This paper proposes a context-based argumentation system based on ASPIC+ frameworks. It considers the changeability of contexts in conversations. Moreover, this system encodes the participants' preferences by giving priority orderings over the set of all the values associated with contexts and norms respectively. People argue for reaching agreements, this paper defines a concept called consensus to represent the agreements reached among the participants.
Formal argumentation systems have been considered as useful tools for resolving
disputes [
Consider the following example from the Aesop’s Fables.
Example 1 (The Fox and the Crow). 1 A hungry Fox was walking through the woods looking for something to eat. Suddenly he found a crow sitting on a branch high up in a tree, with a big piece of cheese in her beak. The Fox decided to get that piece of cheese for himself. He said hello to the Crow and tried to make her talk. However, the Crow suspected that the Fox was after her cheese, so she didn’t return his greeting.
Just before the Crow was about to swallow the cheese, the Fox tries flattery instead: “Oh, crow, how your feathers shine! What a beautiful form and what splendid wings! Such a wonderful bird should have a very lovely voice, since everything else about you is so perfect. Could you sing just one song, I know I should hail you Queen of Birds.”
Pleased with the flattery, the crow forgot her suspicion and also her breakfast. She was so eager to show off her voice, so she opened her beak wide to utter her loudest caw. The cheese dropped down and the Fox gobbled it up immediately. ? This work is supported by the China Postdoctoral Science Foundation [No. 2019M663353], the MOE Project of Key Research Institute of Humanities and Social Sciences in Universities [No. 18JJD720005] and the Fundamental Research Funds for the Central Universities of China [No. 20wkpy104]. I would like to thank the anonymous reviewers for their helpful comments. 1 Several versions of this story can be found on the webpage https://fablesofaesop.com/the-foxand-the-crow.html
In the story, the Crow was certainly not going to open her beak at first because she won’t give her own food to the Fox. But the Fox then sneakily shifted the context of their conversation. Eventually, the Crow decided to show off her singing voice and lost her food for the sake of vanity.
While the change of context is achieved implicitly by tricking in Example 1, in other conversations, such change can also be brought up explicitly. Different contexts of conversations may stand for different culture background, and lead to different social norms and priorities over values. Due to the disagreement between participants, in order to persuade the others, it is common that some participants try to switch the context by proposing new arguments. If they succeed in changing the context, new values may be added and every participant has to reconsider his/her priority orderings over all the values. Consequently, changes in priorities might lead to some consensus. A key problem lies in whether the audiences will adopt the proposed new context.
The current paper argues that the contextual factors are important for obtaining preferences over arguments in argumentation theory, and the alternative contexts should also be compared based on priories. Such comparison can be achieved by associating values with norms and contexts. What’s more, since people argue for reaching agreements, I define a concept called consensus to represent the agreements achieved between participants with different priorities on values.
The remaining of this paper is structured as follows. Section 2 proposes an argumentation system that combines ASPIC+ framework with contexts, values and social norms, then defines the concept of consensus. Section 3 presents some related work, while section 4 briefly concludes this paper and points out the future work. 2
In this section I try to encode the factors influence the participants’ preference orderings over the set of arguments into a formal argumentation system. 2.1
Within different contexts, people may have different priority orderings on values. In Example 1, when the Crow remembered she was hungry, the value of food2 was the most important. However, she was then convinced by the Fox and shifted the context of arguments from ‘hunger’ to ‘longing for compliments’. As a consequence, the Crow forgot the value of her food and lost the cheese for the value of vanity. In other words, in the second context, the value of food was no longer the Crow’s top priority. Suppose that the Crow was wiser, she might think that vanity is far less important than satisfying her hunger and won’t be shifted the context as easily as in the story.
We can see that during a dispute, a participant may implicitly or explicitly propose a new context, then the audience needs to choose whether to shift the main context of 2 The readers may find that in this paper the concept value is used rather generalized. Values are like some crucial factors such as goals or desires related to particular contexts or norms, based on which the participants of a debate can give their priority orderings. arguments based on priorities (may be an unconscious process). When the context is changed, participants’ priorities over values are probably changed concurrently.
To capture these features, the current paper associates values not only to the norms applied in the argumentation, but also to the contexts of arguments. Whether a participant decides to shift the context depends on his priority orderings over values. Moreover, in each context, there is a series of norms. Each norm associated with at least one value3. Each side of participants hold their own priority orderings over the set of all the values in every context. The difference in priority orderings may lead to disagreement on the conclusions.
More specifically, the contexts that appear in a conversation are arranged in the order of their appearance. Each context is consisted of a particular name, a set of norms, a set of values that associated with the norms and the contexts, and a set of participants’ priority orderings over the values. I use ‘N’ to denote the set of norms, ‘V ’ to denote the set of values, ‘P’ to denote the set of different orderings over values, Ci to denote a certain context where the subscript number indicates the particular order of a context. A context is defined as follows.
Definition 1 (Context). Let Ci = ha(vb ); Ni;Vi; Pii (i = 1; 2; : : : ; n) be a context, where – a is the notation for the name of a certain context, and vb denotes the value b corresponding to a; each name of a context is associated with at least one value. – Ni = fn1(vp); n2(vq); n3(vr); : : : ; nk(vx)g is a set of norms, where n j ( j = 1; 2; : : : ; k) is a norm, vp; vq; vr; : : : ; nk(vx) are values corresponding to the norms; each of the norms associated with at least one value. The forms of norms are: j1; : : : ; jn v!u = =v)u j ( v!u and =v)u refer to certain inferences and uncertain inferences respectively; ji and j are elements in the logical language of an argumentation theory); if i > 1, then always let Ni 1 Ni. – Vi = fvb ; vp; vq; vr; : : : ; vxg is a set of all the values that appear in Ci. – Pi = f&1; &2; : : : ; &ng is a set of orderings over Vi in context Ci, where &1; &2; : : : ; &n (n > 1)4 are preorderings on Vi, such that: 1. vn & vm, iff vn is at least as preferred as vm; 2. vn > vm, iff vn & vm and not vm & vn; 3. vn vm, iff vn & vm and vm & vn.
Note that according to this definition, the set of norms in a context Ci is always a union set of N1 to Ni (i > 1), so that when the context be shifted, we won’t lost the norms contained in the previous contexts, as well as their associated values. Accordingly, for the set Vi, if i > 1, then Vi 1 Vi.
In Example 1, there are two main contexts. In the first context, the Crow and the Fox both were hungry and desiring for eating some food. For the Crow, her food is superior to the Fox’s food. So when she possessed a piece of cheese, she won’t open her mouth and drop it to the Fox. For the Fox, on the contrary, food for himself is superior to food for the Crow. I use ‘hun’ to denote ‘hungry’, ‘eat’ to denote ‘eating’, ‘ f c’ to denote ‘food of the Crow’, ‘ f f ’ to denote ‘food of the Fox’, ‘cheese’ to denote ‘a piece of cheese (in the Crow’s mouth)’, ‘open’ to denote ‘open (the Crow’s) mouth’, the first context in the story can be modelled as follows based on Definition 1. 3 While it’s normal that some values are associated with more than one norms or contexts. 4 n is less than or equal to the number of different sides of the participants. V1 = fveat ; v f c; v f f g, P1 = f&1; &2g, the Crow’s ordering 1 (according to &1) : veat & v f c > v f f the Fox’s ordering 1 (according to &2) : veat & v f f > v f c
C1 = hhun(veat ); N1;V1; P1i, where N1 = fn1(v f c); n2(v f f )g, (n1(v f c) = hun(veat ); cheese =) :open, n2(v f f ) = hun(veat ); cheese =v )ff open) v f c 2.2
The idea behind the Context-based Argumentation System (CAS) is that the preferences
on the set of arguments can be obtained based on the orderings over the set of
values. I build the argumentation system on the basis of ASPIC+ framework since some
principles for obtaining preference relations have already been given in its design [
Definition 2 (CAS). An argumentation system based on contexts is a tuple CAS = (L ;¯; R; C ; n), where: – L is a logical language. – ¯ is a function from L to 2L , such that: 1. j is a contrary of y if j 2 y and y 2= j; 2. j is a contradictory of y (denoted by ‘j = y’), if j 2 y and y 2 j 5; 3. each j 2 L has at least one contradictory. – R = Rs [ Rd is a set of strict (Rs) and defeasible (Rd ) inference rules of the form j1; : : : ; jn (v!u) = =(v)u) j ( (v!u) and =(v)u) refer to strict inference rules and defeasible inference rules respectively; ji; j are elements in L ); Rs \ Rd = 0/ . – C = fC1;C2; : : : ;Cng is a set of contexts, where Ci = ha(vb ); Ni;Vi; Pii (i = 1; 2; : : : ; n), such that: 1. fx(vy)jhx(vy); Ni;Vi; Pii 2 C g L and if 8a(vb ) 2 fx(vy)jhx(vy); Ni;Vi; Pii2 C g, 9a0 6= a such that a0(vb 0 ) 2 fx(vy)jhx(vy); Ni;Vi; Pii 2 C g, then a(vb ) = a0(vb 0 ); 2. Ni Rs [ Rd ; 3. Vi L . – n is a naming function such that n: Rd ! L .
The definition of C shows that different contexts in an argumentation system are conflicting with each other. In other words, each side of the participants can only choose one main context among all the alternatives based on their priority orderings over the set of values.
As in ASPIC+, arguments can be constructed from a set of premises, namely a
knowledge base K L . A tuple of an argumentation system and a knowledge base
K is called an argumentation theory (AT) [
j 2 j and for all :j 2 L , we have j 2 :j. Definition 3 (Arguments). An argument A on the basis of a CAS = (L ;¯; R; C ; n) and a knowledge base K = Kn [ Kp has one of the following forms: 1. j; if j 2 K with: Prem(A) = fjg, Conc(A) = j, Sub(A) = fjg, De f Rules(A) = 0/ , TopRule(A) = unde f ined. 2. A1, : : :, An (v!u) = =(v)u) y, if A1, : : :, An (n
strict/defeasible rule Conc(A1), : : :, Conc(An) (v!u) = =(v)u) y in R with: Prem(A) = Prem(A1) [ : : : [ Prem(An); Conc(A) = y; Sub(A) = Sub(A1) [ : : : [ Sub(An) [ fAg; De f Rules(A) = De f Rules(A1) [ : : : [ De f Rules(An) ([fConc(A1), : : : , Conc(An) =(v)u) yg); TopRule(A) = Conc(A1) : : :Conc(An) (v!u) = =(v)u) y.
Based on Definition 3, the following arguments can be constructed in the argumentation theory. Here I use a subscript ‘0’ to indicate the argument constructed only by the specific names of contexts.
A0 : hun(veat )
v f c A2 : A0; A1 =) :open
A1 : cheese
In ASPIC+, three kinds of attack relations between arguments are allowed: attack
on the conclusion of defeasible rules (rebutting), attack on the defeasible rules
(undercutting) and attack on the ordinary premises (undermining). Based on [
1. A undercuts B on B0, iff B0 2 Sub(B) such that TopRule(B0) = r 2 Rd and Conc(A) 2 n(r) (‘n(r)’ means that rule r is applicable); 2. A rebuts B on B0, iff Conc(A) 2 j for some B0 2 Sub(B) of the form B010; : : : ; B0n0 ) j; A contrary-rebuts B iff Conc(A) is a contrary of j. 3. A undermines B on B0, iff B0 = j and j 2 Prem(B) \ Kp, such that Conc(A) 2 j; A contrary-undermines B iff Conc(A) is a contrary of j.
According to [
Definition 5 (Defeat). Let A, B be arguments, B0 2 Sub(B). A defeats B iff A preferenceindependently attacks B on B0, or A preference-dependently attacks B on B0 and B0 A.
ASPIC+ offers two principle (i.e. the last link principle and the weakest link
principle) for ‘lifting’ the preference ordering over arguments, which are based on the
priority orderings over the sets Rd and Kp. Due to space restriction, the introduction
of these two principles are omitted. For further details, please refer to [
Definition 6 (Priority Ordering). Given a CAS = (L ;¯; R; C ; n), Ci = ha(vb ); Ni;Vi; Pii (i = 1; 2; : : : ; n) is a context in C . According to an ordering %2 Pi on Vi, such that – > is a preordering on Ni, for na(vp); nb(vq) 2 Ni: 1. na(vp) > nb(vq), iff na(vp) 2 Rs or vp & vq; 2. na(vp) > nb(vq), iff nb(vq) 2= Rs and vp > vq; 3. na(vp) nb(vq), iff vp vq. – > is a preordering on fx(vy)jhx(vy); Ni; Vi; Pii 2 C g, for g(vp); h(vq)2 fx(vy)jhx(vy); Ni;Vi; Pii 2 C g: 1. g(vp) > h(vq), iff g(vp) 2 Kn or vp & vq; 2. g(vp) > h(vq), iff h(vq) 2= Kn and vp > vq; 3. g(vp) h(vq), iff vp vq.
For the evaluation of arguments, we need to construct abstract argumentation frameworks (AFs) based on the set of all the arguments constructed in the argumentation system and the set of defeat relations between arguments. According to the definitions of contexts and CAS, it is possible to obtain more then one AF within a context since different participants may have different orderings on the set of values6. Using ‘CO’ to denote the set of attack (or conflict) relations, ‘D’ to denote the set of defeat relations and ‘D ’ to denote the set of Ds in a context, an abstract argumentation framework in CAS can be defined as follows.
Definition 7 (Argumentation Frameworks). Let CAT = (CAS; K ) be an argumentation theory in CAS, such that: 1. a structured argumentation framework SAFCi in the context Ci = ha(vb ); Ni;Vi; Pii is a tuple hAi;COi; Pii, where – Ai is a set of arguments constructed from K based on CAT and Ci; – COi is the conflict relations between elements in Ai; (A; B) 2 COi iff A attacks
– Pi = f<1; <2; : : : ; <ng, where < j (i = 1; 2; : : : ; n) is a preordering on Ai lifted from the corresponding ordering & j in Pi; 2. given an SAFCi = hAi;COi; Pii based on the context Ci, let Di = fD1; D2; : : : ; Dng, such that D1; D2; : : : ; Dn are sets of defeat relations according to <1; <2; : : : ; <n2Pi and Definition 5. FCi = hAi; Dii is a sequence of AFs in context Ci, where FCi j =hAi; D ji(1 j n) is an AF in the sequence FCi .
According to the above definition, how many AFs that can be obtained in a context Ci depends on how many different orderings over the set of Vi are given.
Consider Example 3, argument A2 and A3 are conflicting with each other according to Definition 4. Based on the Crow’s ordering &1, A2 is strictly preferred to A3 (i.e. A2 A3), while based on the Fox’s ordering &2, A3 is strictly preferred to A2.
Fig. 1 shows the AFs based on the Crow’s priority order (on the left) and the AF based on the Fox’s priority order (on the right). For the sake of illustration, I label the values contained in an argument alongside it. 6 Therefore, they may have different preferences on arguments, which will lead to different defeat relations.
Arguments in an AF are evaluated based on argumentation semantics [
Given an argumentation framework F =hA ; Di, for E A , we say: E is
conflictfree if and only if @A; B 2 E such that (A; B) 2 D; A 2 A is defended by E if and only
if 8B 2 A if (B; A) 2 D, there 9C 2 E such that (C; B) 2 D; E is admissible if and only
if E is conflict-free, and each argument in E A is defended by E; E is a preferred
extension if and only if E is the maximal (w.r.t. set-inclusion) admissible set [
According to Fig. 1, we can get one preferred extension fA0; A1; A2g based on the AF on the left side, which reflects the Crow’s preferences over arguments; we can get another preferred extension extension fA0; A1; A3g based on the AF on the right side, which reflects the Fox’s preferences over arguments. It can be seen that in the running example, the reason the Fox and the Crow can not reach an agreement in the first context is that they have different priority orderings on the set of values, which causes different preferences on arguments and eventually leads to different extensions of arguments and conclusions. In this subsection I define the concept consensus and illustrate the ideas behind it through the running example. According to Example 1, in the second context proposed by the Fox, the Crow was longing for compliments because of vanity, so she chose to open her mouth for showing off her singing voice and forgot her desire for eating. Meanwhile, the Fox still thought eating is the most important. Let ‘lc’ denotes ‘(the Crow is) longing for compliments’, ‘van’ denotes ‘vanity’, ‘so’ denotes ‘showing off’, ‘voi’ denotes ‘(the Crow’s) singing voice’, ‘sing’ denotes ‘singing for the Fox’, the second context in the story can be modelled as follows.
C2 = hlc(vvan); N2;V2; P2i, where N2 = fn1(v f c); n2(v f f ); n3(vso)g, (n1(v f c) = hun(veat ); cheese =) :open, n2(v f f ) = hun(veat ); cheese =v )ff open, v f c n3(vso) = lc(vvan); voi =v)so sing) V2 = fveat ; v f c; v f f ; vvan; vsog, P2 = f&01; &02g the Crow’s ordering 2 (according to &01) : vvan & vso > veat & v f c > v f f the Fox’s ordering 2 (according to &02) : vvan vso veat & v f f > v f c
Z. Yu
Moreover, a strict rule ‘sing ! open’ and a transposition7 of this rule ‘:open ! :sing’ are added into the argumentation theory. Thus with respect to context 2 of the story, the following arguments can be constructed.
A0 : hun(veat )
Based on the AF in Fig. 2, we can get one preferred extension fA1; A3; B0; B1; B2; B3g, while based on the AF in Fig. 3, we can get the following four preferred extensions: fA1; A3; B0; B1; B2; B3g, fA1; A3; B0; B1; B2; B4g, fA0; A1; A3; B1; B2; B3g, fA0; A1; A3; B1; B2; B4g. It can be observed that now the Crow and the Fox have got one identical preferred extension fA1; A3; B0; B1; B2; B3g, and the corresponding conclusions are ‘a piece of cheese in the Crow’s mouth’, ‘open the Crow’s mouth’, ‘the Crow is longing for compliments’, ‘the Crow’s singing voice’, ‘singing for the Fox’. These conclusions together can be seen as an agreement the Crow and the Fox had reached in the story, which lasted until the Crow found that she was fooled.
I define a concept called consensus to indicate the agreements in a CAS. The intuition behind this concept is that in a multi-participant argumentation, if there is an extension of arguments in a context that can be accepted by at least two sides of participants with different priorities on values, then this extension can be regarded as a 7 In ASPIC+, the set of strict rules should be closed under transposition/contraposition. consensus among these participants. Based on this intuition, a consensus is defined on the basis of argumentation semantics. Let ES denotes a set of extensions under certain argumentation semantics S , a consensus based on S is defined as follows. Definition 8 (S -consensus). Given a CAT , Ai is all the arguments that can be constructed in a context Ci = ha(vb ); Ni;Vi; Pii, while FCi = hAi; Dii is a corresponding sequence of AFs. Let FCi j and FCi k be two AFs in the sequence, if 9E1; E2 Ai, E1 2 ES i j and E2 2 ES i k, such that either E1 E2 or E2 E1, then both E1 and E2 are S -consensus among participants who give the ordering & j and &k. 3
There are some works considered contexts based on other structured argumentation
systems. For example, based on the Assumption-based Argumentation frameworks (ABA),
Zeng et al. presented an approach for making context-based and explainable decisions
in paper [
In paper [
Compared with the current paper, except for the different technical tools applied, none of the above mentioned works focuses on the changeability of contexts in disputes.
As for the notion of consensus, paper [
In this paper I propose a context-based argumentation system called CAS, which takes the changeability of contexts in conversations into account. The system encodes the participants’ preferences by giving orderings over the set of all the values associated with contexts and norms respectively. What’s more, this paper defines a concept called consensus based on the existing argumentation semantics to reflect the agreements reached among different participants.
CAS is built based on ASPIC+ frameworks. One main reason is that ASPIC+ has offered some principles for preferences lifting. Moreover, ASPIC+ provides more flexible choices for modelling nature language argumentation, since it allows users to model more types of attack relations and supports both credulous and skeptical reasoning.
Due to space constraints, the properties of the proposed argumentation system are not discussed. I’d like to explore and summarize the features associated with the CAS in future research.