals the intended argument from which was constructed. If we are to draw accurate conclusions regarding the acceptability of moves in DF , then is not enough to express both efects. Hence, we use No to validate that Ag′’s interpretation of is not acceptable, and (as we will see below) we connect (through a support relation) to the move , which contains , in order to show that the acceptability of (i.e., ) influences the acceptability of (i.e., the intended argument from which was constructed). In this way, changes in the status of do not influence the status of which remains unacceptable in DF since Ag′’s understanding of was mistaken. In all other cases Re does not change.
If is an intend move with content , then a support relation between and is added to S . Intuitively, prior to , replies to (and so challenges) , by requesting the sender of to provide the intended argument from which the enthymeme of was constructed (i.e., extends ). satisfies this request (and so replies to ), and thus is moved to support against ’s challenge (essentially questions why was moved, whereas justifies why was moved). A meant or agree move also provides the intended argument of a move (where is targeted by a move , the defeat relation (, ) is challenged by a move ℎ requesting the interpretation of ’s content by the sender Ag of , a move replies to ℎ providing Ag’s interpretation of ’s content and replies to ) and so we add a support relation between and . Finally, similarly to a meant move, an agree move has two efects on DF : by moving the sender Ag of confirms Ag′’s (i.e., the other participant’s) understanding of Ag’s enthymeme and, thus, we interpret as a positive reply to , thus supporting it; secondly, using , Ag reveals the intended argument from which was constructed. Hence, we use Yes to validate that Ag′’s interpretation of is acceptable, and we connect (through a support relation) to the move , which contains , in order to show that the acceptability of (i.e., ) influences the acceptability of (i.e., the intended argument from which was constructed). In this way, changes in the status of do not influence the status of which remains acceptable in DF since Ag′’s understanding of was correct. In all other cases S remains the same. For an example of a DF see Figure 3. Yes
Re
S m3
Re m2
T T
m1 No
Re m6 Re m5
T S m7 m0
Definition 5. Let DF 0 = ⟨︀ M0 , T0, Re0, S0⟩︀ be the dialogue framework of a dialogue 0 = [0] where DF 0 = ⟨0, ∅, ∅, ∅⟩. Let ℓ = [0, . . . , ℓ] expand 0. Then, for each DF = ︀⟨ M , T, Re, S⟩︀ , where 0 ≤ ≤ ℓ we have that:
We define a complete labelling on DF of a dialogue as a function assigning in to a move in if: 1) every move replying to is out; 2) every move targeting , and such that the defeat relation from ’s enthymeme to ’s enthymeme is not targeted by an in move, is out, and; 3) every move that is supported by is in. A move is assigned out if either a move replying to is in, or a move targeting (such that the defeat relation is not targeted by an in move) is in, or a move that is supported by is out (intuitively, the content of a move supported by is an enthymeme of ’s content and so ’s acceptability influences ’s acceptability); a move is undec if it is neither in nor out.
Definition 6. Let DF = ⟨︀ M , T , Re, S ⟩︀ . We define a complete labelling on DF to be a function L : M → {in, out, undec} such that, for every ∈ M : 1. L() = in if for all , ∈ M : (a) if ( , ) ∈ Re then L( ) = out, and (b) if ( , ) ∈ T , and ∄ ∈ M such that t() = ( , ) and L() = in, then L( ) = out, and (c) if (, ) ∈ S , then L( ) = in; 2. L() = out if there is some , ∈ M such that: (a) ( , ) ∈ Re and L( ) = in, or (b) ( , ) ∈ T , and ∄ ∈ M such that t() = ( , ) and L() = in, and L( ) = in, or (c) (, ) ∈ S and L( ) = out.
If L is a complete labelling on a DF , then in(L), out(L) and undec(L) respectively denote the set of all moves labelled in, out and undec. Below we also define preferred, preferred and stable labellings on DF (for an example see Figure 3).
Definition 7. Let DF be the dialogue framework of a dialogue and let L be a complete labelling function on DF . Then: 1. L is a preferred labelling function on DF if there does not exist a complete labelling function L′ on DF such that in(L) ⊂ in(L′); 2. L is the (unique) grounded labelling function on DF if there does not exist a complete labelling function L′ on DF such that in(L′) ⊂ in(L); 3. L is a stable labelling function on DF if undec(L) = ∅.
We define an argumentation theory instantiated by a dialogue as a tuple AT = ⟨AS , ⟩, where the logical language , the contrariness function ( · ), the naming function nom, and the strict rules of AS are those shared by the participants of . The defeasible rules of AS is the set of the defeasible rules revealed during , excluding the inference rules of enthymemes in assumed moves as these are assumptions which may be mistaken (in case they are correct, they are repeated within the content of an agree move following an assumed move). A premise of an enthymeme belongs to the set of premises in AT (i.e., ) if is a leaf in and if the intended argument from which was constructed has been revealed by the sender of , then is also a leaf in .
Definition 8. Let = [0, . . . , ℓ] be a dialogue between Prop and Op, where AS Ag = ⟨, ( · ), Ag, nom⟩ is the argumentation system for Ag ∈ {Prop, Op}, Ag is the knowledge base of Ag and Ag = ∪ dAegf . Let DF = ⟨︀ M , T , Re, S ⟩︀ be the dialogue framework of . The set of defeasible rules in is DefDRules() = ℓ ⋃︁ {︀ labc()() ∈ d*ef | l() ̸= assumed, c() ∈ E * and ∈ Nodes(c())}︀ . =0 The set of premises of Ag in is denoted DPremAg() =
⎧ ⋃︁ ⎪⎪⎨ =0 ⎪⎪ ⎩
⃒⃒ s(m) = Ag, l() ̸∈ {w.d.y.i., w.d.y.a.b., assumed}, ⎫ labc()()⃒⃒⃒⃒⃒⃒ Leaves(c()), aanndd ∄n̸∈ L(ne a≤∈veℓs)(csu(cht)h)at ( , ) ∈ S ⎭⎪⎪⎬⎪⎪ .
The set of all premises in is denoted DPrem() = DPremProp () ∪ DPremOp (). The argumentation theory instantiated by a dialogue is AT = ⟨AS , ⟩, where AS = ⟨, ( · ), ∪ DefDRules(), nom⟩ and = DPrem().
The argumentation theory of Ag instantiated by a dialogue is AT A g = ⟨AS A g, Ag⟩, where AS A g = ⟨, ( · ), Ag ∪ DefDRules(), nom⟩ and Ag = DPrem() ∪ Ag.
If a dialogue is exhaustive and its participants honest, the dialectical status of the moves in
the DF (determined by a complete, preferred, grounded and stable labelling) is sound and
complete with respect to the dialectical status of the arguments in the AF instantiated by the
contents of the moves made in the dialogue (determined by a complete, preferred, grounded, and
stable labelling, resp.). Essentially, if ’s participants are honest, they only assert an enthymeme
constructed from the intended argument of their move and such that it does defeat the intended
targeted argument (i.e., what the sender assumes to be the target move’s intended argument)
according to the + definition of defeat. It also means that whenever they reveal their
understanding of the argument from which their counterpart’s enthymeme is constructed,
the argument does indeed extends , and whenever they reveal the intended argument of their
own move, this is indeed the argument they intended. If is exhaustive, then any available
move that can be made, is indeed made. In other words, if a participant can move an argument
(constructed by the contents of the moves made in the dialogue) as a defeat against the content
of another move made in the dialogue, or question a defeat relation or reply to a move, they
indeed do so. Due to lack of space we cannot present here the technical details for our claim,
but for a better understanding of our claim the reader can look at Example 2.
Example 2. Figures 4.(a) and 4.(b) respectively show the DF of Table 2’s exhaustive and honest
dialogue 2, and the AF AT instantiated by AT . We also show a valid complete (resp. preferred,
grounded and stable) labelling of DF ’s moves given by the labelling function L on DF , and a
valid complete (resp. preferred, grounded and stable) labelling of arguments in AF AT given by
the labelling function L′ on AF AT . In both frameworks the topic of is labelled out, (since
0 is labelled out in DF and IntArg(0) is labelled out in AF AT ) which means that does
not belong to the complete (resp. preferred, grounded and stable) extension [
This paper introduces a novel dialogue system that allows participants to seek and provide clarifications regarding possible misinterpretations that arise from the use of enthymemes, and instantiates a dialogue framework that is used to determine the dialogical status of the dialogue moves. As far as we are aware, there are no other works that handle misunderstandings that may occur between the participants due to the use of enthymemes and also propose a mechanism for determining the status of the dialogue moves. Our work paves the way for a sound and complete dialogue system that accommodates use of any kind of enthymeme, and where any kind of uncertainty that arises from their use can be resolved. This is important since it ensures that the dialogue can be played out such that an enthymeme moved in the dialogue is only justified in the case that its intended argument is justiefid by the contents of the moves made in the dialogue. In future work we will explore how enthymemes may be used to give a strategic advantage to a participant. 2We assume that and are preferred to since owing money takes precedence over eating at a restaurant, and is preferred to ′ since ′ is just a statement. Yes
Re
S m3
No
Re
m2 Re
T
m1 T Re m5
T S m7
m0 m6 (a) b ⇒ a a b (c)
A A′