A DISCRETE MODEL OF DEGREE CONCEPT 
IN NATURAL LANGUAGE 
Shin-ichiro KAMEI and Kazunori MURAKI 
Infi>rlnntiion Technology Rese~m:h Laboratories, NEC Corporation 
4-1-1, Miynzaki, Miyamae-ku, Kawasaki, 216 JAPAN 
k~mtei@hum, cl.nec.co,j p, k- mur~Lki((.~-hum.cl.nec, co.j i) 
Abstract 
Degree words in natural lauguage, such as 'often' ~md 
%ometinies,' do not ha.ve (ieln)t,~tions in the reid world. 
This causes some interesting clmracteristics tor degree 
words. For exami)h; ~ the correspondence between the. 
English word 'often' :rod the intuitively corresponding 
aa>nese word is not ebvhms. This paper proposes 
a conceptual representat;ion to describe a wide range 
of linguistic phenomelul, which are rehtted to degree 
concepts in natural language. 
1 Introduction 
Degree words in natural l;mguage, which are exempll- 
lied by the tbllowing, exist across p~rts of speech and 
across specific languages. 
(:) a. qua.ntifiers: 
b. adw.'rbs: 
e. adjectiw!s: 
~/li I lllally) SOllle I t'e~,v) llO 
always, oftell~ SOlIletil\[les~ 
seldom, never 
tall, short 
Degree words haw,. S()llle interesting characteristics. 
First, quantities in the real workl which can be repre 
sented by degree words w~.ry pragmatically, depending 
on speakers, situa.tions, etc. (Fauconnier, 1.975). This 
means that degree words do not have denotations in 
the rea,1 world. IIowever, lllalty degreL' expressions are 
used in daily life and it; is not tblt thai, they are par- 
ticula.rly incomprehensil~le. The authors do not think 
thaC to underst~md the lneanhtgs of degree words is to 
understan(\[ tit(; rea.l quantities in the real world. 
Second~ it is difficult to compare: degree words in dif- 
ferent languages, in the case of the English non-degree 
word ~dog, ~ we may think that the word semantically 
corresponds to the ;lapanese word 'iml' because these 
two words reli!r to ~he same object 'clog' in the real 
world. However, this correspondence is not true of de- 
gree words. Tim English woM %ften' intuitively cor- 
responds to Lit('. Japanese word %hitmshib% ~ but this 
correspondence is not ol)vious. 'Fhat is because these 
words do not have. denota.tions in the real world. 
These chttra.cteristics a, re related to the b~vse of Ma- 
chine Tr~tnslation and its dietionaries. Even when the 
real quantity, which is reti?rred to by ~t degree word in 
~t text, is not clearly understood, it is usually believed 
that it is possible to translate the word into ~tnother 
language. When building bi-lingual dictiomtries, it is 
necessary to consider the correspondence between (lm 
gree words in each htnguage. A new reference frame- 
work is needed by which to investigate to what extent 
the two words correspond to each other. These issues 
are. also related to conceptual descriptions in btrge scale 
knowledge l)aae projects, wlfieh have started recently. 
Third, degree words tuwe some ch~racteristics which 
are indepelMent from in~rts of speech. One of the phe- 
nomena degree words have in comnlon is modification 
restrictions between degree words and rlegree intensi.- 
tiers. Each degree word ha.s its own modification re- 
striction (Bolinger, 1972; Quirk, 1985; Kamei, 1988, 
1990). For example, %IP and qm ~ can be modified 
by 'Minos(,' but 'tall,' 'short,' hnany~' and 'few' can- 
not usually be so h~mdled. On the other hand, %all,' 
%hort,' 'many,' and 'ihw' rail be modified by ~ve.ry,' but 
'all' a,nd %0' cannot. 'Some' trod 'sometimes' cannot 
be modified by either %ery' or ~ahnost.' 
Previous researchers pointed out a lot of important 
linguistic phenonmna~ which ~re related to degree words 
but, the issues described abow ~. were left uninw~stigated. 
Barwise trod Cooper (1981) investigate relations I)c~ 
taween determiners in English and ge,mrMized quantl- 
tiers in logic. Howew.'r, they did not focus so much on 
degree words, such ~s ~tdjeetiw'.s and adverbs in gen- 
eral. it is still undetermined how to fully comprehend 
such words as 'ma.ny' and % i'ew.' Oazdar (1979) and 
IIirsehberg (1985) introduced ideas of a linear ordering 
of degree words and treated ~ wide range of phenom- 
elut related to degree words. However, they directly 
htmrlled real words and treated ~positive words ~ such 
as %11' and hnany,' ~nd qmgative words ~ such as flew' 
and %o ~ separ~ttely. Relations between the positive and 
negative words were not clear. In order to con,prehend 
these, unsolwM linguistic phenomena~ the authors pro- 
pose a semantic model of degree, concepts. 
775 
2 Discrete Degree Primitives 
and a List Expression 
This section introduces discrete degree priinitives and 
a list expression 'co represent mea.nings of degree con- 
cepts. From the perspective of quantities in the real 
world, hnany' and 'some' are sfmilar. However, The 
modification restriction be.tween degree words and de- 
gree intensifier shows that e~mh word is normally mod- 
ified by intensifiers selectively. This suggests the exis- 
tence of DISCILETE degree concept primitives, which 
are independent from parts of speech. The authors 
introduce five basic semantic primitives (~A,' ~M,' ~S ,' 
'F,' and 'N') indicating degree that are abstracted from 
the meanings of %.11,' hnany,' 'some,' ~fcw,' and ;no.' A 
list of degree primitives is used to describe meanings 
of degree words in terms of relative positions in the list 
expression. 
{A, M, S, l>, N} 
The list expression above is a basic list of the discrete 
model. The ,utthors divide meanings of degree words 
inte two parl;s. For example, 'tall' and %hort' can be 
divided into the semantic axis regarding 'tallness' and 
the degree concepts 'many (much)' ~md 'few (little).' 
Tables 2 and 3 represent the htter part of meanings of 
degree words. In these lists, '-' means that the value 
in that pa.rtict,la.r position is lacking. 
Table 1: List Examples (1) 
Basic list {A, M, S, F, N} 
~dl, always {A, -, -, -, --} 
many, often {-, M, -, -, -- } 
some, sometimes {-, -, S, -, -} 
few, seldonl {., -, -, F, -} 
no, llever {--, --, --, --, N} 
Ttd)le 2: List Examples (2) 
Basic list {M, S, F} 
tall {M, --, -} 
not tall and not short {-, S, -} 
short {-, -, F} 
The authors think that degree words are identified 
by their relative positions in the list expression. It is 
true that quantities in the real world, which are ex- 
pressed by degree words, are continuous, tIow('.ver, the 
authors think that language treats degree concepts in 
a discrete way. T~d)le 3 shows modificatiou restrictions 
on degree intensifiers using the primitives. In this ta- 
ble, %' shows that the intensifiers can lnodify the (le- 
gree primitives, and '-' shows the intensifiers cannot 
modify the prinfitives. Note that these primitives are 
not real words and that they consistently describe re- 
lationships that are independent from parts of speech. 
These are important differences between this model 
and previous research reports. 
Table 3: Modification Restriction of Degree Prinfitives 
and Intensifiers 
Intensifiers 
Booster 
Compromlser 
l)iminisher 
Approximator 
Maximizer 
Examples 
very, extremely 
pretty, somewhat 
a little, slightly 
ahnost~ nearly 
absolutely 
Degree Primitives 
A M S P N 
- + - + - 
- + - + - 
- + - + - 
+ + 
+ + 
3 A Dual List Expression of De- 
gree Concept 
It is pointed out that degree words convey non-literal 
'conversational' meanings when they are used. The dig 
ference between a literati meaning and a converst~tional 
meaning is called 'CouversationM Imf)licature' (Grice, 
1967). This section explains how this model treats this 
aspect of degree concepts. 
3.1 Question and Answer 
(17o exemplify Conversational Implicature, let us con- 
sider the following sentence, which includes a number. 
(3) I solved three of the problems. 
A natural interpretation of this sentence is "I solved 
just three of the prol)lems, not all or four or two or 
one or none of them." Kowever, in a logical way, this 
statement is true, when "I solved FOUR of them." For 
example, if the border line between success and failure 
of a test is three, this sentence is naturally spoken, even 
when, in fact, the person solved four of the i)roblems 
(Chomsky, 1972; Ota, 1980; Ikeuehi, 1985). The fol- 
lowing is a Yes/No question corresponding to sentence 
(3) and its answers. Interestingly, both of the answers 
below are possible in this case. 
(4) A: Did you solve three of the problems ? 
B: - Yes, in fact i solved four. 
- No, I solved four. 
In order to handle these phenomena, more complex 
states than just 'three' for the meaning of the nu,nber 
thre.e are needed. The authors think that five states 
are a.ctually needed fbr clarity: (1) All problems are 
solved. (2) The mmlber of solwxl problems exceeds 
the number which aI)pears in the sentence (=three in 
this case). (3) The number of solved problems is ex- 
actly the number which appears. (4) The number of 
776 
solved problems does not total tilt: mnnber which ap- 
pears. (5) No l)roblenls are solved. The authors intro- 
duce the five primitives, 'A,' '>n\] ~: 1 h' 'In,' and 'N,' 
corresponding t/) these five states, respec.tively. A list 
expression is introduced as follow.'< 
{A, >:1, <u, N} 
'l'he live sta.t, es are represent, e.d with rela,tive l>O~d • 
tiens shown in Talfle 4. 
Table 4: List Examph'.s (3) 
Bast<;list {A, >n, ::n, In, N} 
all 
> three 
three 
< three 
ltOlrl~ 
{A, , -., --, -} 
{-., >::, --, - , -} 
{-, -, , -} 
{-., -, -, in, -} 
To expre~,~ tit('. Conver~;ational lmlfliea, ture, the au- 
thi)rs ri,presiutt the mea.ning <>f tit(: number l>art in sen- 
tence (3) with a dual list. 
I 
The upper row (the direct meaning row) it: this tel> 
resentation shows the state wherifin the number of 
solved probh;m.~ is the number that appear.~ in sen- 
tcnce (3). The h)wer row (the possible interpretation 
row) expr(~sses the l)os>dbh: rm:nl><'.rs of solved l)rol)- 
ferns, whet: aeutence (3) is q)oken. For examph', this 
statement is false, wh,m "I solved TWO o'f theln." 
Logica.iiy, however, this star(relent is TR.UI~, when "l 
solved FOUR of theuC' The dual llst; represents the 
first \]>hen(line:m::, The ditl'erence I:)etwee:~ the two 
rows, 'A' and '>u' in this case, cxl>resses the l>OSSlbili - 
ties of C<>nviusatl<ma.l lnll:)liea,ture. \¥hen this sentence 
is Sl,ok(m , tim degree part <>\[ this seiltenee conveys th(; 
meatfings which correslx)nd to BOTH of the rows in 
the dual list. That is, it not (rely is indicated by the 
upper 'dir<!cC row, but also by the lower 'possil>h/row. 
In an atIirmativc sen~,enci:, t.he upper ~direct' lrleilAl- 
ing may be dominant, ilowever, in the case of an inter- 
rogative sentence, the h)w(:r 'pos.~ibh? meaning plays a 
more imp/)r taa:t role. This model explains the two pos- 
~dl)h' a n,~wers in ntwranee (d) in a. simph' way, In Fig. 
:\[, the meauing o\[' the que.,a.i/)n is expre,~sed with a dual 
list,. The meaning of the real sit, uathm (the iii('.anillg 
of 'fonr' in this ci~se) is express(!d with a single list (in 
the nfiddle), because it; is not. an interpretation, but is a 
situa.th)m ~¢'Vhell Cmnl)ariu Z the upl)er row of the ques- 
t.ion and t;he row expressing the ~dtuation 'tour,' there 
iS Ill) cotnnlon vii,lit(!, rF\]lel'e is tto intersection between 
them. 'i\['his case c(>rre.N)(>nds to the ~tnswer with ~No. ~ 
\;ghen comparing the lower ~l)ossible' row and the sit-- 
uation, t;here is an inti;rsecl, ion, that is, the wdue ~>n.' 
Therefore the answer is ~Yes.' This intersection opera- 
tion is a simple and naturM way to calculate possible 
answers to a question which includes a number. 
Question ('3' ?) Situation ('4') Answers 
A'>n':=lh-'-- YES 
Figure 1: Intersection Operation for Q and A 
3.2 Negation Operations 
This section introduces Neg~tion Operations, which 
axe. deti.ed on the dual list representation. Sentence 
(7) is a negative smite,we which correslmnds to sen- 
tence (3). A negative sentence like this tuus several in- 
terpretations which previous research has pointed out 
but has not been able to treat satisNctorily. This 
model calculates all the possible interpretations of a 
negative sentence fi'om the representation of the origi- 
na.l Mlirmative sentenc.e. 
(7) 1 didn't solve three of tile problems. 
One possible interpretation of sentence (7) is that 
there ~rc three problems that "i (lid uot solve" (Inter- 
pretation A). In tllis interprets.lion, the n/tmber %hree ~ 
is not under the influence of the negation; the number 
in out of the scope of negation. '1'o obtain this interpre- 
tation, it is not necessary to change the dtml list for 
the originM affirmative sentence (6). It is necessary 
to change the meaning of the values from the num- 
ber of tim solved problems to the number of unsolved 
problen:s in the representation of the original Mfirma- 
tiw~ sentence (Fig. 2). The lower row expresses the 
possibility that the number of the unsolved problems 
exceeds t hl.et:. 
Affirmative. Negative 
~" --,--,: in,--,-- ), ~ ~" --,--,=11,--,-- 
) (A,>ll,=n, , ) 
Solved probhm:s Unsolved 1)robh'.ms 
Intc'rpretation (A) 
Figure 2: One Nega.t:ve l::terpretaion from Aith'mative 
Dual List 
Where the musher (=three in this case) is within 
the scope of negation, the negative sentence requires 
other interpretations. 
777 
(8) A: Did you solve three of the t)rot)lems? 
B: No, i didn't (get to) solve three of 
the problems. 
.--- Interpretation (B) 
1Lesponse B might mean that some of the 1)roblems 
were solved, but that the munber did :tot reach three. 
This interpretation can be obtained from tile model 
shown in Fig. 3, and the negation operation is shown 
in Table 5. 
A ftirmative 
--,--,=n,--,-- } 
A,>11~=11,- ,- 
l (1) row Reverse each 
{a,>n,-,<n,N } 
} / \ 
(2) COMMON (2) DIFFERENT ,/ \ 
Z-,>,,,-,-,-; 
--,--,--,<n,NJ tA,>n, , , J 
Interpretation (B) Interpretation (C) 
Figure 3: Two Negative Iuterl)retatimls from Attirma- 
tire Dual List 
1. Reverse e~mh atlirmative row. 
2. Select the COMMON part of the two rows. 
The result is a new possible interpretation row. 
3. Omit t, he edge values (A aud N). 
The resnlt is a. new direct ineauing row. 
Tabh~ 5: Negation Operation tbr Interpretation B 
Step 1 in Table 5 rea.lizes a primitive negation oper- 
s, tion on each row. This i,~terpretation of the negative 
sentence is consistent with the negations of both the 
direct meaning and the possible implic~tion. Step 2 
realizes this condition. This interpretation usually im- 
plies that there are some solved problems. This means 
the negation usually does not deny the. existence of tile 
solw~d prol)lems, ttowever, in a logical way, no prob- 
lem being solved is a possible situation. Step 3 realizes 
this condition. 
(9) A: l)id you solve three of tke problems? 
C: No, I didn't solve THREE of 
the probh;ms: I solved ALL of them. 
- Interpretation (C) 
The above is a possible utterance, which requires 
~nother interpretation. Table 6 shows the way to cal- 
culate this interpretation (I:~terl)retation (C)). 
1. Reverse each affirmative row, 
2. Select the DIFFERENT part of the two rows. 
The result is a new possible interpretatiou row. 
3. Omit the edge wdues (A and N). 
The result is a new direct meaning row. 
Table 6: Negation Operation for Interpretation C 
This interpretation differs from interpretation B, 
only at Step 2, that is, 'to select the DIFFERENT 
p~rt of the two rows.' This means that the interpreta- 
tion is consistent with only the negation of tile direct 
meaning, and does not satisfy the negation of the pos- 
sibh." implication. Step 2 realizes this condition. This 
exemplifies that the Conversational Implicature can be 
canceled. In speech~ stress is put on THREE and ALL 
in this interpret,~tion, and this linguistic phenomenon 
is accounted for in Step 2. 
4 Negation of Degree Expres- 
sions in Natural Language 
In this section~ the dual list representation and the op- 
erations introduced in the previous section are applied 
to degree words other than numbers. 
4.1 'All,' ~no,' ~some,' and ~not all' 
Here, we will apply the same model to the relations 
between 'all/~some~' 'no,' and blot all' in natural lan- 
guage. Sentence (10-1) logically entails sentence (10- 
2). Sentence (10-2) usually implies senteuce (10-3). 
Itowever, sentence (10-3) contradicts the original sell- 
fence (10~1). A careless mixture of logical implication 
and usmd implication in language makes the inference 
of (10-3) from (10-1) unreasonable (t-Iorn, 1972; Ota, 
1980; McCawley 1981). 
(10-1) All students are intelligeut. 
(10-2) Some students are intelligent. (10-3) 
Some students are NOT intelligent. 
The discrete model is ~ usethl tool for describing 
these relations. List (11) is used to express relations 
1)etween 'all,' 'some,' 'no,' ~tnd 'not all (= some ... 
not).' In this case, only three primitives are used. 
(H) {a, s, N} 
In this list, the w~lue ~S' corresponds to the state 
wherein there are SOME students who are intelligent 
and SOME other students who are NOT intelligent. 
The me,~nings of these words are also expressed with a 
dual list. Figure 7 graphically represents this. 
778 
~11 
A,-~- 
"enta~il" 
(second row) 1 
SOllte 
lie 
/ "ent;dl" 
dii\[orent 
(second ~ow) ilot MI 
=: SOllle llOt 
l?igure 4: ~AII,' 'some,' '1/O,' aLlld 'll.Ot adl' 
Ill Fig. 4, the second 'possible' rows of Call' ~nd 
~sorne' ha.ve an intersection at the wdue ~A.' ~No' au,d 
'not ~dl' hatw; a, simila.r intersection. This re~dizes en- 
t~dhnent between tile two concepts. \]."igure 4 also ex~ 
presses the differeuce between 'eontra.ry' and %:ontr~v~ 
dietory.' If '~dl' is true,qlo' is faJse, if 'no' is tru% 
qdP is fa,lse. Both exl)ressions etLlllll)t be trite at the 
same time. Itowever, these two CAN BE FALSE art 
tile s~mm time, beca.use it is 1)ossible that some stu- 
dents a.re intelligent auld seine students are not. The 
term %:ontra.ry' expresses this rela.tion. ()t~ the other 
h~tnd, 'MI' and hu)t all' h~w', :~ ditli~rent rel,ttionship. 
'.\['hes(~ two Call.liO~ \])e trl.le at tim same time, and e.~Lll.- 
not 1)e false at the SS,llle ~il\]\[le. ~No 1 and ~SOllle' h~Lve 
the S&llle constra.int. The term 'contradictory ~ in Fig. 
4 expresses this rela.tion. 
An iml>orta.nt poiut her(~ is that the s;mm oper~ttion 
of nets.lion, ~I'able 5, used fl)r numl)ers will also obt;dn 
tile representattion of h~ot all' front that of bdl' ill Fig. 
4. Tim other nets.glen operation, q'able 6, produces 
nothing in this (:ase. (Fig. 5). The negation oper~tions 
a.re basic tuld general. 
Note th~tt 'S' in list (11) in this section mentions only 
the existence of intelligent st.ud~mts and non-intolllg('nt 
students. In Section 2, the same symbol 'S' was used 
for the rnea.ning of 'seine' which is relatively defined 
ill the {A, M, S, F, N } (list (2)). \]h, tha.t ease, the 
value ~S' r(;presents a quantlt~ttlve aspect of {SOllle.' A 
relation betwc'eu ~S' in list (11) and %S+ ' = ~S' in list 
(2) is described as iollows: 
S- ( M, S +, F ) 
YIowever, the authors used the same vadue ~S' in 
both list:;, 1)ecaa:se the difl'erence~ between these two 
~S's is represented by the set of values in list expres.- 
sions. These two 'S's correspond to ambiguities which 
the. word 'some' in natural la:Dguage has. 
(2) COMMON ,/ 
{-,S,-~ 
-,S,NJ 
not all = Soille ,., It(It 
all 
(1) row Rew, rse catch 
{-,s,N } 
{-,S,N } / \ 
(2) DIFFERP~NT 
Figure 5: Neg~tion Operation executed on 'ALL' 
4.2 ~Not many' and 'not a few ~ 
This section tel)resents lne~tnlngs 'nt~tny,' h~ few,' ~nd 
q'ew', ~nd applies the negation operations on these con- 
cepts. Figure 6 shows du~d list representations for these 
three concepts. This figure shows that the ditference 
between 'few' and 'it few' is in the lower possible mei~n-- 
ing row. It is the first time tht~t ~he difference betweem 
the two is explicitly shown. 
(a0,t,~lly A,M, , , 
,, 3,- (1:,) :t few 1, , 
A.,M,S,I ,- 
(,:)few { , , ,F,-},, ,, ,t ,N 
Figure 6: 'Many,' 'A Few,' atnd 'Few' 
Figures 7 and 8 show interl,ret~ttions for 'not lltgny' 
~u:d q,ot t~ few,' which ~tre c~dcub~ted from the me~tn~ 
ing of hna.ny' ~utd %t fe.w' using tile neg~ttion ol>ertttions 
introduced in the previous section. The neg~tion Ol'- 
erations produce two possible interpretations for qlot 
m~uy.' However, the direct me~udng row for one inter~ 
pretattion is htcking. Tills shows t, hatt this interpret~- 
lion is logically possible, lint unusu\[d (this interpret~t- 
tion is hdl'). The other is *t usual interpret~tiou of blot 
lnatny.' q'he dual llst of the usmd interpretation shows 
that blot m:~ny' does not clahn 'few,' but it rues.us less 
th~.ul j'ttst ~soi:te ... tier. ~ Th(~ s~ttite negation operations 
also I)roduce m(,anh:gs of 'not a few.' The dual list 
of its usual interpretation shows that ~not a few' does 
not ('.l~dm hmtny,' but it meatus more tha.D, just ~some.' 
Note that the dual list ret,resentattion and tile negation 
oper~ttions on it exphdn vagueness of htot nt~tny ~ atnd 
'not a few', a~s well as ~unbiguities of their interpretat- 
lions. 
779 
nl&lly {-,M,-,-,-; 
A,M, , , J 
(1) row Ileverse each 
{A,-,S,~ ,N } 
{-,-,S,S,N } / 
(2) COMMON / 
{-,-,S,F,- l 
-,-,S,F,NJ 
Usual interpretation 
\ 
(2) DIFFERENT \ 
{7,' ' ' '-} 
Unusual interpretation 
Figure 7: Not many 
~:L few 
-,-,-,F,- 
A,M,S,F,- } 
(1) Reverse each row 
{A,M,S,-,N } 
/ 
(2) COMMON / 
{-,,, ,;} 
Unusual interpretatiou 
,N } \ 
(2) DIFFERENT \ 
A,M,S,-,-J 
Usual interpretation 
Figure 8: Not a few 
This paper introduced eight basic degree primitives 
for degree concept, that is, 'A,' 'M,' 'S,' 'F,' 'N,' '>n,' 
'=n,' and '<n.' ttowever, the authors do not claim 
that these eight primitives are sufficient to indicate 
all degree concepts. Instead~ the authors clMm that 
people comprehend degree concepts in a discrete way, 
and that degree concepts are identified by their rela- 
tlw~ positions in the fl'amework of understanding. Con- 
sider the following cxan~ples concerning ,~nother degree 
concept 'several,' which differs from these eight degree 
concepts. 
(13) They legally have several wives. 
Quantities, which are refl.'rred to by 'sew'.ral' and 'a 
few,' seem to be close. It is often said that quantities 
ret>rred to by 'several' include fiw.' or six, and more 
tha.n the quantities referred to by '~t few.' However, 
sentence (13) shows th~tt 'several' means more than 
one in this case. Previous researchers have not been 
successflfl in describing the diflhrence between 'sevend' 
and 'a few.' The authors think that 'sew.'ral' should 
be in a list including 'several' and 'one,' while 'a few' 
should be in a list which contains 'a few' ~tnd 'many.' 
'Several' implies 'not one,' while '~ few' implies 'not 
many.' An important point is that the ditference be- 
tween 'several' and ~a few' is not the exact quantity 
involved, but a framework of understanding, that is, 
the set of vahms in the lists and their relative posi- 
tions. 
4.3 'OR' in Natural Language and 
Negation 
It has been shown that the logical operator ~OR' 
has characteristics similar to degree concepts (Gazdar, 
1979). This is because 'of in natural language gener- 
ally has two interpretations, the 'inclusive or' and the 
'exclusiw'. or.' This section applies the same model for 
degree concepts to a logical operator 'OIL' and 'or' in 
natural language. 
It is difficult to conceptualize the ne.gation of 'or' in 
natural language, in a usual sense, although negation 
of 'and' is easy. Logically, however, the negation of the 
logical operation 'OR' (that is, 'Inclusive or') is 'NOR.' 
However, in a sense in natured language, 'AND' instead 
of 'NOR' can also be a negation of 'Oil..' 
and or nor 
{ (+ +), (+-4-+), (- -) } 
((--,(+-/ +),:}Exc. OR 
+ +), (+ -/- +), Inc. OR 
{(+ +) ,-, (--) } 
{ , , (---) } 
COMMON DIFFERENT { } 
nor and 
Fig. 9: Negations of 'OP,' 
Figure 9 shows the relationship between tile inclusive 
and exclusive 'or' and their negations. The authors use 
three states: (,-+), (+-/-+), and (--). 'Exclusive 
or' is a direct meaniug of 'or' ~tnd 'inclusive or' is a 
possible interpret~tion of 'or' in this framework. The 
same negation operations will produce the two neg> 
780 
th)ns of 'or 1' tha.t is, both NOR and AND. Tim direct 
mea, ning rows in the two interl)ret;~tlons of nega.tions of 
'or' ha.ve no values. This corresponds to the Met tha.t 
it. is difficult to consider the nega.tion of %r' in mttura,l 
language. No~(; that the dual list fbr 'or' and the du;d 
list. for 'somo' in Fig. 4 ha.ve an ido.ntical structure. It 
is equally explained that the nega.tion of 'some' is difli- 
cttlt to consider in na.tural language., while the neg~tion 
of %11' is easy. 
5 Conclusion 
This pa.per has present(!d ~ new model for degree con- 
cepts in na.l, ur;d langtutge. The characteristics for the 
model are: (1) The discrete degree primitives. (2) Tim 
list represent~tion of degree concepts. (;3) q'iu', duaJ list, 
representa.tion \[br possibilities of Conw~rs~tiona.1 hnpli- 
ca.tuft. (4) The intersection operation on the list for 
realizing t;nt;a.ihnent of two concepts. (5) The nega£iol| 
ol)era.tions on the dua.1 list to <dculaw~ a,ll the possil)le 
interl)reta.\[ions of nega.tion of degree concepts. 
The model describes, cah:nhttes, ;utd eXlflah> a, wide. 
r~u~ge of linguist, it l)hcllotnelta, related to degree cm> 
eepts, such as (\]) Mo(lillc:~tion lh~strictions between 
degree intensifiers and (h'gre(~ words ~mross i);xrts of 
speech. (2) All tam l)OSslble aw;wers to a question 
which c.ontains a. qua ntit;aMve word. (3) All the po> 
sil)le hlterl~retatious of w'tg~tioa of quantitative words. 
(4) 3'Ira diti'erenee 1)¢~t;ween 'few' and % Jew.' (5) 't'he 
Vtl~llel\[ess o12 e/tl)ll(?llliSni t)\[ lt(~ga, l;it)ltS of dcgr(,e wOl:ds~ 
stteh ils eliot llI~t, lly' altcl in()t. ;t t'ew.' ((J) ~\['he difficulty 
elf a.pi)lying negation fitr stmte (lnanl.il.a.tive WOL'(\[S, Sl/ch 
People use a lot of degrt~e words and communicam~ 
wii.h each othc'r in da.ily life, even when qua.ntities 
which are exprcsstxl by thenl ma.y trot \])e precisely un- 
derstood. '\['he authors I.herefore think tlmt natural 
la.nguage in itself has ;t I)ISCItETI3 framework of de- 
gree. concel)t , and tha.t 1)(tth the Sl)eaker aud tit(.' hearer 
IlIIIsL ha,ve a COI|IIlIOlt fr3,11te of lllldel'stallding~ before 
holdhtg at. sl)ecific conversa.tiou. 'l)t) under.M.a.nd degree 
coneel)ts ix to understand their relative positions in a 
discret;e t'rz~nw, of unde.r.~tandS.iW;. 'Phis is the authors' 
viewpoin(; Olt (\[(!gl'oe cOllct~pt COllllll/lnication. 
'.i?he corresl)ond(mce 1)ctween the Fnglish woM %t= 
t, en' ~utd the ,ltq)anese wor, l 'shihashib;¢ has been es- 
tM)lished a, nd is genera.lly conced('d as 1)tint ai)prol)ri- 
ate. llowever, th;~g is not jltst be(:a.use these two words 
refer to the same rea.l qua.ntity. \'Vh;tt is (:OlnlilOli l)(;- 
tw(~t;n the. two is rehtfive position in the multi-window 
device. 't'h~t establish(:s the correspondence, for the 
nmanhlgs o\[' the two. 
This mt)del also describes ph(utom(m~v related to 
'OIU in logic a, nd 'or' in m~turM language. This sug- 
gests theft the model represents substantial structures 
in mttural la.ngu~tge and is ~t suitalJle tool for naturM 
l~nguage understanding. The authors holm that this 
model will be one of the possible extensions of the first- 
order Logic. 
Acknowledgments 
The authors wish to exi)ress appreci~tlon to the b~te 
Stephen Cudhea, NEC Corpora,ti(m, whose chtssifict> 
tion of English adverbs mi~de this reset~rch possible. 
The numerous research data were obtMne.d during one 
of the authors' stay at Computi,tg l~ese~trch Labora.- 
tory (CRL), New Mexico Sta.te U,fiversity. The au- 
thors tlmnk Yorick Wilks, who was Director of CI/L, 
'\]?akMfiro W*tkam, David Parwell, John B~mMen, ~md 
Stephc'n Helmreieh at CRL for their suggestions. 
References 
\[I\] Ba,.wise, 3. and l/,. Cooper (1981). 'Generalized quan- 
ti\[iern and natural language.' Language and Philoso- 
tlhy .I.2, pl ). 1,39-2119. 
\[2\] Bolh,ger, I). L. (1072). Degree "~oo','ds. Mouton. 
\[3\] Clmmsky, N. (I 972). St~tdies ~m semantics in genera- 
~'UI~ 91'(ullg?lga'l'. MolltoiL 
\[4\] li'auconnler, G. IL (1975). q~raglnatic scales aim logi- 
cal struct.m'e.' IAnguisbic Inquiry 6, pl ). 357-375. 
\[5\] G;~zclar, G. (1979). lh'(tgmalics: lmldicalure, l"res'ap- 
posiliol~, al,l Logical Form. Academic Press. 
\[6\] C, rice, I1. I'. (1975). ' \[,ogic ~tll(1 conversat.ion' In P. Co\[c 
and J. 1,. Morgan Eds., Speech ~tcis, pp. ,t5-58. Syntax 
;~nd Semantics 3. Academic Press. 
\[TJ llirschberg, ,1. 1/. 11985). 'A Theory of Scalar hnplica- 
tin'(? Ph. 1). dissertathm, University of Pemlsylwmla. 
\[81 Horn, L. 11. (1972). 'On the semantic l)roi)ertles of log- 
ical ol)erators in l!',nglish' l{cl~roduced by the Indiana 
University Linguistics Cliff) (1976). 
\[9} Ikeuchl, M. (1985). Mei,shi&'a no genh'.i hyougen 
(Noun l, hra~,: sl, e, cifyi',~g e:cprcsaiona), (in Japanese). 
TMslmkan. 
\[10\] Kaluei, S. imd K. Murakl (1988). On a Model ofl)e- 
grce Exprcsslon, (in Japanese). NLC 88-6. '\]?he lnstl- 
l, ui.c of Electronics, lnformatAOll and C)omlnunlcatlon 
I')ngineers. 
\[I t\] K,unel, S., A. Okumura, and K. Muraki 11990). Syn- 
tax of Â"mgllsh Advcrl), (in Japanese). Proceeding of 
the .lOlh Conference of lnfo'rmatiou Processing Soci- 
ety of Japan, Vol. l, PP. 417-418. 
\[12\] Mc(Jawley, ,I. \]). (11981). ltveryU~ing that l;inguists 
have Always Wanted to Know about l, ogic but were 
ashamed to ash. The Universlt.y of Chk:ago Press. 
\[13\] ()t% A. (1980) llitel no I.mi (Meaniw/s of Negation), 
(in Japallese), %dshukan. 
\[14\] Quirk, R, S. Greenbaum, G. Leech, and J. Svartvik. 
(1985). A Comprehensive Grammar of the English 
La~l'lMt(Je. Loltgtlt3,11. 
\[151 Yagi, T. (1!187). 7'eido hyougen to hikab~ kouzou 
(1)c..qree c:~:pre.ssions and comparative slruct'~tres), (in 
,/~q)anese). Taishukan. 
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