Structured Argumentation:Restricted Rebut vs.Unrestricted Rebut*

Zhe Yu

Department of Philosophy,Zhejiang University Center for the Study of Language and Cognition,Zhejiang University zheyu@zju.edu.cn

Kang Xu

Zhejiang University of Water Resources and Electric Power xukanguuu@163.com

Beishui Liao

Department of Philosophy,Zhejiang University Center for the Study of Language and Cognition,Zhejiang University baiseliao@zju.edu.cn

Abstract.Inrecent years,formal argument ationhasbeen an increasingly active research topic in the field of logic and artificial intelligence.One of its aims is to bridge the gap between human reasoning and computer-based reasoning.For this purpose,several argumentation formalisms have recently been proposed,including ABA,ASPIC+,ASPIC−,etc.Different design choices are implemented in different systems.This paper focuses on two opposite design choices,namely restricted rebut versus unrestricted rebut,and carries out an empirical research.The empirical results show that unrestricted rebut is more likely to be accepted by human users.It suggests that the current formalisms should be improved and that a better way to combine naturalness and rationality is needed.

1 Introduction

Human reasoning is usually subjective and defeasible.For instance,in common sense reasoning,knowledge used as premises is often uncertain and incomplete.So,different arguments based on this kind of knowledge are often in conflict,and cannot be accepted all together.When confronted with a set of conflicting arguments,agents evaluate them according to their preferences and principles that are usually subjective.Meanwhile,when new information arrives,some arguments and the conclusions they support may be withdrawn.So,the reasoning process is intrinsically dynamic and defeasible.Since traditional computer-based reasoning is typically based on classical monotonic logic which can only deal with certain and complete knowledge,it does not match the properties of human reasoning.In oder to cope with this problem,a series of nonmonotonic formalisms have been proposed since 1980s,including default logic([2]),circumscription([17]),auto-epistemic logic([16]),etc.In recent years,formal argumentation as a nonmonotonic formalism is gaining momentum,thanks to the fact that human reasoning can be naturally modeled as arguments and their interactions.More specifically,since human reasoning is typically realized by exchanging and evaluating arguments,it is natural to model formal models based on this mechanism.

In the existing literature,there are basically two lines of work in argumentation studies:informal argumentation and formal argumentation.On one hand,informal argumentation attaches attention to modeling and evaluating arguments in natural argumentative discourse with the help of informal tools like argument schemes,e.g.the Pragma-dialectics by F.van Eemeren and R.Grootendorst([22]),the New Dialectic by D.Walton([23]),etc.In recent years,there are also a lot of Chinese researchers working on this topic,including M.Xiong([25]),R.Jin([13]),Y.Xie([24]),et al.On the other hand,formal argumentation is a nonmonotonic formalism to model various kinds of reasoning,from epistemic reasoning to practical reasoning,from individual reasoning to multi-agent reasoning,etc.In the area of formal argumentation,there are two main research directions:abstract argumentation([1,15])and structured argumentation([10,12,18,19,9,21]).The former is mainly about handling conflicts between a set of conflicting arguments to obtain sets of arguments that can be accepted together,while the latter focuses on how to construct arguments from underlying knowledge base and to identify defeat relations between arguments,such that the conclusions obtained from an argumentation system satisfy somerationality postulates([7]),which guarantee that the outcome of the system is logically reasonable.

In this paper,we focus on structured argumentation.A structured argumentation system is mainly composed of the following parts:a logical language to represent underlying knowledge,a definition of an argument,a definition of the conflict/defeat relation between arguments,and an approach to evaluate the status of arguments.In all these parts,there are different design choices,which may affect the conclusions of an argumentation system.For instance,when arguments are in conflict,under which conditions an argument rebut another argument?In existing literature,there are two different optionscalledrestrictedrebutandunrestrictedrebut.([4,7,5,6])Theformeronlyallowsa rebutting on an argument whose top rule is defeasible,while the latter allows a rebutting on all defeasible arguments,i.e.,the arguments contain at least one defeasible rule,no matter where the defeasible rule is located.The intuition behind unrestricted rebut is that a conclusion is defeasible if and only if it has been derived using at least one defeasible rule.The intuition behind restricted rebut is that in order to argue against a particular derivation,one has to argue against its premises.([4])Now,an important issue is that adopting different design choices may lead to different problems.As observed in[19]and[6],a formalism with unrestricted rebuts may violate some rationality postulates,while it is not natural in many cases to adopt restricted rebut when considering human reasoning.

Given the pros and cons of adopting restricted rebut and unrestricted rebut,the goal of this paper is not to develop a new system to cope with those problems,but to analyze two systems using different definitions of rebutting,and conduct an empirical study.We expect that the empirical results can provide some hints for future development of structured argumentation formalisms.

The initial idea of the present study was first presented on MIREL workshop 20161Workshop on MIning and REasoning with Legal texts-December 14th,2016-Nice(France)..A revised questionnaire survey was carried out then.In the current survey,we modified some expressions of the questions to make them more precise and clearer.An additional question is added for each case to get more parameters.Furthermore,we increase the number of subjects,and add one more case which represent a generalized version of unrestricted rebut in the questionnaire.Moreover,each part of the paper has all been extended.

The structure of this paper is as follows.In Section2,we provide some basic notions of formal argumentation,and analyze the pros and cons of applying restricted rebut and unrestricted rebut.In Section 3,we introduce an empirical study and its results.In Section 4,we conclude the paper with a discussion.

2 Formal Argumentation Systems:Restricted Rebut vs.Unrestricted Rebut

In the current section,we introduce some notions concerning formal argumentation,and analyze the pros and cons of applying restricted rebut and unrestricted rebut in a formal argumentation system.In general,a formal argumentation system can be illustrated in Fig.1(taken from[14]),which is composed of a knowledge base representing by a(logical)language,a set of arguments constructed from this knowledge,a set of conflict/defeat relation between arguments,and an approach to evaluate the status of arguments.A pair of a set of arguments and a set of defeat relation(also called attack relation)between arguments is called an argumentation framework.The evaluation of status of arguments can be done at an abstract level where the structures of arguments and the origins of attack relation can be ignored.After evaluation,a set of arguments that are acceptable together is called an extension of the argumentation framework.For each extension of arguments,the associated set of conclusions is the output of the system.

To model a formal argumentation system,one important issue is to guarantee that the outcome of the system satisfies some rationality postulates and human reasoning intuitions.However,it may be the case that when some postulates/intuitions are satisfied,others are violated.In this section,we introduce two such systems(i.e.,ASPIC+andASPIC−)and analyze their pros and cons.2Apart from ASPIC+and ASPIC−,there are also other structured argumentation systems(like DeLP,ABA and deductive argumentation)where the issue of restricted versus unrestricted rebut plays a role.Readers are referred to[3]for more information.

2.1 ASPIC+and ASPIC−

TheASPIC+andASPIC−frameworks originate from the EuropeanASPICproject,and are formulated in a series papers,including[5],[18]and[20],etc.In all these for-malisms,arguments are constructed from a set of strict rules and defeasible rules,as well as a set of premises.One basic difference betweenASPIC+andASPIC−is that the former only allows a rebutting on an argument whose top rule is defeasible,while the latter allows a rebutting on all defeasible arguments.In the following,we provide the definitions ofASPIC+,while the corresponding definitions ofASPIC−are omitted.

Figure 1:A typical working mechanism of an argumentation system

InASPIC+,the main components are a logical languageLunder negation¬or a more general notion of conflict(i.e.contrariness),two sets of inference rules,and a knowledge base.Firstly,a triple called an argumentation system consists of three parts:a languageL,a set of rules,and a partial function mapping some rules to distinct names,which are also elements inL.In terms of[20],we have the following definition.

Definition 1An argumentation system is a tupleAS=(L,R,n),where

·Lis a logical language closed under negation“¬”;

·R=Rs∪Rdis a set of strict(Rs)and defeasible(Rd)inference rules of the formφ1,...,φn→φandφ1,...,φn⇒φrespectively(whereφiandφare meta-variables raging over wff inL),andRs∩Rd=∅;

·n is a partial function such that n:Rd→L.

We writeψ=−φwhenψ=¬φorφ=¬ψ.

Secondly,the premises of an argument are from a knowledge base,which can be defined as follows.([20])

Definition 2A knowledge base in anAS=(L,R,n)is a setK⊆Lconsisting of two disjoint subsetsKn(the axioms)andKp(the ordinary premises).

The combination of an argumentation system and a knowledge base is called an argumentation theory,denotedAT=(AS,K).

GivenanAT,arguments can be defined by chaining inference rules fromASintoinference trees.Letall the formulas ofKused to build an argumentαdenoted byPrem(α),the conclusion ofαdenoted byConc(α),all the sub-arguments ofαdenoted bySub(α),all the defeasible rules ofαdenoted byDefRules(α)and the last rule ofαdenoted byTopRule(α).Formally,we have the following definition.([20])

Definition 3An argumentαon the basis of an argumentation system(L,R,n)and a knowledge baseK⊆Lis defined as:

1.φifφ∈Kwith:Prem(α)={φ},Conc(α)=φ,Sub(α)={φ},DefRules(α)=∅,TopRule(α)=undefined.

2.α1,...,αn→ψifα1,...,αn(n≥1)are arguments such that there exists a strict ruleConc(α1),...,Conc(αn)→ψinRswith:Prem(α)=Prem(α1)∪...∪Prem(αn);Conc(α)=ψ;Sub(α)=Sub(α1)∪...∪Sub(αn)∪{α};DefRules(α)=DefRules(α1)∪...∪DefRules(αn);TopRule(α)=Conc(α1)...Conc(αn)→ψ.

3.α1,...,αn⇒ψifα1,...,αn(n≥1)are arguments such that there exists a defeasibleruleConc(α1),...,Conc(αn)⇒ψinRdwith:Prem(α)=Prem(α1)∪...∪Prem(αn);Conc(α)=ψ;Sub(α)=Sub(α1)∪...∪Sub(αn)∪{α};DefRules(α)=DefRules(α1)∪...∪DefRules(αn)∪{Conc(α1),...,Conc(αn)⇒ψ};TopRule(α)=Conc(α1)...Conc(αn)⇒ψ.

Foranyargumentα,wesayαisstrictifandonlyifDefRules(α)=∅,andPrem(α)∩Kp=∅;defeasibleifandonlyifDefRules(α)̸=∅,orPrem(α)∩Kp̸=∅.Givenasetof argumentsE,we useConcs(E)={Conc(α)|α∈E}to denote the set of conclusions supported byE.

Based on the above formal definitions,let us consider the following example.

Example 1Letα,βbe two arguments as follows:

·α:Tom is probably a bachelor because he goes to pubs frequently;

·β:Tom is probably married because he wears a ring.Since someone who is married is not a bachelor,Tom is not a bachelor.

LetL={r,m,b,p,¬b,¬m},wherer,m,b,pdenote ‘wear a ring’,‘be married’,‘be a bachelor’and ‘go to pub frequently’respectively.

Rs={m→¬b}∪{b→¬m}3Rsis assumed to be closed under transposition in ASPIC+/−([19,5]),thus we have“b→ ¬m”transposed from“m → ¬b”.,Rd={p⇒b;r⇒m},andK={p,r}.Then,we may construct the following arguments:

As illustrated by the above example,not all arguments can be accepted together.For instance,sinceαandβhave contradictory conclusions,they can not be accepted at the same time.So,given a set of arguments,before their status are evaluated,all the conflicts between them should be identified.In terms of[20],three kinds of conflict between arguments are defined as follows.

Definition 4Letαandβbe arguments.

1.αundercutsβonβ′if and only ifConc(α)=−n(r)4‘n(r)’means that rule r is applicable.for someβ′∈Sub(β)such thatTopRule(β′)=rwherer∈Rd;

2.α(restrictively)rebutsβonβ′if and only ifConc(α)=−Conc(β′)for someβ′∈Sub(β),andTopRule(β′)∈Rd;

3.αunderminesβonφif and only ifConc(α)=−φfor any ordinary premiseφofβ.

Note that in Definition 4,the notion of‘rebut’is restricted.For unrestricted rebut,the second item of Definition 4 is modified to the following:

2.′αunrestrictively rebutsβonβ′if and only ifβ′is defeasible andConc(α)=−Conc(β′)for someβ′∈Sub(β).

Then,it is said that an argumentαattacks an argumentβif and only if(1)αrestrictively(unrestrictively)rebutsβ,or(2)αundercutsβ,or(3)αunderminesβ.Meanwhile,whether an attack fromαtoβ(on its sub-argumentβ′)succeeds as a defeat may depend on the relative strength ofαandβ′.Letbe a binary ordering on the set of all arguments that can be constructed on the basis of an argumentation theory.In terms of[20],the definition ofdefeatis formulated as follows.

Definition 5αdefeatsβif and only ifαundercutsβor successfully rebuts or successfully underminesβwith respect to.

Note that the term ‘defeat’in Definition 5 is called ‘attack’in Dung’s abstract argumentation,where each attack succeeds as a defeat.([8])Hence,the term ‘defeat’in this paper is corresponding to the term ‘attack’in[8],while the term ‘attack’in this paper is a rebutting,an undercutting,or an undermnining.LetAbe a set of arguments constructed from an argumentation theoryAT=(AS,K)andRbe the set of defeat between arguments.Then,we call a tupleFAT=(A,R)be an argumentation framework.

Example 2Continue Example 1.LetA={α1,α,α′,β1,β2,β}.Assume that all arguments have equal strength.LetR1andR2be the sets of defeats by applying restricted rebut and unrestricted rebut respectively.It holds thatR1={(β,α),(β,α′),(α′,β),(α′,β2)}andR2=R1∪{(α,β),(β2,α′)}.Two argumentation frameworks(A,R1)and(A,R2)are illustrated in Fig.2.

Figure 2:Argumentation framworks corresponding to restricted/unrestricted rebut

2.2 Abstact argumenation:eveluating the status of arguments

After an argumentation frameworks has been constructed from argumentation theories,the status of the arguments can be evaluated according to the extension-based approach or labelling-based approach in the field of abstract argumentation([8]).

In the extension-based approach,given an argumentation framework(A,R)andE⊆A,we say:Eisconflict-freeif and only if∄α,β∈Esuch that(α,β)∈R;α∈AisdefendedbyEif and only if∀β∈Aif(β,α)∈R,then∃γ∈Esuch that(γ,β)∈R;Eisadmissibleif and only ifEis conflict-free,and each argument inE⊆Ais defended byE;Eis acomplete extensionif and only ifEis admissible,and each argument inAthat is defended byEis inE;Eis agrounded extensionif and only ifEis the minimal(with respect to set-inclusion)complete extension;Eis apreferred extensionifandonlyifEis amaximal(with respect toset-inclusion)completeextension;Eis astable extensionif and only ifEis conflict-free andEdefeats each argument that is not inE.

Example 3Continue Example 2.It holds that

(A,R1)has two preferred extensionsE1,1={α1,β1,β,β2}andE1,2={α1,β1,α,α′},while(A,R2)has three preferred extensionsE2,1=E1,1,E2,2=E1,2,andE2,3={α1,β1,α,β2}.

It turns out thatConcs(E1,1)=Concs(E2,1)={p,r,¬b,m},Concs(E1,2)=Concs(E2,2)={p,r,b,¬m},Concs(E2,3)={p,r,b,m}.

2.3 Properties of structured argumenation systems

As presented above,given an argumentation theory,by means of argument construction and evaluation,we get a set of extensions,each of which is a set of arguments that can be accepted together according to some criteria.Then,a set of conclusions of the system is obtained.Now,the question is whether this set of conclusions can satisfy some rationality postulates?In[7],the authors introduce the following postulates:

·Subargument Closure:for every argument in an extension also all its subarguments are in the extension.

·Closure under strict rules:the set of conclusions of all arguments in an extension is closed under strict-rule application.

·Direct consistency:the set of conclusions of all arguments in an extension is consistent.

·Indirect Consistency:the closure of the set of conclusions of all arguments in an extension under strict-rule applications is consistent.

RegardingASPIC+andASPIC−,it has been shown that the latter satisfies the postulates only under grounded semantics,while the former satisfies the postulates under any complete-based semantics.

Example 4Continue Example 3.It holds thatConcs(E1,1),Concs(E2,1),

Concs(E1,2)andConcs(E2,2)are consistent,both directly and indirectly.However,althoughConcs(E2,3)={p,r,b,m}is directly consistent,but it is not indirectly consistent,inthattheclosureofConcs(E2,3)under strict-rule application is equalto{p,r,b,m}∪{¬b,¬m}which is inconsistent.

SinceASPIC−may violate some rationality postulates,it seems that it is better to adopt restricted rebut in a structured argumentation system.However,as pointed out in[5],this problem is controversial.This is illustrated by the following example to argue that it seems more natural to apply unrestricted rebut in daily life human reasoning.([5])

Example 5In this example,John and Mary have the following arguments.

John:“Bob will attend conferences A and I this year,as he has papers accepted at both.”

Mary:“That won’t be possible,as his budget of£1000 only allows for one foreign trip.”

LetL={accA,accI,budget,attA∧attI,¬(attA∧attI)}whereaccA,accIandbudgetdenote ‘paper accepted at conference A’,‘paper accepted at conference I’and ‘budget is£1000’respectively.LetRd={accA⇒attA;accI⇒attI;budget⇒¬(attA∧attI)},andRs={attA,attI→attA∧attI}.There are the following arguments:

When adopting restricted rebut,M2does not rebutJ5.It turns out that Bob will attend conferences A and I,which seems counter intuitive.

Besides the restricted/unrestricted rebut mentioned above,in[11],the authors propose a disjunctive rebut by generalizing the unrestricted rebut,which could be described as:An argumentαgenerally rebuts an argumentβif and only ifβis defeasible and for someβ1,...,βn∈Sub(β),Conc(α)(“¯”denotes the contrariness).By the definition of this kind of rebutting,in the above example,M2rebutsJ5,since it claims that the conclusions of two defeasible sub-arguments ofJ5,which are“attA”and “attI”,cannot hold together.

3 Empirical Study

Since theoretically there are pros and cons to apply restricted rebut and unrestricted rebut,in this paper we conduct an empirical study to examine whether human intuitions are more in line with restricted rebut or with unrestricted rebut.

In this survey,the subjects are provided with a couple of cases,each of which consists of an argument and a counterargument from a dialogue.The counterargument attacks the first argument under unrestricted rebut but not under restricted rebut.The respondents are asked whether they feel that the counterargument is a legitimate response to the first argument or not.If the answer is YES,then their intuitions are more in line with unrestricted rebut;otherwise,their intuitions are more in line with restricted rebut.We also designed an example of the generalized rebut by[11].Furthermore,in this survey,we add a question for every example:if a respondent feels that the counterargument is a legitimate response,then he is asked whether he think the counterargument actually attacks the first argument.Based on these ideas,a questionnaire used in this survey looks like the following.

Questionnaire5Since the survey was conducted among Chinese students,the questionnaire was designed in Chinese.What we show in the paper is the English translation of the questionnaire.

In each of the following six cases,you will see two arguments A and B.Intuitively,do you think that B is a legitimate response to A?If your answer to this question is YES,then do you think B actually attacks A?

1.A:“We found Steven’s DNA at the scene of the crime,and we also confirmed that he has a similar previous conviction,so Steven is probably the murderer.”

B:“Steven is not the murderer,because eyewitness Branden testified that Steven was not at the scene when the murder happened.”

2.A:“Jessica is a fan of two popular Korean bands,EXO and Bigbang.Both of them will hold concert series separately at nearby cities in next few weeks.So,Jessica will attend at least two concerts soon.”

B:“That won’t be possible.She has been assigned too much work recently,so that she doesn’t have the time to attend two concerts.”

3.Assume that“if a man is tall,rich and handsome,then he is a‘tall rich handsome’6In China,“tall rich handsome”(“高富帅”in Chinese)is a popular expression that stands for a very desirable and admirable man(much like “Mr.Right”in English speaking countries),while an occupational chauffeur is normally not such a man..”

A:“Lee is not only tall and handsome,but also rich,if you ever seen the luxury car he drives.So,Lee may be ‘tall rich handsome’for girls.”

B:“No,he is not.He is an occupational chauffeur for the car owner.”

4.Assume you agree that“crime is always forbidden”.

A:“Lying leads to crime.Crimes should be banned.So,lying should be forbidden.”

B:“Sometimes telling white lies may help others,there is no reason to for bid people from doing something helpful.Thus lying should not be forbidden.”

5.A:“Every human is mortal.Lu Xun(a great writer from China)is a human.Lu Xun is mortal.”

B:“Lu Xun is immortal,because I am happy today.”

6.A:“I have a gift for good friend.Lee and King are both my good friends,if I give a gift to Lee,then I should also give a gift to King.”

B:“But there is only one gift.Either you cannot give a gift to Lee,or you cannot give a gift to King.”

In this questionnaire,we formulate six different cases for different scenarios so that we may not rely so much on a single case.In the first case,there is an attack fromBtoAunder both restricted and unrestricted rebut.In the fifth case,there is no attack fromBtoAunder restricted or unrestricted rebut,sinceA’s argument is strict.In the cases 2,3 and 4,there are attacks under unrestricted rebut but not under restricted rebut.In the last case,the counterargument attacks the first argument as:for two argumentsαandβ,ifα′,α′′∈Sub(α),whileConc(α′)=p,Conc(α′′)=qandConc(β)=¬p∨¬q,thenβgenerally(disjunctively)rebutsα.

What follows is the formal structure of the arguments in the questionnaire.7Here att1and att2denote for the attack relations under restricted rebut and unrestricted rebut respectively.

1.LetL={D,m,pc,tB,¬as,¬m},whereD,m,pc,tB,andasdenote ‘DNA of Steven’,‘Steven is the murderer’,‘Steven has similar previous conviction’,‘testimony of Branden’,and ‘Steven was at the scene’,respectively.K={D,pc,tB},Rd={D⇒as;as,pc⇒m;tB⇒¬as;¬as⇒¬m},andRs=∅.Arguments are:

2.LetL={fE,attE,fB,attB,attE∧attB,w,¬t,¬(attE∧attB)},wherefE,attE,fB,attB,w,andtdenote ‘fan of EXO’,‘attend the concert of EXO’,‘fan of BigBang’,‘attend the concert of BigBang’,‘too much work’,and ‘afford time’,respectively.K={fE,fB,w},Rd={fE⇒attE;fB⇒attB;w⇒¬t;¬t⇒¬(attE∧attB)},andRs={attE,attB→attE∧attB}.Arguments are:

3.LetL={t,h,lc,r,t∧r∧h,MR,c,¬MR},wheret,h,lc,r,MR,andcdenote‘tall’,‘handsome’,‘drive luxury car’,‘rich’,‘Mr.Right’,and ‘chauffeur’,respectively.K={t,h,lc,c},Rd={lc⇒r;c⇒¬MR},andRs={t,r,h→t∧r∧h;t∧r∧h→MR}.Arguments are:

4.LetL={l,c,f,wl,h,¬f},wherel,c,f,wl,andhdenote ‘lying’,‘crime’,‘be forbidden’,‘tell white lie’,and ‘help people’,respectively.K={l},Rd={l⇒c;l⇒wl;wl⇒h;h⇒¬f},andRs={c→f}.Arguments are:

5.LetL={LX,h,m,hp,¬m}whereLX,h,m,andhpdenote ‘LuXun’,‘human’,‘mortal’,and ‘I am happy today’,respectively.K={LX,hp},Rd={hp⇒¬mortal},andRs={LX→h;h→m}.Arguments are:

6.LetL={one_g,fL,fK,gL,gK,¬gL,¬gK},whereone_g,fL,fK,gL,andgKdenote ‘have a gift to good friend’,‘Lee is good friend’,‘King is good friend’,‘give a gift to Lee’,and‘give a gift to King’,respectively.K={one_g,fL,fK},Rd={one_g,fL⇒gL;fL,fK,gL⇒gK;one_g⇒¬gL∨¬gK},andRs=∅.Arguments are:

By using the above questionnaire,the survey took place among 230 undergraduate students(101 males and 129 females)whose majors covers computer science,law,linguistic,economics,management,museology,engineering,medical science,business administration and advertisement.The ages of the students were between 18 and 22.And,88 percent of these respondents had no logic background(the rest had accepted a short-term course ‘introduction to logic’),so that they could represent the general audience in argumentation.

Figure 3:Result

The results are showed by the table and graph in Fig.3.

For the case 1,88.7%of the respondents agree that B’s argument is a legitimate counter reaction,and 84.8%of them think thatBactually attacksA’s argument(75.2%of all the respondents).For the case 2,3 and 4,there are respectively 71.7%,72.2%and 83.9%of students agree thatB’s argument is a legitimate counter reaction,and respectively 91.5%,94.6%and 83.9%of them think thatBactually attacksA’s argument(65.7%,68.3%,70.4%of all the respondents respectively).For the case 6,there are 85.2%of the respondents agree thatB’s argument is a legitimate counter reaction,and 81.1%of them(69.1%of all the respondents)think thatBactually attacksA’s argument.

For the case5,only12.2%of the respondents think thatB’s argument is a legitimate reaction,and only 4.8%of the respondents thinkBattacksA’s argument,which means that people’s intuitions are in line with that we cannot attack a strict argument..Especially,none of those respondents who have accepted logical training answers YES to the first question of this case.

According to the empirical result,most of the respondents agreed that in the cases 2,3,and 4,arguments ofBattack arguments ofA,which reveals that people’s intuitions are more in line with unrestricted rebut.What’s more,the answer to the case 6 shows the disjunctive rebut as we shows is in line with people’s intuition.

4 Conclusions

In this paper,we have introduced the basic notions of formal argumentation systems,and analyzed the pros and cons of adopting restricted or unrestricted rebut in an argumentation system.Based on the theoretical analysis,we have conducted an empirical study.The results show that according to people’s intuitions,it is more natural to adopt unrestricted rebut.

These results are especially relevant when argumentation has dialectical aspects,for instance,when there is a bilateral discussion going on.This might be because human reasoning intends to look an argument as an integral entity,not only considers its toprule.People analyze conflicts between arguments unconsciously.When they put forward counterarguments, they know somehow the argument that they are attacking is not strictly strong, no matter where the weak point is (at the top rule or any other parts).There stricted rebut constrains rebutting only on the arguments with defeasible top rule,so that it forces people to modify the way they give their opinion,to make their counterarguments suit the requirement of an argumentation system for making an attack,which is unnatural.On the other hand,although unrestricted rebut is more intuitive than restricted rebut,it comes at a price:when unrestricted rebut is applied,the formalism will be guaranteed to satisfyClosureandIndirect Consistencyonly under grounded semantics([5]).

Since there are advantages and disadvantages of applying restricted rebut or unrestricted rebut and the unrestricted rebut is more natural to people’s intuitions,it is worth to further develop argumentation formalisms that are not only consistent with people’s intuitions,but also general enough to cover most of argumentation semantics.According to the working mechanism of formal argumentation systems introduced in section 2,in order to develop a well-behaved system,the requirements of different components need to be considered systematically.More specifically,at the language level,the requirements are mainly about the consistency of conclusions,while at the argument level,the requirements is mainly about when an argument can be attacked and when an argument can be accepted.How to better combine the requirements at two levels to improve the properties of an argumentation system is an open problem,and will be our future work.

What’s more,the presented survey considers only a few types of argumentation,and the respondents only cover undergraduate students.In further studies,we plan to extend the survey by including more kinds of argumentations as well as by including more varied respondents,so as to further explore the relationship between natural language ar-gumentation and the formal argumentation formalisms,which can bring more guidances for the improvement of formal argumentation systems designing.