Arguments - Semantic Scholar

正ICETRANS． INF＆SYST．， VOL．E88−D， NO、9 SEPTEMBER 2005

2021 PAPER Special section on software Agent and’ts Apρlica tions

Applying Logic of Multiple-Valued Argumentation to Eastern

Arguments HajimeSAWAMURAandTakehiaTAKAHASHI, Nonmembers

SUMMARY ln our fbrmer paper． we formalized a Logic of Muitiple・

reSearCh・where interaction among agentS is eSSen！ial and

valued Argumentation（LMA）・n an巴xpressi・e㎞・wledge rep祀5en励on

social mechanisms such as negotiation， persuasion， con−

language， Extended Annotated Log三c Program皿ing（EALP）， in order to ：胆ke it pessib］e to construct argum聞ts under unce血in infbrma［ion． In

sensus attamment， etc． are emphasized as well as self−

吐Us paper， Wヒcon丘rm expressivity and applicabiEity by applying LMA te

delibera目on［3］，［15ユ．

afgumentS refiecti ng East巴mers’preference over argurnenta吐on a5 we1］as

    We have developed a Iogic of multiple−valued argu・

Eastem tlought a皿d philosophy． In doing so， we exploit a wide variety of

mentation LMA theoretically［19］． We believe thaロhis is

c。mplete latUces as皿血一values． sh・wing曲e fleXibility and adaptabitiry。f

也・first the・ry・f a血11−・iress argurnenta廿・n in v・hich argB−

LMA toΨarious multiple−valuedness required in argument且tion under un− ce血in information． ln particular， vve c・nsid識signifi伽t specialization

ments among agentS can・be made using uncertain knowl−

of LMA to T加a工e而ma with an Eastem mind． Through various argument

edge． In面s paper， we will cenfirm its expressivity and ap−

exampl巴s， it is shown that LMA allows fbr a kind of pluraiistic argumenta−

plicability by applying LMA to arguments reflecting East−

tion， or a fUsion of Eastem and West巴m argumentation． key w・rdS：agent，・argumentat輌・n． multipte・v・∼uedne∬， tetralenvna， easr−

eηnthought

enners’ 垂窒???窒?獅モ?@over argumentation as wdl as East醐 thought and philosophy． In doing so， we exploit a wide vari−

ety・f complete lat目ces as㎞由一values， showing也e・fiexibil− 1．  Int『ediユctiOR

ity a皿d adaptability of LMA to various multiple−valuedness

required in argumentatien und訓nc斑迦㎡brmation． This Traditional syrnbo韮c AI models of reasoning have been ba− sicaUy fbunded upon logic such as first order predicate ca1−

cuius，品d later have㎞ed to seek m・re expressive 1・gics since the f・rmer is inadequate for dealing with situati・ns

which are incomplete， uncertain， or dynamicaUy changi皿9．     On也e other hand， studies on argumenta毛］on and infor−

mal Iogic haΨe been do鵬with也eir own purposes， some− t口工1es criticizing traditienal logic as saying that it is not re−

ally a true legic but simply one of several mathematica三the−

odes． Argumentatio皿， at丘rst glance， seems to be synony−

mous with logic in some logic textbookS． In compariso蝋o Iogic， argurnentS are not proefs in the traditional s酷se， but

proofs that can faiL They are deft∋asible or ovem11ed proofs

when faced t・・也er王㎎umen題虻are bette…s仕・ng・・ 出a皿the o亘9三nal argument・Put it differently， a proof is in− variably a static obj ect， and an argument is a proof of dy−

namlc皿at蹴Therefbre， argumentation is more suitable to describ㎞g and processi皿9廿1e dynalnic and cha皿gi皿g nature of infor nation in a networked dist】dbuted informadon envi＿ ro㎜ent， or to problem solving in ill−fbmed unce血in prob− lem domains where modeling is cli伍cult to make．

    Recently it has been recognized by computer scien軸

6sts・and in particuIar AI researchers that argumentation has clear correlation with the interests of the newly巴加erg一

三ng丘eld of soc呈al computation such as multi−agent systems

aspect of LMA is most advantageous to other approaches to multiple一Ψalued a㎎umentation such巨s［4］，［18］． Hae皿i et 品・uses the term’ argumentation’in his probabilistic argu− mentation systems［6］， but it is not a dialectics・−based ftame−

work as opposed to LMA． as well as o也er argumentation

systems wi甑d坤ctical pro。f theory． lt should be noted 出at LIVLA is different f『om our fommer paper［20］in that it is

a4−valued paraconsistent argumentation system wjth differ− ent phiiosophy in its fbn皿aliza廿on， paying attention to the

notion of difference as a momentum of argum聞tation．     Tbe paper is oエga」nized as follows． In Sect．2， we out−

1ine the underlying language for the legic of Inultiple−valued

argumentation， Extended Annotated Logic Programming （EALP）， toge出er with its interpretation． EALP is a very expressive， in the sense that it allows to r巴present different

kinds of uncertainty such as vagueness a皿d inconsistency（or

paraconsistency）i蝋em 3 of multi−valuedness、 and in。om− pleteness with the help of de』1t nega加n・1n Sect・3， we

ouUine the logic of inultiple−valued argumentation（LMA）．

in Sect．4， we illustrate two arg迦e斑example iロwhich a popular set of trut｝1 values， the closed interΨal猟［0．1］of real mlmbers， and a power set of dates as a co：nplete lattice

are employed fbr argumentation under unceltain informa− tien． In Sect．5， we describe a signi｛icant specialization of

L蜘o愉由e㎜awi血輪stem mind． We discuss in particular the third and fbuf曲h le】㎜as of Tetrale］㎜a which

   Manusc坤t民ceived Novemもer 24，2004．    Manusc『t revised May 16．2005．

   tTh・ a画・rs眠wi臨M9・t・U㎡vers触NUg・幅一甑95［ト、 2181Japa11．

are said to characte］rize Eastern thought and logic・In Sect・6，

we take up two argument examples to曲ow伽t LMA a1− 1・ws・f・r・a・kind・f pluralistic・argumentati。叫胴血si・n of

  a）E・mai1：sawamura＠ie．niigata−U．ac」P

Eastem and Westem a培umen血d皿． They also坦us蜘e the

 b）E−mail：takeh」isa＠csje、niigata．u、acjp

uses of由e敵d and fb賦h le…．血the釦al sec目on we

   DOI：10．1093／ietisy／e88−d9．2021

summarize the paper by d巴sc曲ing曲antages and implica一

Copyright＠20e5 ’1 he lnstitute of Eleetronics， lnformation and Communicatien Engineers

IEICE TRANS．聴＆SYST，VOLE89−D， NO．9 SEPTIEMBER 200S

2022

tjons of LMA，

in the ru］es can evaluate to the elements of 7’．     The head of a rule is called a conclusion of a rule． An−

2・ Extended Annotated Logic Programs

notated o正）jective literals and annotated default literals in the

body of the rule are called anteceゴβπr50f the rule and as−

EALP is an underlying knowledge representation language

sumptions of the rule respectively． We identify a distributed

that we formalized for our logic of multiple−valued argu−

EALP with an a8ent， and treat a set of EALPs as a mul’」一

mentation LMA． EALP has two kinds of explici｛ negation：

α88n砧yぷf醐．

Episternic Explicit Negation ‘一’and Ontological］巳xpl三cit Negati・n‘一一’， and the default negation ‘not’． lhey are sup−

2．2  1nterpretation

posed to yield a momentum or driving force for argumenta− tion or dialogue in LMA、 Here， we outline EALP．

1）efinitio皿4： （Extended annotated Herbrand base）． The set of all annotated literals constructed f士om an EALP」P on

2」  Language

acomplete latticeワーof tuth values is called the extended

annotated Herbrand base璋． Definition 1：（Annotation and a皿netated atoms［9］）． We  assurne a complete lattice（？一，≦）of truth values， and denote

1）efinition S：（lnterpretation）． Let？一 be a complete lat−

 its least and greatest element by⊥and T resp㏄tiΨely． The

tice of tru血values， and P be a皿EALP． Then，也e interpre−

 Ieast upper bound operator is denoted by u． An annotati皿

tation on P is the subset∬⊆・確of the extended annotated

 is ei吐Ler an element of「アー（constant annotation）， an annota−

 tion variable on「r， or an annotation term． Annotation teml  is defined recursively as」follows：an element of？−and anno−  tation variable are annotation terms． In addition， if t t，．．．，tn

i鵬a皿notatiOn terms， then f（tl，＿，tn）is an a皿otation term．

Here， f is a tota1 continuous fUnction of typeワm→7−． ．      If A is an atomic fbrmula andμis an annotation， then

Herbrand base畔of P such that for any annotated atom A，

  1．lf A：μ∈Iandρ≦μ， then A：p｛≡1（downward     heredity）；

  2．if A：μ∈Iand A：ρ∈1， then A：（μUρ）∈…1（tolerance     of difference）；

  3．If∼Alμ∈∫andρ≧μ， then∼A：ρ∈∬（upward     heredity）．

・A：μis an annotated atom． We assume an annotation func−  tion r：ワ’→「1−， and define th at r（ノ1：μ）：＝A：（「μ）． rA：μ

 is called the epistemic explicit negation（e−explicit negation）

ofA：μ．      1n this paper， the e−explicit negatien ”A：μis embedded

into an annotated atom、4：「μ， and implicitly handled．

Defin」籠on 2：（A皿otated literals）． Let A：μ be an anno−

    The conditions l a皿d 20f De丘nition 5 reflect the defi− nition of the ideal of a co111Plete lattice of truth values． The

ideals−based semanticS was丘τst introduced for the interpre−

tadon of GAP by Kifer and Subrahmanian［9］． Our EALP for argumentation also employs this since it was shown that the general semantics with ideals’ 奄刀@more adequate輪也e resUicted one simply with a complete lattice of truth val−

tated atom・Then∼（A：μ）is the ontological explicit nega−

ues［19工．

tion（o−expliciE negation）of A ：μ． An annotated objective

    We de丘ne three notiens of inconsistencies correspond−

Uteral is either∼A：μ or A：μThe symbol・’・is also used

i皿gto three concepts of negation in EALP．

to denote co皿Plemel1町y an皿otated objective literals． Thus ∼∼

`：μ＝A：μ

    If L is an annotated objective literal， then not L is a

1｝e血nitio皿6： αnconsistency）． Let∬be a皿interpretation．

Then，

default negatipn of L， and called an annotated default litera！．

  L・A：μ∈Iand rA：μ∈1⇔Iis epistemologically

An annotated literal is either of the form not L or L．

    inCOnSiSte11t（e−inCOnSiStent）．

The tems， epistemic n’egation and ontological negation， originate from Kifer and Lozinskii［8］． Note， however， that

the rneaning of our ontological explicit negation is different from their ontological negation， being properly adj usted to argumentation as can be seen in the succeeding sections．

Defi皿ition 3：（Extended Annotated Loglc Progralns （EALP））． An extended a皿notated logic prograrn（EALP）is aset Qf annotated rules of the fbrm：H← Ll＆．．．＆」Ln， where」H is an annotated obj ective litera1， and Li（1≦i≦n） are annotated literals in which the annotation is either a con−

stant annotation or an annotation variable．

  2，A：μ∈land∼A：μ∈∬⇔lis ontologicaUy incothsis−     tent（o−incOn畠iStent）．

  3、A：μ∈∬and not A ：Pt∈∬， or∼A：μ∈1and not∼A：     μ∈1⇔・「is inco皿sistent in default（d−inconsistent）、

    When an interPretation I is o−inconsistβ且t or d− inconsistent， we simply say l is inconsistent． We do not

see the e−inconsistency as a problematic㎞consistency since by the condition 20f Definidon 5， A：μ∈1 and ”A：μ＝、A：

「μ∈∬imply A：（μ］「μ）∈1 and we think A：μand rA：μ are an acceptable’differentia．

    Let∬be an inte甲retadon such that∼A：μ∈∬． By the condition l of Definition 5， fbr anyρsuch thatρ≧μ，

    For sirnplicity， we assume that a rule with annotation

．if・4：ρ∈1．then 1 is o−i丑consistent・In other words，∼A：μ

variables or obj ective variables represents every ground in−

rej ects all recognidonsρsuch thatρ≧μabout A． This is the

stance of it． In面s assumption， we restiict ourselves to con−

underlying reas皿for adopting the condition 3 of Definition

stant annotatiens in this paper since eΨery annotation term

5．These notions of inconsistency yield a Iogical basis of

SAWAMURA and TAKAHASHI： APPLY ［NG LOGIC OF MUI：皿PLE−VALUED ARGUMENTAnON rD EASTERN ARGUMaNTS

2023

attack relations described in the multiple−valued argumenta一

argument set， and x be an attack relation onル8s． S⊆Ar8s

丘on of Sect．3．

is confi三ct−free wrt・x⇔Sd㏄s not contain arguments、4r宮1

De丘nition 7：（Satisfaction）， Let I be an interpretation．

and Arg2 such that（Arg 1，Ar82）∈x，

For any annotated ebjective literal H and annotated literal」L

Theorem 1：Let x and y be attack relations on Args． If

and Li， we define the satisfaction relation denoted by ‘t’as

x⊇ythen JArgs，x／y i S confiict−free wrt． x．

follows．

    For the proof、 readers should Tefer to our fomer pa−

●1ト」乙⇔L∈∬

per［19】and technical paper cited in it， as well as for other

・1 1・ L，＆…＆Ln⇔1 P Li，＿，1トLn

proofs to be omitted in the rest of this paper．

・1FH←L1＆…＆Ln⇔ 1トHorl≠

  L1＆…＆Ln

3．Multiple−valued Argumentation

3．2 Dialectical Proof Theory Justified arguments can be dia王ectically detemined from a set of arguments by the dialecticai proof theory． We give the

sound and complete dialectical proof出eory fbr the abstract

In formalizing logic of argumentation， tbe most primary COnCem iS the rebUttal relatiOn amOng肛gUmentS SinCe it

argumentation semantics JA瑠エ．エty・

yields a cause or a momentum of argumentation or dia−

Definition 11： （x／1」−dialogue［13］）．  An x／y−dialogue

10gue． The rebuttalτelation for two−valued argument mod−

is a finite nonempty sequence of moves mov召1 ＝

els is Hlost simple， so that it naturaUy appears betWeen the

（」P互αyeri，Argi）， （i ≧  1） such that

contradictory propositions of the form A and rA． In case of

multiple−valued argumentation based on EALE much com− plication is to be involΨed into the rebuttal relation under the

different concepts of negadon． One of the ques目ons arising 丘om multiple−valuedness is， fbr example， how a literal with

truth−valueρcon丘onts with a literal with㎞th−valueμin the

 1．Player輌＝P（Proponent）iff匡is odd；and Play已ri＝0     （（）pponent） ⇔  匡is even．

 2・If P」ayeri＝Pla：ソerj’＝二P（i≠」）then Argi≠Argj．  3．If、Playeri＝・」P（匡≧ 3）then（Argi，Ar8i＿1） ∈y；and if

    P栂εrド0（f≧2）血en（Arg輌，Argi．1）∈エ．

involvement with negation． In this section， we wi110utline

1｝efinition 12： （x／y−dialogue tree［13］）． An xly−dialogue

our argumentation f『amework．「lb begin with， we describe

tree is a tree of moves such that every branch is an x／y−

abstract argumentation framework．

dialogue， and for al1 meves mov召i＝（P， Argi）， the children

of movei are all血ose moves（0，Ar8輌＋lj）（」≧1）such也at

3．1 Abstract Argum」entation Framework

（A㎎’・1．」、Argi）∈x・

1〕面nition 13：（Provably x／yrjustified［13］）． An x tY− 1｝e血ni目on 8言（Abstract attack relation［5ユ）． Let Ar8s be aset of abstract arguments． An attack relation x on Args is abinary relation on Args， i． e．， x⊆Args× Args・．     We define the argunientation se皿antics as the least fix−

dialogue D is a winning x∫y−dialogue⇔Ule terlnination of Dis a move of proponellt． An x／y−dialogUe tree T is a win−

ning x／y−dialogue tree⇔eΨery branch of T is a winning エ／y−dialogue． An argument A r8 is a provablyエ／y−j ustified

poiqt of the fUnction which coHects all acceptable argu−

argument⇔吐1ere exists a winningエ／y−dialogue tree with

ments．

Ar8 as its root．

De触ion 9：（鞠一acceptable and j副ied a㎎㎜ent［5】）・

Theorem 2：Let Ar85 be an abstract argument seしThen

Let x and y be attack relations on Ar8∫・ SupPose Argl−E

A㎎∈Ar8s is proΨably x／yづustified⇔Ar8 isエ／yrjusdfied・

Args and S⊆Ar8s． Then」Argl is x／y−acceptable wrt・∫ if fbr every Ar82 ∈Ar8s such that（Arg2，Argi） ∈xthere

exists、Ar83∈Ssuch吐1at（A rg3，Ar92）∈y・     The fuuction ｝「A rgs，xノァ InapPi皿g from P（A rgs） to

P（Args）is defined by FA ，g醐（∫）＝｛A・8∈A・8・1A・8 isエ／y−acceptable wrt・ S｝・ We denote a least fixpoint of 1『Args，xly by JA rgs．x∫γ ’An argtl−

ment A rg is x∫y−justified if A rg∈Jxty；an argument iSエ／y−

overruled if it is attacked by a x／yづustified argument；and an argument is xノンーdefensible if it is neither x／y−j usti fi ed

’nor x／y−ovemlled，

    From here on， we concredze the abstract argumentati加 framework to introduce various notions． proper to．a logic of

multiple−valued argum号ntation LMA．

33 Annotated Arguments Defi皿ition 14：（Reductant and Minimal祀du幽nt）． Sup− pose P is an EALE and Ci（1 ≦ 匡 ≦ R）are annO− tated−rules in P of the’ ?b窒香F」4：ρ輌←Ll＆．．．＆Lh∫， in which A is an atom． Letρ＝・u｛ρ1，∴，Pk｝・Then the fbllowing annotated nlle is a reducta祉of P・ A：

   ’W・・writ・曲ply・F．ノ， and J．ly f・・馬綱1y蝋』．x∫，

ρ←Ll＆＿叫＆…＆Lf＆一・＆Lい・edU・t・nti・

when A r8ぷis b恒ious・SinCe Fエiy is monotonic・1it has a least

called a hUn㎞al reducta皿t when there d㏄s not exlst non−

fixpoint， and can be eonstructed by tbe iterative method［5｝

empty proper subset S⊂｛ρ1，＿，Pk｝such that P＝〕S・

Definition 10：（Conflict・free［5］）． Let Argsbe an abst］ract

Definition 15：（An皿otated a㎎凹皿lents）・ Let P be an

狂…1CE TRANS．！NIF．＆S yS’r．， VOL．E8S−D，NO．9 SEPTEMBER 2005

2024

EALP． An a皿otated argument in P三s a丘nite sequence Arg ＝ ［rl，．．．，rn］of rules i！】Psuch that曇）r every輌（1 ≦

3．5 Argumentation Semantics for EALP

f≦n），

By Theorem 1， we㎞ow JA r8s，aly（a＝attack， a⊇y）is 1．ri輌s either a rule in」P or a min三ma五reductant in P．

confiict−free wrt． attack． Funhermore， we have，

2．For every annotated atom A：μin由e body of ri， there   exists a rk（n≧k＞のsuch that A：ρ（ρ≧μ）is head of

Theorem 3：工f a set of arguments S is con伍ct−free wrt． the defeat relat▲onship， then s is conHict−free wrt． the attack．

  η ・

3．F（）r every o−explicit negation∼A：μin the body of ri，

Proposition 3： ・IA rgs，ば15。＝JArgs．d∫凹＝JIArgs，d／d・

  there exists aηヒ（n≧た〉輌）such that∼A：ρ（ρ≦μ）is   head of rk，

4．There exists no proper subsequence of［r1，＿，ηJ

    In this paper， we employ／Args，dlsu to specifシthe set of

justified．arguments since the stricdy undercut is smallest or

  which meets from the first to the th三rd condi目ons， and

simplest in these attack relations， and straightfbrward．

  三ncludes r1．

    Let P be an EALR andルtAS be a set of EALPs． Then we de丘ne the sernantics of argumentation on P by JArgsp，dtエu

Asubargumelnt of an argument Ar8 is a subsequence ofArg．

（Jp fbr short）， and defi ne也e semantics of argumentation

The conclusions of n11es in Ar8 are called conclusions of

on MAS by JA rgs”As ，dtsu（JMAs fbr short）． When we do not

Ar8， and the assump加ns of rules in Ar8 are ca11ed assulnp−

need to teU apart P a皿dルtAS，we simply denote Ar8ぷp and

tions ofA㎎． We wri te concl（Ar8）for 1山e set of conchlsions

Arg5、MAs by Ar8s， and denote Jp and JMAs by J． 」♪and

andαぷ∫η1（Arg）fbr the set of assumpt三〇ns of」4 rg． We denote

JMAs can be dialectically determined by也e dialecdcal proof

the set of a1！arguments in P by Argsp， and de五ne the set of

theory specified in Sect．3．2．・

all arguments in a set of EALPs MAS＝｛KBi，…，KBn｝by Ar8工〃AS ＝ArgSKBs U−・UAr8SKe，（⊆Ar8s」K、BtU…vrrBn）・

3．6 （Snictly）Well−B ehaved EALP

    Anon−minimal reductant usually results in having a longer anteceden杜han a m三nimal reductant． if an argument

We scrutinized the argumentation semantics fbr EALP in

is made by non−minimal reductants， the argument may be

various directions， and found some logica1 anomalies（inter−

redundant in its content． Fur也ermore， a non−minimal re−

ested readers should refer to our foπner paper［19］）． For

ductant occasionally tends to be irrelevant rule（refer to［19］

ex ample， even if the j us面ed arguments are conflict一丘ee，

fbr details）．

the interpretation satisfying血em is not always consistent・ Tb avoid them， we introduced subset3 of EALP， called we皿一

3．4 Attack RelatiOn

behaved and strictly weU−behaved EALP．     1皿what follows， we assu皿e that agents’㎞owledge bases are sUictly weU−behaved EALPs fbr argu皿entation．

Definition 16： （Rebut）． Arg l rebuts Arg2⇔tbere exists

Alμ1∈cohct（Argi）and・’一 A：μ2∈concl（Arg2）such・that

Definition 20： （（Strictiy）Well■behaved EA］LP）．

μ1≧μ2，0r exists∼〆いμ1∈concl（Arg1）a皿d A：μ2∈i

Let P be an EALP白n a complete latticeワー of truth values，

concl（Arg2）such tha［μ1≦μ2．

∼A：μbe a conclusion of a rule in、P， and S⊆「アーbe a fi皿ite

subset such that S ≠φ． Then， if P satisfies the condition，

Definition 17：  （Undercut）． 」4r81 undercuts Ar82⇔

μ≦〕S⇒］ρ∈Sμ≦ρ，Pis called well−behαved EALP．

吐1ere ex三sts A ：μ1 ∈ coπd（Argi）and not A ：μ2 ∈

    haddition， if an assumption not A：μof a rule in P

α∬m（Arg2）such血atμ1≧μ2，0r exists∼A：μ1∈

also satisfies the above condition， P is called stri‘tly well−

concl（Argi）and not ∼ A ：μ2 ∈ assm（Ar82）such that

behaved EALP．

μ1≦μ2．

Proposition 1：  For any Ar8i and Arg2 in Ar8s「， if Argi

4．lllustratiΨe Argument Examples

rebutS Arg2，1 such that 1トArgs is o・inconsistent． And if Argi undercuts Arg2， I such that lトArgs is d−inconsistent．

In order to see potentials of LMA for extensive』applic ations，

we、illustrate two argument examples that exploit usual and Defi皿ition 18：  （Strictly undercut〕． A rgi strictly under−

cuts Ar82⇔Argi undercuts Arg2 and Arg2 does not under− cut Argi．

1）efinition 19：  （Defeat）． Argi defeats・4r82⇔Ar8 i un− dercuts Arg2，0r Argi rebutS Ar82 and／lr82 does not undeエー cut Aア8い

unロsual uses of complete lattices respectively・

Example 1：Let us consid￠r an argumentation about the pros and cons of the death penalty of Murderers・SupPose acomplete latticeワー＝恨［O，1］2 ・where（＃1，ρ1）≦（μ2，ρ2）

⇔μ1≦μ2andρ1≦m，…md in（μ，P）∈∼「，μ and P represent the degrees of an af丘rInation岨d a negation re−                                    ■                コ

spectively． This tru．th va Iue can represe皿t e−1皿consistent o「

Propositien 2： For any 4 r81 and Arg2 in Args， if／Arg i at−

unknown state and degree of tru廿1 togethe： Let MAS ＝

tacks or defea臨Ar82， l such that 1トAr85 is o−inconsistent

｛KB，，KB2，、KB3｝be a set of EALPs， a皿d each、1（’Bi be a

or d−inconsistent．

knowledge base such that：

SAWAMURA and TAICAHASHI：APPLYING LOG rC OF MULTrPLE・VALUED ARGUME㎜工ON T［）EASTERN ARGUMENTS

2025

Fig・1 R・画。・・血・ng．arg・r・entS， wh・fピu‘・・ta・d・・f・・u・d・ユ・nポピ鋼d・f・・民b・tta1・th・ broken lines sta皿d fbr由e op已ra面on of the reducta皿t、 and a：gume耐s framer］in a吐Uck line are justifi巴d

argumellts・

（Arg21 with a c。nclusion with annotaUon（02，0．8））， an oP・

κB1ニ｛ ∼agree（叔の：（α0、0・8）←

position of the death penalry is acknowledged to some de・ gree at the same time（A rg22 with a conclusion with anno−

      声1ロre（∫amily， murderer）：（0．8，0．0） ＆

tation（02，0．4））」’Westerners generally might prefer a defi−

      ば已∫’re（∫antily， death）：（0．7，0．0），

nite cenclusion of either true or false． However， as a whole，

』’ε（『醐ity， murderer）：（1．0．0．0）←

we would say that吐1e agent society of血鶴agents三s in−

      一一 aUowげamily、 murderer）：（0．5、0・0），

clined to institute the death penalty in this argument among

de5ire（∫酬り，， death）：CO・7，0・0）←・

∼伽三re（∫amily， death）：（0．0，0．8）←， ∼atlew（∫㎝輌砂， murゴerer）：（05．0．0）←  ｝，

K・B2＝｛ agree（deロth）：（02，04）←ロ’oπε（ば召ath． guilt）：（0・0・05）・ atone（d「eath， guitt）：（0．2、0．8）←

      皿ot remorse（dε口d）：（1、0，0．0）， α8reε（dξath）：（0．0， LO）←

      n・t−・ll・w（fan・ily， murd・…）・（α8・0・0）． L

由ose three a、gents． The exalmple would suggest a possibility

and feasibility of argurnent systems 1ike LMA as a function fc）r E−government in the future．

    In Example 1， assume 一・（μ，ρ）＝（LO一μ， LO一ρ），

and replace every o−expUcit negation‘∼’in each KBi by e−expUcit negation‘「’． Then， Arg is will nor more re− but Arg33， resulting in、Arg34 being juStified， and thus the

conclusion of Arg34 mea皿s”a bereaΨed f㎞且y desires and does not desire廿1e death penalty with amotations O．6碑一 spectively’， The degree凸f opPosition of the death p￠nalty

κβヨ＝｛ agree（death）：（0，0，0．6）←desire（∫伽記ン， death）：（0．0，0．6），

desire（∫‘vnily， death）：（0．OJ・0）←  ・       notα5幽8￡（伽’好・unily〕：（0・7，0・0）・ agree（d8ロth）：（0．6，0．0）←       加甚ε（familン， murderer）：（0・6． O・0），

加f・げamity，murderer）・（o・9・02）←｝・

Figure l shows eΨery possible argument and every poss1−

ble attack relation among them． The just通ed arguments are JKBs＝｛Ar8n，A㎎12，Ar813，Ar814， Ar821，Ar8ユ2・Ar831・

increases to O．6（Arg34）in comparison with the use of th巳                                               か ontologic…il negation． From dle viewpoint of Westem lo91c・

廿ris is a contradiction that Westerners hate most． However，

thanks to the paraconsistency LMA has， it avoids ex falso quodlibet． T is not a truth value that sirnply turns knowl− edge base into a meaningless one． Instead， It represeptS an ・pi・t・mi・・t・t・・n血・p・・P・・iti・n”血・d・a也P・nalty・i・・titu− tion，’agellts bear i皿L］ULA・

     Thus agents can incorporate what也ey intend to do in

argumentati。n int・血・廿㎞・wledge・bas・by・・1㏄血g ap− propriate explicit negation accordmg to a topic of argumen−

A・83、｝．rn a literal sense， the justifi・d argumentS・w・uld b・

able to read as］bHows：‘‘lt is sure吐lat a bereaved fam− ily hates a mロrderer（Ar831 with a conclusion with an、no−

ta口on．

Ex・mpl・2：L・t・us・c・n・id・・紐卿m・ntati・n・b・ut th・

tation（0．9，0．2））， and so agreement to the death penalty

mont hity schedule management・Here we use an unconven‘

is pardy possible（Arg32 with a conclusion with allnotatiol1

tional complete lattice of truthΨalues which is the power set

（0．6，0．0））．In addition， because a bereaved faエnily ca皿no匡

aCCept・ll・Wing・mU・d・・e・（A・gll Wi血aC・nCIU・i・n Wi也 a㎜o城on（0．5，0．0））， and desires the death penalty（Ar8日 with a conclusion with annotation（0．7，0．0））， a complete op−

position of tiユe death penalty is never acknowledged（A rg i4 with a conclusion with annotation（0．0，．0．8））． However， be−

canse there is no evidence to show that a dead ma皿ls m remorse fbr the crime， and the death can not atone a 9uilt

P（｛1，．．．，31D of the set of the monthly dates， with the or−

d・・by也・・et一桓・1u・i・n・r・1・ti・n． Tbe蜘胆・。魎・t°m w。rk（α）、｛5，6｝・eads・・Ag・・t・w・・k・・n也・5由鋤d血・ 6也」，lt・asse，t、・th舳・p・・P・・id・n w・rk（のis血・・㎡y㎞

ace血in dme interΨaL∼work（ロ）：｛5，6｝醜ds‘‘Agent a  does not work on the 5th and也e 6th”． We define the epis一

蜘cexpli・imegad・n・・a・・t・be「μ＝｛1・…・31｝一μ・紐d 也u、rw。rk（α）：｛5，6｝民ad・“Agent・w・rkS・n也・d・tes・・“

IEICE TRANS」NF．＆SY∫T．，VOしE8S−D， NO．9 SEPT直MBER 2005

2026

cept the sth and the 6th．” The di ffe rence and signi丘cance

the 26th、”ConsequenUy， the first argument of Agent m is

between the ontological and ep三stemic explicit negations is

justified．

obvious． Under ｛his complete lattice of truth values， we con−

sider MAS ＝｛K、Bm， KBa，KBb，、KBo｝， where KB of each agent isパn EALP，

    These idiosyncratic uses of complete lattices of differ−

ent type make our points：（i）the va泣ty of the expressive− ness of EGAP， and（ii）the versatility of the multiple−valued

argumentatlon．

κ8用＝｛ ∫輌用ish（proノεc’）：16｝←

5・Specializing LMA to Tetralemma vvith an Eastern    Mind

    werk（a）：｛314，5｝＆arr輌Vt！（‘omponent）：‘5｝，  tS・orA：（ロ）：｛3，415）←，

arrive（component）：15｝←，  paγ（upchar8e）：｛8｝← ｝，

fOi〃∼＝．（｛⊥， t， f，丁｝，≦）． where∀エ， y∈｛⊥， t， f， T｝エ≦

y⇔x＝y∨エ＝⊥Vy＝T， is a well一㎞own complete

KBa＝｛

lattice（depicted in吐）e left of Fig．3and in the right is added

 ∼wロrk（の：｛51←not wθrA（占）：15｝＆みo’iday：｛5｝．

acomplete Ia垣ce of ideals，∫（TO￠t！R）， constructed丘om ダ0百R）．From here on， we specialize LMA by idendfying

 ・一 work（a）：｛12｝←

    noいvork㈲：口2｝＆heliday：｛12｝，  hot輌day：｛5． 6， 12、 13｝←｝、

也e comp｝ete lattice FO〔頂wiぬ恥ale㎜a wi也an Eastem mind， so that LMA allows for argumentadon of the kind

KB占＝｛

seen in廿1e Eastem tradition of culture．

 rwork｛占）：112，19，261←，  蘂’ゴ直y：｛5．6、12，13｝←1，

5．l Tetralemma KB。＝｛ 一・ arrive（companent）：｛5） 一一 not paXup‘harge）：ip｝．

In the early philosophical literature and text of 13uddhism，

也e notion of four altemative positions（ca噂ko亘in Sans㎞t；     Where KB．，κβロ，KBb a皿d KBσ stand for knowledge bases of a manager agent m， employee agents a， b and a sub−

shikttfunbetsu in Japanese）appears vely o丘en in argume皿ts on met且physical questions such as whether Nirviin4 is an ex−

contractor agent o respectiΨely． Agent m’s argument which

istent， whether Ta由互gata existS after death， in such a way

has the conclusi皿finish（proノ松ロ）：｛6｝（the project will fin−

that all conceptually imaginable pos垣ons are exhaustive

ish on也e 6th）is justヨfied by the dialectical proof theory as

and exclusive such as”Nirv亘皿a is an existent， Nirvana is a                         ロ                                                           ’

shown in Fig．2．

non：existent， Nirvfina is both an eXistent a皿d a皿on−eXistent，

    In the winning dialogue tree， initially Agent m says‘‘廿

Or Nirv亘na iS neither an eXiSt￠nt nOr a IIOn−eXiStent．”

acomponent wiII arrive on the 5th， and Agent a works on

    These represent four logical possibilities of the form：

the 3th， the 4th and the 5th， th印the pr〔）j ect wiU finish on

（1）athrmation，（2）negation，（3）both affillnation and nega−

the 6th．”Then Agent o defeats it as follows‘‘I will be not

tion， and（4）neither aMrmation nor negation． The ancient

able to bring a component on the 5也ifthe additional charge

Indians believed that the truth with regard to any］matter Iay

is not paid．”But Agent m sUictly undercutS o’s argument by

in one of these alte而atives． Over the last few decades， the

saying 4‘I will pay it匡o you on the 8th．” For the first argument

Iogical structure of the four altema且ve positions has been

of Agent m， Agentロalso defeats by saying‘‘the 5th is a

the subject of a considerable amount of discussion and con−

holiday， and if the coworker b does not work， I do not want

troversy， and also of some speculatio4 in 吐te context of

to work on the 5th．’， However Agent b sUictly undercuts it

what is sornedmes called East−West philosophical’ Dcompar− ison． Nowadays”the four alternatives（positions）”is also

by saying‘‘I wi皿work on days except the 12th，191血and

termed”tetraiemma”or”tetrachotomジin the literature［1］， ［7］，［14］，［16］，［21］．

PAg己川m

    Tetralemma is an epistemic state or way of recogniz− ing things， beings， objects， propositions， etc． We use similar patterns of expressions very often in our daily life， fbr exaln−

ple，‘‘Animals have moral rightS， Animals do not have moral

dghts， Animals have moral rights and do not， or A垣mals ．neither have moral rights nor do not．”

T snictly

sajctly

undercut

undercut

f

t

⊥ Fig 2 The vvinning dialegue tree in Exarnple 2・

ITl 1f1

ltl

」

φ

Fig．3  0『O伽R and」r（チーon？R）．

SAWAMURA：and TAKAHASHI：APPLY ING LOGIC OF MVLTPLE−VALUED ARGUMENT灯10N TO EASTERN ARGUMENTS

2027

    In his book， Logos and i尤㎜a［21］．緬uchi

dialectics．1t is a dialecUcal contradiction but not a contra臼

TokUryu contrasted Wes〔em and Eastern thought by char−

diction in formal logic， being a fbrm of the un並y of opPo．

acterizing the first as being determined by logos and the laト

sites as a diaiectical law in Hegelian and Marxist dialectics．

ter as being structured by the principles of the tetraleユ

The eastem style of民asoning is basicaUy dialectical from

thereby conceiving of logos as a method of exclusion and

scratch in the sense that it focuses on contradictions and how

of le㎜a as a way of撫dly inclusion（eΨen of the血d−

to reso1Ψe them or虹anscend也em or伽d the tru由in both．

dle）， The tetralemma， in fact， is closely related to由e view of

In either world， we need to invent concepts to bo吐l discard

e皿ptiness in Buddhism and the culturahdiosyncrasy such as

and absorb those contradictory propositions． 「rurning our

holisticΨiew on world， society and ecosystem in nature［12］．

eyes to the development in other traditional sciences， we can

    It would be helpful to spe田late about the meanlngs

see that dialectical thought unites various opPosite concepts，

of the third and fourth lernmas in more details since they

principles， and theories through mediating logical links in

are beyond simply saying either true or false， so that we

could exploit them in applications to knowledge representa−

tion and argumentatlve reasonlng・

higher syn吐1etic constructions．

    We use」the top symbol T for such an epistemic state or way of recognizing things， beings， objectS， propositions， etc．， and annotate propositiens with the symbol T as in［2］．

5．2The Third’Lemma：‘it is and it is not’ 5．3The Fourth Lemma： ‘it neither is nor is n。t’

The出ird lemma obviously seems to violate the law of non− contradiction by Aristotle． It． however， often appears in口le

Like血e third lemma， the fourth lemma al so appears very

sutras−of B uddhism and the eastern tradition of thought and

o丘en in the sutras of Buddhism and the eastern tradition of

culture． Daoists and Zen Buddhists， for example， see the

thought and culture， such as in Nag萌una’s

two sides of any apParent contradiction existing in a har−

Malamadhyamakakarik5， which is one of the Inost inHuen−

m皿y，opposed but｛nterconnected， interpenetrating， and in−

tial work in the history of Indian philosophy［11］， and in the

terdependent． in everyday situ ations where experienc己or

Heart Sutra．

des辻e is dorninant， easterners are tolerant of contradictions・

    Murt三interprets it 1ike thi s． It represents the fUll con−

They haΨe thought that 1由e law of noncontradiction appUes

sciousness that no correspQnding aff…rl皿ation is aΨailable・

only to the realm of concepts and abstractions， which are

This is an、 extreme fbrm of non−committal． It is not an at−

merely re且ections of thiIlgs． In his recent book［12］、 a cul−

titude of decision， but of doubt．The competence of thought

1皿ral psychologist， Nisbett has delnonstrated tendency aLnd

is not questioned and reason is not缶anscended． It may cor−

evidence which indicate various di｛丑≧rences of cognition and

respond to the agnosdc position［10］・According to廿1e in−

reasoning between Easterners andi Westerners， including the preferen直al exaエninalion of accepta皿ce or avoidance of con− tradictions．

terPretation of Jayatilleke， there is left a part of the deter−

minable constituting the un、iverse of discourse which is re−

』ed to by血e飴u曲le⊇since the second altern ative is

    Let us see solne multi−faceted皿eanings of the third

the contrary and not吐1e contradictory， and the曲d asserts

lemma since we think they are he1pful to understand and

that也e subject has a combination of some of the contrary

use it加actual knowledge representadon as weU． Mur丘［10］

characteristics［7］． Here is a good exalnple illustrated by

says出at血e醐le㎜a represen馳at we ha⊇e con− sciousness of the olle−sidedness of Inere ls or mere ls not・ f・・examp1・， G・d exi・t and d・e・n・t・J・y・till・k・［7］s4Ys

1由at the historical examples show that the second alternatlve is the contrary and not the contTadictory of the五rst・for ex一

迦P1・， w・・tv・、 e田t，㎞・w1・dgev・…nd・ct Th・n・也・ t1丘rd asserts that the subject has a combina直on of the con一 剛charact・ristics． F・r・exampl・， when the st・tem・nt’u㎡一． verse is both finite and infinite’is made， it is intellded that

・the universe is丘nite in one dimension（in some respects）  and infinite in another． Here is a typical locution fbr‘it is and is not， in our daily life（or rather， polltlclans may use 廿ris）：When they arei asked a hard question to dea1 witll・the

answer would be‘yes and no’． Also in our daily life， we

s・me直mes rhet・亘cally use由・・xym…n・It is a phrase that

combines two words that seem to be the oPPosite of each  other． For ex泣皿Ple， an open secret， too much of a good  thing（bear’s service）， a hapPy scream， a deafeni皿g silence

and so on． Oxymoron unit問two opPosipg words into an expression with a single meaning， without contradiction・

     The third lemna also can be found eve蜘Western

Jayatilleke｛刀：Aperson is happy， A person is not happy， A・Pers・n i・b・螂・pPy and u血・pPy…A・Pe・s・n i・n・ithr「 hapPy nor unhapPy． HapPiness in this context is a det官rm』■

n・t・q・ality・h砲・t・ri・ing・p・rs・n’・hed・垣・t・n・・Wh・n we remove the qualities of‘‘happiness∼’‘Mnhappiness，” or a mixture of the two， we are left with‘‘neutral hedonic tonel”

So a person who is‘‘neither happy nor unhappジcompnses th・・clas・・f卿le exp・essing・a・neutral・h・d・皿i・t加e・Such  aclass need not necessadly l］e a null class、 although it could

be so sometimes． Hele is anether similar type of 10cutions・ whlch can be seen in our dai正y life as we111 His act is neither

軸t…w・。ng，・・w・ar・・ei血・・fo・n…g瓜nst』ct・      The founh iemma has had a wide spectrurn Of皿6an−

i。g…epi・t・㎡・・t・t…堀・g・， being・，・切ec底興 P・・P・・iti・n・． It w・uld・b・w・血y鋤d u・・fU1 t・1ist up’t

as fbllows fb抽e knowledge representatiop and argumenta’ tion described later：no com㎡血ent， no commenL no con°  cem， no infonnation， an attitude of doubL a neutral or t「an−

、ce。d，血1・pi・t・血・・田t・， and・vi・w・f E岬血・ss□e  Buddhist technical terms fbr the lack of independent exls° t，n、，， i。h，，ent・xi・t・n・e，・・es・en。・i噛・g・・W・d・gene「一

1EICE TRANS． INF＆SYST， VOしE呂8−D， NO．9 S EFrEMBER 2005

2028

飢e these i・t・由・b・仕・m・ymb・1⊥・吾d血e c・nt・・t i・s・p−

Aristotie

posed to speci fy a meaning of it．

    It migh由e instructiv飢o take up contemporary expres− sions that capture o酷aspect of the meaning of⊥． For ex−

ample’we have three ways to pronounce‘‘s垣p」medar・in

EALPl ω∫坤一med・t：t←・ゴ・拍r已・戸ng−rules：t    吻latentdoping」utes：t（2り’dence）

EastCm Ag字nt

Gi田ee

（2）訂r輌」ρ」veedal：t←−violate」ゴoρ∫n8＿ruleぷ：⊥

   Fig．4

Western and Eastern argumc…nts against Aristoae．

   violate」ゴoρ加8」r“∬e∫：⊥（doubt）

③5ゆ一m・d・t・t←n・’一・輌・互at・d・P加8．・ule・・⊥    （w占εrε∫』∬醐μ∫・π∫5vわ」・舵」ゴ・ping−ruies：⊥．）

（actualiy defensible）：an Western anaiytic argum6nt and an

    Fbr the issue on WMD， there are two approaches to

Eastern holistic one， where thc second rule in Galileo’s ar−

reSO1Ψe：

gum剛is a reducta皿t made丘om his knowledge base． Note

（1）ロr血7ck−lrag：f←has＿wθ｛2、pons＿η1α35＿destruc’輌o／1：⊥

that Galileo made his argument by reductio ad absurdum

   （Frαηcε醐Ger加卿’s・sta刀cの

f（〕rwhich吐1e default negation‘not‘has a crucial role in

（2）一・匡’酩加91f←has−w・αρ・ns．ma∬」ie蹴‘ti。n：t

the rule representation． Furthelτnore， we note that the head

  （USA・and・UK’王ぷr伽cの

∼aristo’lehyp：tin the first nlle of Ga田eo’s argument does not undercut the assumption∼aristbt正ε一hyp：fof the

6．PEuraliStic er MulticU ltura 1 Argument Examples

second rule， that is， Galileピs aエgument is coherent or not self−defea1］ng， and Eastem agent does not undercut the as＿

  Wb i11ustrate two pluralistic or multicultural arguments  based on the speciaIization of LMA in the previ皿s sec廿on．  Exampl e 3：Let us consider the Wes tern and Eastern arg u−

 ments against Aristotle． Ari stotle believed that吐1e heavier

・b・凪yi・・也e趣te・it飽ll・t・出e g・・und． We simply・ep−

 resent this as aristot∼ehyp：tGalileo’s logical argument  against吐Us pr㏄eeds as］f（）110ws：‘‘Suppose that we have two bodies， a heavy one ca皿ed H and a light one called L． Under

Aristodピs assumpUon， H will fh11 faster than L． Now 8up−

pose that H a皿d L are joined togetheL Now what happens？ Wel1， L plus｝l is heavier that H so by the initial assu皿ptio皿

it should飼l faster that H alone． But in the joined body， L is lighter and will act as a‘brake’on H， and L plus H wilI

sumpロon∼αr輌∫ro’」ぱyp：fof the second rule in Gahleo・

argument．（lnterested readers should refer to［19エfbr the technical terms used．）     In血i s example， all the argumen ts by Aristotle， Galileo

and Eastem agent become defensible． IncidentaUy， let us consider a Uttle modi血ed version of the example． We first

change Aiistotle’s belief as fbllowsl aristotleky、p：t←

not∼em」pirically．factual l t． AIld we make one more agent apPear on the stage， who is a mOdern scientist having

a丘rm belief on verificationism．

MOdern scientist’s knowledge： ∼empi・ically−factua∼：t←π・t・scientげically．verified：t． The叫it is obvious that Aristotle’s argument is overruled， Galileo ’s and a modern scientist’s ones are justified， and

制slowe抽at H alone． Hence it follows from the initiai

Eastem agent’s one is sti皿defbnsible．

assumption that L plus H w沮both faster and slower that H

    In today’s globalized world， such a pluralistic or mul−

alone． Since血is is absurd， the mitial assumpdon must be false．”On the other hand， Eastemers prefer a more hもlis− tic or dialectical argument like this：“Aristode is based．on

dcultur司卿ment of co㎜on interest is getdng mo祀血一 portant to us than ever・This is no exception even in agents’   ら

SOClety・

abelief也at the physical obj ect is free from any influences of other contextual factors， which is impossibIe in reality、”

Example 4（Nobunaga’dying words：Zehinioyobazu）l This

［12］

is a familia・hist・rical event， typically reflec直ng也e meaning

    These are well translated into EALP as fbUows：

of⊥， and is the Iord Nobunaga’s dying words when he was

GaIileo’s knowledge：

’attacked and assassinated by his vassal Ak㏄hi in the histor−

∼aristetie−hyp：t←faster（L＋H， H）：T

icaI event at TempIe Honnoliji in the Warring State Period

faste・（L＋H， H）：t←n・t・arist・tl・」ryP：f

of the Japa皿ese history． It is‘Zbhinioyobazu’in Japanese

fdi1ぷ∫εr（L＋」「f，晶「）：f←5∬Ol伊er（L＋」Y，、晶0：t

which mea皿s that he is in such a cognitive or epistemic state

sl・wer（L＋・H， H）：t←brake（L， H）：t

that Akechi’s rebellious act is neither right nor wrong， sim−

占嬬ε（L，胡：t

Ply and best expressed as Akechi」rebellious act：⊥，

Eastem agent’s㎞owledge： ∼融ぷ’otlehyp：t←dis’rust．decontextualization：t

Nobunaga’s knowledge base Kbi＝〔

distr“st．decon蹴tualization ： t

r1：A輪cみ匡．reb松lliousact：⊥←

    Figure 4 depi6tS a dialogue仕ee const皿cted with the

     evade：f＆counterattack：fl

dialec面cal proof出eoly」Obviously， AAris，θr1百is defbated（re−

r2：evade：f←b． esiedged：t＆set＿，fire：t，

but）by A1ロェ’ern and AEa富fε打、， and turns out皿ot to be justi−

r3：besiedged：t←goed．battle（Akechi）：t，

fied by two culturally diHもrent kinds of counter−arguments

r4：good．加’tle（鯖εcみO：t，

SAWAMURA and TAKAHAS田l APPLYING LOG IC OF MUじTIPLE・VALUED ARGUMEN冊ION TO EASTERN ARGUMENTS

2029 Satellites’ argument

Nob皿aga’s ar“gumentS Def㎏nsible Akechi reb曲ous act：．L

一counterattack：f

       r！

1

    evade：f

drilled：r

covnterattack：f

besiedge（S：t

high＿al1巴gi皿ec：1

i，12

defeat

Ir13］1巴ss＿S臨田巳lt

   r3

Akechi−“reh巳IliOUS＿actし」L         rS d°（故㏄垣 ﾉ皿ga−ac°：t Justified

Fig． 5

Nobunaga’s dying words：khinioyobanu．

rs：set−fire：t，

gins where logos ends just as fUzzy logic begins at the East−

r6：counterattack：f←−less−satell三言ε：t，

erp edge where Western logic ends． Ofcourse we adminhat

r7：」ε∬．satellite：t，

there is more East in the ．West than meets the eye． We bo

rg：Akechi−rebellious．act：⊥←

lieve that the specializatiofi of LMA to Tetralemma b血呂s

     ゴo（Akechi，1Vo占unaga』！cr）：t，

us four advantages．

rg：do（Akechi， Nobunaga−act）：t←      power．gam已（Akechi）：t， rlo：porver．9傭ε（Akechi）：t．｝

The re吐nue’s knowledge base Kb2 ＝｛

r11：＾」‘ount已r＿attack：：t←drilled：t＆high」71iegia［nce： t，

rl2：ばrilled：t←le∬．satetlite：t，

  ●AfUsion of Eastem and V泥stem㎎umenta目on．   ●A combination of Eastern and Western rationalisrn．

  ●Away of handling the inexpressible or inexplicable     that can be viewed as another vvay to 9rasp things with−

    out representadon and reason about・

●An approach to bridging the gulf of incomm巴nsurabil−     iry between East and West in agentS’world．

rl3：le∬．satellit召：t，

r14：h輌ghUt∼legiance：t←le∬．satellite：t｝・

    For LMA， we have taken a syncretic or㏄1㏄tic ap−

    Neadng his death， it is said that Nobunaga accepted

proach to the fUsion of Eastern and Western argumentation．

his death ca1正dy and u仕ered‘Zehinioyobazu’in response

We血ink that it makes the best of both worlds． Both the

to his retinue． Akechi」二ebeUious act：⊥is a justi負ed conclu−

comb桓ation of Western and Eastem rationalism would be

sion in the dialogue between No『bunaga and his re血lue with

usel仙1 fbr d㏄ision−making in tlle agent society as well as

也e knowledge bases Kb i and」（カ2 respec目vely． Figure 5

in the human society． Only either ef the皿will lead to the

displays two possible argument trees， but it is the lower ar−

gument only也at is justified、

cu1＿de＿sacs，

    In traditional science， it is well㎞own that Bohピs

    For other pluralistic or multicultural argument exam−

quantum physics alld Jungl psycology were infiuenced or

ples， interested readers should refc…r to［17］although its ar−

inspired by Eastem thoughし In statistics， the late Prof． T、

gumentation framework is based on a different philosophy．

Kitagawa丘1troduced a］rr…ladve logic fbr statistical i！血rence

whose idea comes加m麺dian logic． In coInputer and mfor−

7．  Conclusion

                                           ， madon science， the original of objectS in software epgineer・

ing ca皿be fOund 1〕o血in Mstotle，s theory of objects and in

Very few attempts have been made at fU11−dress multiple一

曲ed肛gument fr㎜ewo止s so加， i皿w姐ch argumen麺e to be constructed based on uncertainry that is an essential

ancient Indian philosophy． What we haΨe done in th且s pa− per is another significant case infiuenced or inherited frorn

Eastem thought．

nature of infotmation． We have proposed LMA：Logic of Multiple−valued Argumentation as a logica1 basis for皿ulti−

agem dialogue and reasoni且g， under uncertain knowledge．

Its m輌dvantages are surnmarized as follows・

Acknowledgme皿ts                                           ，

The authors would like to thank the anonymous revlewers who made helpfU1 suggestions to this PaPe「・

  ●LMA has potentials for extensive applications by ex−     PloiUng various co皿plete latdces・

Re艶rences

  ●The building bl㏄ks of LMA would be helpfUI even     when you elnploy血zzy logic， beUef net， etc・as un−

［1］V．・Bh・・adw・j叫F。m皿d堀i輌i・1・伽坤・・血dim圃塒

    derlying tools for uncertain knowledge・

   of Ad冊皿ced StUdy，1990．

    The tetralemma−based logic of argumentation now be一

   ㎡ng”TheoL ConlpuL Sci．，ΨoL68． pp．35−54．1987．

12ユH．A． Bl輌d V，S． Subrahm輌1，“P腿剛sistent 1°9ic P「09「a「n一

IEfCE TRANS．INF．＆SYST．， VOL．E99−D， NO．9 SEPTEMBER 2005

2030

  〔3］C．1．Chesnevar， G． Maguitman， and R．P． Loui，’‘Logical models of

Hajime Sawamura  was bom in l949・He

     argumcnt，”ACM CompuしSurv．IvoL32， pp．337−383．2000．

received the B．E．， M．E， and Doc【or of Engi−

  凶 C．1．Chcsfievar， G． Simari， T， Alsinet． and L Godo，“A！ogic pro−

ncering degrecs加m Hokkaido Universi【y in

     grammmg frarnework for possibilistic argumen【ation withΨague

！972，1975and 1993 respectively． During l980−

     knowlcdge、11 Proc． In【1， Confer印ce on Uncertainty in Artjficial In−

1996，he was wi山Institute fbr Social InfOrma・

     【elligcnce（UAI2004）， pp．76−84，2004．

tion Science， Fujitsu Laborateries Ltd．， where

  ［5］P．M． Dung，”An argumentation sernandcs fbr logic programming

he was a seniorresearch fe∬ow ofcomputational

     w▲【hcxp】icit negation．”Proc，10th Int． Conference on Logic Pro−

logic group． During I990−1991and in 2000、 he was a visiting fellow of Austrain National Uni−

     gra㎜ingl PP，616−630．1993・

  ［6］R，Hacnni，J．Kohlas，and N． Lehmann，’‘Probabnisdc argumentadon

「t

versity， and in 2002−2003 a visiting回。w of

Ph▲losophy departrnent， Victoda University ef

     systems、”In Handbook of Dcfeasible Reasoning and Unce宜ainty      Managemen【Systems， voL5：Algorithms for Uneertainty and De−

Welling【on、 New ZeaIand．  His research intercstS include computational

     feaslble Reasoning， cd． J． Kohlas and S．Mora1、 pp．221−2B8． Kluwer．

10gic， logical fo岨Clation of computer sottware and artificial intelligence．

     2000．

正］eis a member of rPSJ， JSAI JSSST， and Philosophy of Science Society

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 ［9］M．Ki歓and VS． Subra㎞manian，‘Theory ofgeneralized annotated      l。gic programm lng nnd itS apPlicati・ns、“J・Lo9・Progra皿・． v。L12，      pp．335−397，1992，

［10］T．R． Murd， The Central Phnosophy of Buddhism−A Study of the      Madhyamika System， George Allen and Unwin l．960．

［11］N5g珂una， The Fundamental Wisdom of血e Middle Way，      Nfigdrjuna’s MO］amadhyamakakdrikE， translated and commented by     J．L， Garfield， Oxfbrd University Press，1995．

［12］R．E． Nisbett， The Geography ofThought：How Asians andWestern−     ers Think Diffe間tly＿and Why、 The Free Press，2003．

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［14］P．T． Raju，“Th巳principle of four−cDrnered negation in indian phi｝os・     ophy，”Rev， Mctaphysics， vo1，7， pp．694−713，1954．

［15】C．Reed and T，J． Nom）an， eds．． Argumentation Machines， Kluwer

    Academic Publishers，2004． ［16］D，S． Ruegg，‘The uses of【he four pesitions of the catulko巨al】d     由eproblem of the．desc可tion ofreali【y in mahayana buddhism，‘’」．     Indian Phjlosophy， voL5， pp．1−71．1977．

〔17j H， Sawamura and E． M訂es，“How agents should exploit tetralemma     with an eas【enl mind in argumen【adon，”in IntelligenヒAgents and     Multi−Agent Sys￠ms V皿． ed． M Barrey and N． Kasabov． LNAI     3371，pp．259−278， Springer，2004，

［18］M、Schroeder and R． Schweimeier，‘’Fuzzy argumentation and ex−     tended logic p《ograrnming，’， Proc． ECSQARU WOrkshop oo Ad−     vances in Argumentation，pp．1−13，2001． ［19］ T．Takaliashi and H． Sawamura，”A logic of multiple−valued ar−

    gumentation、”Proc． Third Intemational Joint conference om Au・

    tonOmous Agen6 and Muhi Agent Systems（AAMAS12004），     pp．80〔｝−807．ACM，2004．

［20］Y．Umeda， T．τ歓ahashi， and H． Sawamura，”An argumentanon     frarnewo【k based on paracensistent logic，”工Artificia1血telligence     ofJapan， voL19レno．2， pp．83−94，2004．

［21】T． Yamauchi， Log。s and Le㎜a， Iwariami．1974．

Takehisa］P】kahashi  was a master course studentofGraduate School ofScience andTech− nOlogy， Niigata UniverSity、 He now wOrks fOr Niiga【a CannOl】Co．