1967 International Conference on Computational Linguistics 
Axiomatic Characterization of Synonymy and Antonymy 
H. P. Edmundson 
University of California, Los Angeles 
i. Introduction 
i.i.- Background 
This work is a continuation of research reported in the paper 
Mathematical Models of S~non~, which was presented at the 1965 
International Conference on Computational Linguistics. That paper 
presented a historical summary of the concepts of synonymy and 
antonyms. It was noted that since the first book on English syno- 
DS, which appeared in the second half of the lath century, dic- 
tionaries of synonyms and antonyms have varied according to the 
particular explicit definitions of "synonym" and "antonym" that were 
used. The roles of part-of-speech, context of a word, and substi- 
tutability in the same context were discussed. 
Traditionally, synonymy has been regarded as a binary relation 
between two words. Graphs of these binary relations were drawn for 
several sets of words based on Webster's Dictionary of S~non~ms and 
matrices for these graphs were exhibited as an equivalent represen- 
tation. These empirical results showed that the concepts of synonymy 
and entonymy required the use of ternary relations between two words 
in a specified sense rather than simply a binary relation between two 
words. The synonymy relation was then defined implicitly, rather than 
explicitly, by three axiams stating the properties of being reflexive, 
symmetriC, and t/~ansitive. The entonym¥ relation was also defined by 
three axioms stating the properties of being irreflexive, symmetric, 
and antit/~ansit~ve (the last term was coined for that study). It was 
noted that thes~ six axioms could be expressed in the calculus of re- 
lations and that this relation algebra could be used to produce short- 
er proofs of t~eorems. However, no proofs were given. In addition, 
several gec~aet~ical and topological models of synonymy and antonymy '..J~ 
were posed and examined. ,~ 
It was nOted that certain of these models were of more theoretical 
than practical interest. Each model was seen to be simple in that it" 
could be expressed from mathematically elementary concepts, end each 
stressed certain aspects of the linguistic object being modeled at the 
expense of others. However, there seemed to be little theoreti~al 
preference among them. Their adequacy as models could be measured by 
their generality and predictive power. In terms of these criteria the 
algebraic model, whether expressed in terms of relations, graphs, or 
matrices, seamed to have the most usefulness. In part, this was due 
to the fact that one geametrical model, although highly suggestive, 
did not include a precise specification of the origin, axes, or co- 
ordinates for words in an n-dimensional space. Similarly, one topo- 
logical model required a closure operation for each of the intensions 
or senses and had no linguistically interesting interpretation. 
1 
1.2 Summary 
The present paper investigates more thoroughly the characterizations 
of synonymy and antonymy initiated in Edmundson (1965). In section 2, 
synonymy and antonymy are defined jointly and implicitly by a set of 
axioms rather than separately as before. First, it is noted that the 
original six axioms are insufficient• to permit the proofs of certain 
theorems whose truth is strongly suggested by intuitive notions about 
synonyms and antonyms. In addition, it is discovered that certain 
fundamental assumptions about synonymy and antonymy must be made ex- 
plicit as axioms. Some of these have to do with specifying the domain 
and range of the synonymy and antonymy relations. This is related to 
questions about whether function words, which linguistically belong to 
closed classes, should have synonyms and antonyms and whether content 
words, which linguistically belong to open classes, must have synonyms 
and antonyms. Several fundamental theorems of this axiom system are 
stated andproved. The informal interpretation of many of these 
theorems are intuitively satisfying. For example, it is proved that 
any even power of the antonymy relation is the synonymy relation, 
while any odd power is the antonymy relation. \ 
In section 3, topological characterizations are posed and examined. A 
neighborhood topology is introduced by defining the neighborhood of a 
word. It is proved that this definition satisfies four neighborhood 
axioms. Also, a closure topology is introduced by defining the 
closure of a set of words. It is proved that this definition satis- 
fies the four closure axioms. 
2. Algebraic Characterization 
2.1. Introduction - Relations 
Before investigating antonymy and synonymy, we will estsblish some 
notions and notations for the calculus of binary relations. 
Consider a set V of arbitrary elmnen~s, which will be called the uni- 
versal set. A binary relation on V is defined as a set R of ordered 
pairs <x,p, where x,y s V. The proposition that x stands in re- 
lation R toy will be denoted by xRy. The dcmain~Y(R), range ~(R), 
and field ~ (F) of relation R are, respectively, defined by the sets 
\[x:(~y)(xRy)\] ; (y:(~Lx)(xRy)} ; \[x:(~y)(xRy)} U (y:(~x)(xRy)\] 
The complement, union, intersection, and converse relstions are de- 
fined by 
x~y = -.x~ ; x(RUS)y - x~vxSy ; x(RnS)y " x~x~; 
xR'ly -- yRx 
The identity relation I and null relation ~ are defined by 
xIy ~ x=y ; ~y - (x~x),V~(y~y) 
The product .and power relations are defined by 
xRISy = (.~z)\[xRz ^zSy\] ; R n =- RIR n'l n~ 1 
Inclusion and equality of relations are defined by 
RC S =- xRy ==> xSy ; R = S m R c SA S c R 
Later we will use the following elementary theorems which are stated 
here without proof: 
Theorem: R g S ==> R "I c_ S "I 
m -- Theorem: R c S ~> S c R 
Theorem: (R'I) "I = R 
Theorem: (RIS)IT : RI(SIT ) 
Theorem: (RIS) "I = S'IIR "I 
Theorem: IIR = RII = R 
Theorem: s -r => RIs=RIT ^ SIR=TIR 
2.2 Axioms and Definitions 
Under the assumption that synonymy and antonymy are ternary relations 
on the set C of all content words, the following definitions will be 
used: 
xSiY = word x is a synonym of word y with respect to the 
intension i (or word x is synonymous in sense i to 
word y) 
xAiY -= word x is an antonym of word y with respect to the in- 
tension i (or word x is antonymous in sense i to word y) 
We will assume that the synonymy and antonymy relations are defined 
Jointly and implicitly bythe following set of axioms rather than 
separately as in Edmundson (1965). 
Axiom 1 (Reflexive) : (Vx)\[xSix\] 
Axium 2 (Symmetric): (Vx)(Vy)\[xSiY => xS;Iy\] 
Axium 3 (Transitive): (Vx)(Vy)(Vz)\[xSiY A YSiZ :> 
Axi~n 4 (Irreflexive) : (Vx) \[x~ix\] 
Axiun 5 (Symmetric): (Vx)(Vy)\[xAiY => xA;ly\] 
Axi~n 6 (Antitransitive): (Vx)(Vy)(Vz)\[xAiY A YAiZ 
Aximm 7 (Right-identity): (Vx)(Vy)(Vz)\[xAiY A YSiZ 
Axiom 8 (Nonempty) : (Vy) (:~x) \[xAiY\] 
xSiz\] 
~> xSiz\] 
~> xAiz\] 
The properties named in Axiams 6 and 7 were coined for this study. 
The above eight axioms may be 
as follows: 
Axicm I (Reflexive) : 
Axiom 2 (Symmetric): 
Axiom 3 (Transitive) : 
Axicm 4 (Irreflexive) : 
Axiom 5 (Symmetric) : 
Axiem 6 (Antitransitive) : 
Axiom 7 (Right-identity) : 
Axiom 8 (Nonempty) : 
expressed in the calculus of relations 
I~Si 
sl =- si 1 
~i = S i 
Ai c_ A; 1 ' 
Ai I Si c_ Ai 
(Vy)\[A(y) ~ ~\] where A(y) = {<x,y> : x E~(A)} 
3 
This relation algebra will be used to produce shorter proofs, al- 
though this is not necessary. The consistency of this set of aximms 
is shown by exhibiting a model for them; their independence will not 
be treated. 
In addition to the synonymy and antonymy relations it will be 
useful to introduce the following classes that are the images of 
these relations. The synonym class of a word y is defined by 
si(Y ) '= \[x : xSiY\] 
which may be extended to an arbitrary set E of words by 
si(E) =- {x : (.~y)\[y ~ ~. ^ xSiY\]\] 
Similarly, the antonym class of a word y is defined by 
ai(Y) --- {x : xAiy\] 
which may be extended to a set E of words by 
ai(E ) m {x : (~y)\[y e E A xAiY\]\] 
2 • 3 Theorems 
For ressons of notational simplicity, the subscript denoting the 
intenslon i will be omitted in the sequel whenever possible. How- 
ever, the theorems must be understood as if the subscript were 
present. 
As with any symmetric relation, it is possible to get stronger re- 
sults than Axi~ 2 and Axiom 5. 
Theorem: S "1 = S 
Proof: 1 S c S-1 by Axiom 2. Hence S "1 c_ (S-1)-I = S. There- 
fore S" = S_by definitiQn of equality. 
Theorem~ A "I = A 
Proof: Same as above theorem using Axi~n 5. 
Also we get a stringer result than the transitivity property of 
AxiQm 3: 
Theor .em: ~ S 
Proof. ~ c_ S by Axiom 3. Hence S = SII c_ SIS = ~ by Axiom 1. 
Therefore S 2 = S by definitio~ of equality. 
In fact, by induction we have the generalization: 
Proof, 8n= .1 s (sl~ "2) .... = sl(sl(sl"'Is)'") =s. 
It can be shown that anton¥~ and sync~n~ are distinct: A ~ S. In 
fact we have the stronger result: 
Theorem: A ~- 
Proof: Assume A ~ 7. Hence A n S ~ ¢ or (~x)(~M)\[x(A 0 S)y\]. 
Then x~7 ^ xSy implies xAy ^ ySx by Axicm 2. So xAx, which 
contradicts x~x by Axi~ 4: I ~ ~. Therefore A c_ ~. 
because of Axiom 8, can we get a stronger result than the anti- 
transitivity p~oplFty of Axiom 6. 
Theorem: A = S 
O~I :. ,A~-.AI S bYl~imm 7. Hence.A 2 = AIA ~-- AI(AIS ) = A'II(AI s) = 
" IA)IS s~ce A" = A. Now (Vy)(~x)\[xAy\] by Axiom 8. So 
(_vy)(~)E~AI~ ^ ~Ay\]~ by ~i~ 5. H~ce (Vy)E~Iy --> ~A-11~l. 
z nus I c_ A" IA. So A ~ ~__ I IS = S. Therefore A "~ = S since A2 ~ S byAxium 6 and S G A 2. 
The right-identity property of Aximm 7 can be strengthened to: 
Theorem: AIS -- A 
Proof: AIS u A byAxinm 7. NowA =AII U AIS since I u S. 
Therefore A I S A by definition of equality, 
As a corollary we get that S and A ccexnute : 
Corollary. AIS = SIA 
Proof: AIS --A = A "A = (AIS) "l = (A'lls'l) "l -- SIA 
From the above two theorems it follows that: 
Theorem: SIA = A 
Proof: S~A =A IS =A. 
As a special case we ~et: 
Theorem: A 3 =AIA =AIS =A. 
In fact, we have the generalization: 
S if n even Theorem: An = A if n odd 
Proof: For n even, A n = A 2k = (A2) k = ~ = S. For n odd, 
A n = A 2k+l = AI (A2) k = AtS = A. 
Next, several theorems about synonym classes and antonym classes will 
he stated and proved. First, the synonym class of a word is not 
empty: 
Theorem: s(y) ~ ¢ 
Proof: NOW I c S by Axiom 1. So (Vy)\[ySy\]. Hence (.~x)\[xSy\]. 
Therefore, s(y) ~ ~. 
Because S is a symmetric relation, we have: 
Theorem: y e s(x) <~> x e s(y) 
Proof: y e s(x) <-----> ySx <-----> yS'ix <----> xSy <-----> x e s(y). 
Since S is reflexive, symmetric, and transitive, S is by definition 
an equivalence relation on theset C of all content words. Hence, we 
have the important result: 
Theorem: xSy <-----> s(x) = s(y) 
Proof: (------->) Assume xSy. First let u G s(x). Then uSx ^ xSy 
------> uS2y -------> uSy -------> u • s(y). Hence s(x)c_ s(y). Also 
s(y) c_ s(x) by a similar argument. Therefore s(x) = s(y). 
(<==) Assume s(x) = s(y). Then u e S(X) -~-> U • s(y). SO 
uSx -------> uSy. Hence xSu ^ uSy ~-> xS~y ==> xSy. Therefore xSy. 
In fact, we have the stronger result: 
Theorem: s(x) N s(y) = .~" s(x) if xSy 
L ¢ if  -sy 
Hence for a given intension i the equivalence relation S i parti- tions the set C of all content words into subsets that are 
disjoint (i.e., the subsets have no word in common) and exhaustive 
(i.e., every word is in some subset): 
Theorem: C =~ ) si(x) 
x~ 
Second, the antonym class of a word is not empty: 
Theorem: a(y) ~ 
Proof: A~ 8: (vy)(~x)tx~1 ~p~es a(y) ~ ~. 
Note that a word does not belong to its antonym class: 
Theorem: y ~ a(y). 
Proof: Assume y e a(y) so that yAy. But this contradicts 
Axiom 4: yIy ~ yXy. Therefore y ~a(y). 
Next we will establish some relations between synoc~ym classes and 
anton~a classes. 
Theorem: xA~ ~ ~(x) = s(y) 
Proof: (==>) Assume x e a(y). First let u e a(x). 
~owue a(x)AxAy ~ uAx^xAy ~ uA2y ~ u~ 
~-~ u ¢ sCY). Hence aCx) g s(y). Also sCy) c_ a(x) by a 
similar argmnent. Therefore a(x) = s(y). (~) Assume a(x) = s(y). 
But y • s(y) = a(x). Hence yAx. Therefore xAy by Axicm 5. 
In fact, we get the following necessary and sufficient condition 
for equality: 
Theorem: a(x) = a(y) <==~ s(x) = s(y) 
Proof: (~--~) Assume aCx) = a(y). Now a(x) rl a(y) ~_~ 
z 
(~z)\[z~ ^ ~,v\] ~-~ (~z)\[xAz ^ zAy\] ~ xA-y ~ xSy. 
Therefore s(x) = s(y) by a previous theorem. (<~) Ass~e s(x) = 
s(y). Then xSy. First, let u • a(x). Then uAx. Hence uAx A 
XSy ~ uAISy ~ u~y ==~ U • a(y). Therefore a(x)g a(y). 
Also a(y) g a(x) by an identical argument. Therefore a(x) = a(y). 
2.4 Comments on the Algebraic Characterization 
Even though s(y) # ~ since ySy by Axinm i, it may be necesssry to 
add the following axiom: 
Axiom 9: (Vy)C~x)Kx ~ y ^ x~\] 
to guarantee that the domain of the relation S is not trivial, i.e., 
s(y)-Cy\] ~¢ 
Axiom 9 is not necessary if s(y) is permitted to be a unit set for 
certain words. Thus, we might define s(y) = (y) for any function 
word y, e.g., s(and) = (and). But this will not work for antonymy 
since a(y) might be considered empty for certain words such as 
function words, e.g., a(and) = ~. The alternative of defining 
a(y) = ~ is not reasonable since it produces more problems than 
it solves. Axiom 8: (Vy)(~x)\[xAy\] is reasonable if the contraries 
_of words (e.g., nonuse, impossible, etc.) are permitted, i.e., 
y e ~(y). 
6 
The theorems 
= S , A 2 = S , AIS =A , SIA =A 
can he summarized in the following multiplication table for products 
of the relations S and A 
S A 
A 
which is isumorphic to the table for addition modulo 2 
0 1 
1 
Note, even without Axicms i-8, for 
(1) A 2 = S , (2) A S =A , (3) A\[S = A 
that 41) and 42) £mp~ 43), (i) and 43) ~P~v 42), but (Z) and (S) do 
not i~ (1). 
Suppose that for every pair <x,y> Of words in the vocabulary V of a 
language exactly one of the following ternary relations holds : 
(1) x and y are synonymous, xSy 
2) x and y are antonymous, xAy 
3) neither (1) nor 42), xMy 
This can be expressed by 
(Vx)(Vy)\[x,~ e v----->xs~Vx~vvx~\] 
which is an exclusive disjunction. Thus the vocabulary V is 
partitioned as follows: 
V = s(y) U a(y) U m(y) 
This also can be pictured in the lattice of for every word y. 
relations 
U=V~V 
It can be shown that the multiplication table for products of 
the relaticms S,A, and M is 
S A M 
S S A M 
A A S M 
M M M M 2 
7 
3. Topological Characterizations 
3.1. Introduction 
We will now examine two topological models of synonymy. Being 
topological, they concern "semantic spaces" of words without any 
notion of "semantic distance" between two words. Again, we will 
restrict our attention to content words. Topological models for the 
antonymy relation will not be considered. 
3-2. Neighborhood Topology 
The first model considers a neighborhood topology, i.e., a topology 
based on neighborhoods. A set is said to have a neighborhood 
topology if there exist elements x called ~ and sets N x called 
neighborhoods of x Which satisfy the following axlcms: 
Axiom l: (Vx)(~Nx)\[X e N x\] 
~ian 2: (v~ x) (vN x).c.~N''~x. \[~x ~ ~x n N x\] 
Axiom 3: (vy)CVSx)CZs~,)\[y ~ s x ~> ~y ~ ~1 
Axiom ~: (Vx)(Vy)(~x) (aNy) Ix ~ y ------> Nx n Ny -- ~\] 
These axioms can be pictured informally by the following Euler 
N x N~ N x Nx Ny 
Define a neighborhood n~(x) of a word x as any subset of the synonym 
class si(x) o~ x that cSntalns x, i.e., 
X e ni(x ) ~ si(x ) 
Wain, for- reasons of notational simplicity, the subscript denoting 
the intension i will be emitted whenever possible. 
First, neighborhood Axiom 1 is satisfied. 
Theorem: (Vx)(an(x))\[x z n(x)\] 
Proof: By definition s(x) is a neighborhood n(x) of x 
c oalt aining x. 
Second, neighborhood Axiom 2 is satisfied. 
Theorem: (Vn(x))(Vn'(x))(~n"(x))\[n"(x) c n(x) n n'(x)\] 
Proof: For arbitary n~x) and n'(x), let n"(x) = n(x) N n'(x). 
Then n"(x) ~ s(x) since n"(x) = n(x) n n' (x) c s(x) N s(x) - s(x). 
Also, x e n"(x) since x ¢ n(x) ^ x e n'(x) imply x e n(x) n n'(x) = 
n"(x). Therefore, (Vn(x))(Vn'(x))(~n"(x))~"(x) ~ n(X) n n'(x)\]. 
Third, neighborhood Axiom 3 is satisfied. 
Theorem: (Vy)(Vn(x))(~n(y))\[y e n(x) ==> n(y) c_ n(x)\] 
Proof: For arbitrary y e n(x), let n(y) = n(x). But y e n(x) 
implies s(x) = s(y) since y e n(x) c_ s(x) = {z : zSx\] implies ySx 
and ySx implies s(y) = s(x)° Then n(y) c_ s(y) since n(y) = n(x) 
c_ s(x) = s(y). Also y e n(y) since y e n(x) = n(y). Therefore, 
(vy)Cvn(x))Czn(y))\[y ~ n(x) ~ n(y) ~- nCx)\]. 
In fact, the neighborhood topology satisfies Axiom 4, which is a 
separation axiom: 
TheorT: (Vx)(Yy)(~n(x))(~n(y))\[x ~ y => n(x) n n(y) = ~\] 
Proof. Assume x ~ y. Let nCx) = (x} and n(y) = {y}. 
Then x e n(x) ~ s(x) and y e n(y) ~ s(y). Thus n(x) n n(y) = 
{x} n (y} = ~ since x ~ y. 
Therefore, with respect to synonymy, words have a neighborhood 
topology since 
(1) (Vx)CZn(x))\[x • n(x)\] 
(Vy)(~n(y))\[y e n(x) ~ n(y) ~ n(x)\] 
.(Vx)(Vy)(~n(x))(~n(y))\[x ~ y ~ n(x) N n(y) = ~\] 
3.3. Closure Topology 
The second model considers a closure topology, i.e., a topology based 
on a closure operation. A set is said to have a closure topology 
if there exists a unary operation on its subsets, denoted by~ and 
called the closure, which satisfies the following axiums: 
Axiom 2: E c_ E 
Axiom 3: E c E 
~i~ ~: .~'O-'f =~'u ~" 
Define the closure of a set E of words as the synonym class of E, i.e., 
The closure axiums can be shown to be satisfied by using the original 
definition of synonym class 
sCE) z {x : (~y)\[yeE^xsy\]} 
However, shorter proofs are possible by noting that the synonym 
class of a set E of words can be expressed as 
s(E) = y e E s(y) = E (X : xSy} 
First, closure Axicm 1 i~ satisfied: 
Theorem: s(g) = 
Proof: s(~) = sCy) = ¢ 
Second, closure Axium 2 is satisfied: 
Theorem: E =- s(E) 
 oof. = ..Uo ryj 
I 
= E since y • s(y) -~> 
Third, closure Axi~n 3 is satisfied: 
Theorem: s\[s(E)\] c s(E) 
~oof: N~s(s(y))=sCtu:u~1)=tv:v~y\]~ {v:v~\] = 
s(y) since ~ c_ S. Thus sis(E)\] = U s(x) 
x ~ sCE) 
U x U(.(x) - U.(,) yeE y) y~E yeE 
Fourth, closure Axiom ~ is satisfied: 
The=am: sCE u F) = sCE) u sCF) 
~oof: sCE u F) = ~J s(y) y~-E 
UF 
s(E) U S(F). 
i a 
= l~J sCx) : 
x e U s(y) 
~E 
= sCE) 
-- U s(y) u s(y) = 
yeE yEF 
Therefore, with respect to synonyay, words have a closure topology 
since 
(1) s(¢) = 
C2) E ~- sCE) 
(B) s\[sCE}l ~ s(E) (~) 
sCE U F) = sCE) U sCF) 
Note that fram Axioms 2 and 3 we get 
Theorem: s\[sCE)\] = sCE) 
3.~. Ca~nents on Topological Characterizations 
Note that for the neighborhood topology a separation sxicm has been 
added to the t~ree axioms proposed in Edmundson C1~5). Also, the 
neighborhood topology seems more intuitively satisfying than the 
closure topology. However, for the closure topology if we define the 
derived set of a set E of words as the set of all words that 
~are synonymous to some word of E, but not identical to that 
Worde i.e.. 
then we have the followi~ result: 
Theorem: s(E) = E U g' 
which may be given a reasoQahle linguistic interpretation. An 
example is {y}' = s(y) - {y} which was discussed in the sectio~ on 
algebraic characterization. 
4. Conc~sions 
These results support the belief that the algebraic characterization 
is insightful and appropriate. For example, the assumption that 
synonymy is an equivalence relation also has been made, either 
directly or indirectly, by F. Kiefer and S. Abraham (1965), 
U. Weinreich (1966), and others. Since the axiom system defines the 
notions of synonymy and anton~ Jointly and implicitly, it avoids 
certain difficulties that are encountered when attempts are made to 
define these notions separately and explicitly. 
iO 
These topological characterizations provide a no,metric represen- 
tation of what has been called informally a "semantic space". 
Previous attempts to construct a semantic space that is metric 
(i.e., one for which a distance function is defined) have not met 
with much success. The consideration of general topological spaces 
avoids this difficulty. 

References 

R. Carnap, Introduction to Symbolic Logic and Its Applications, 
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H. P. Edmundson, "Mathematical Models of Synonymy", International 
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"Same P. Kiefer and S. Abraham, Problems of Formalization in 
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V. V. Martynov, P~tannJa prikladnoji lingvistyky; tezisy 
dopovideJ mi~vuzovs'koji naukovoji konferenciJi, 
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