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<Paper uid="C67-1025">
  <Title>1967 International Conference on Computational Linguistics Axiomatic Characterization of Synonymy and Antonymy</Title>
  <Section position="1" start_page="0" end_page="1" type="metho">
    <SectionTitle>
i. Introduction
</SectionTitle>
    <Paragraph position="0"> 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, dictionaries of synonyms and antonyms have varied according to the particular explicit definitions of &amp;quot;synonym&amp;quot; and &amp;quot;antonym&amp;quot; that were used. The roles of part-of-speech, context of a word, and substitutability in the same context were discussed.</Paragraph>
    <Paragraph position="1"> 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 representation. 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 entonymY= 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 relations and that this relation algebra could be used to produce shorter 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&amp;quot; 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 topological model required a closure operation for each of the intensions or senses and had no linguistically interesting interpretation.</Paragraph>
    <Section position="1" start_page="1" end_page="1" type="sub_section">
      <SectionTitle>
1.2 Summary
</SectionTitle>
      <Paragraph position="0"> 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 explicit 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 satisfies the four closure axioms.</Paragraph>
    </Section>
  </Section>
  <Section position="2" start_page="1" end_page="7" type="metho">
    <SectionTitle>
2. Algebraic Characterization
2.1. Introduction - Relations
</SectionTitle>
    <Paragraph position="0"> Before investigating antonymy and synonymy, we will estsblish some notions and notations for the calculus of binary relations.</Paragraph>
    <Paragraph position="1"> Consider a set V of arbitrary elmnen~s, which will be called the universal set. A binary relation on V is defined as a set R of ordered pairs &lt;x,p, where x,y s V. The proposition that x stands in relation 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</Paragraph>
    <Paragraph position="3"> 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 ==&gt; 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 ==&gt; R &amp;quot;I c_ S &amp;quot;I m -- Theorem: R c S ~&gt; S c R Theorem: (R'I) &amp;quot;I = R Theorem: (RIS)IT : RI(SIT ) Theorem: (RIS) &amp;quot;I = S'IIR &amp;quot;I Theorem: IIR = RII = R Theorem: s -r =&gt; RIs=RIT ^ SIR=TIR</Paragraph>
    <Section position="1" start_page="1" end_page="3" type="sub_section">
      <SectionTitle>
2.2 Axioms and Definitions
</SectionTitle>
      <Paragraph position="0"> Under the assumption that synonymy and antonymy are ternary relations on the set C of all content words, the following definitions will be used:</Paragraph>
      <Paragraph position="2"> 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).</Paragraph>
      <Paragraph position="3"> Axiom 1 (Reflexive) : (Vx)\[xSix\] Axium 2 (Symmetric): (Vx)(Vy)\[xSiY =&gt; xS;Iy\] Axium 3 (Transitive): (Vx)(Vy)(Vz)\[xSiY A YSiZ :&gt; Axi~n 4 (Irreflexive) : (Vx) \[x~ix\] Axiun 5 (Symmetric): (Vx)(Vy)\[xAiY =&gt; xA;ly\] 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) \[xAiY\]</Paragraph>
      <Paragraph position="5"> The properties named in Axiams 6 and 7 were coined for this study.</Paragraph>
      <Paragraph position="6"> 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</Paragraph>
      <Paragraph position="8"> This relation algebra will be used to produce shorter proofs, although 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.</Paragraph>
      <Paragraph position="9"> 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</Paragraph>
      <Paragraph position="11"> For ressons of notational simplicity, the subscript denoting the intenslon i will be omitted in the sequel whenever possible. However, the theorems must be understood as if the subscript were present.</Paragraph>
      <Paragraph position="12"> As with any symmetric relation, it is possible to get stronger results than Axi~ 2 and Axiom 5.</Paragraph>
      <Paragraph position="13"> Theorem: S &amp;quot;1 = S Proof: 1 S c S-1 by Axiom 2. Hence S &amp;quot;1 c_ (S-1)-I = S. There-</Paragraph>
      <Paragraph position="15"> Proof: Same as above theorem using Axi~n 5.</Paragraph>
      <Paragraph position="16"> Also we get a stringer result than the transitivity property of  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):</Paragraph>
      <Paragraph position="18"> 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).</Paragraph>
      <Paragraph position="19"> Also a(y) g a(x) by an identical argument. Therefore a(x) = a(y).</Paragraph>
    </Section>
    <Section position="2" start_page="3" end_page="7" type="sub_section">
      <SectionTitle>
2.4 Comments on the Algebraic Characterization
</SectionTitle>
      <Paragraph position="0"> 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\] ~C/ 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).</Paragraph>
      <Paragraph position="2"> can he summarized in the following multiplication table for products of the relations S and A  (1) A 2 = S , (2) A S =A , (3) A\[S = A that 41) and 42) PSmp~ 43), (i) and 43) ~P~v 42), but (Z) and (S) do not i~ (1).</Paragraph>
      <Paragraph position="3"> Suppose that for every pair &lt;x,y&gt; 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)\[x,~ e v-----&gt;xs~Vx~vvx~\] which is an exclusive disjunction. Thus the vocabulary V is partitioned as follows:</Paragraph>
      <Paragraph position="5"> This also can be pictured in the lattice of for every word y.</Paragraph>
      <Paragraph position="6"> relations U=V~V It can be shown that the multiplication table for products of the relaticms S,A, and M is</Paragraph>
    </Section>
  </Section>
  <Section position="3" start_page="7" end_page="7" type="metho">
    <SectionTitle>
3. Topological Characterizations
3.1. Introduction
</SectionTitle>
    <Paragraph position="0"> We will now examine two topological models of synonymy. Being topological, they concern &amp;quot;semantic spaces&amp;quot; of words without any notion of &amp;quot;semantic distance&amp;quot; between two words. Again, we will restrict our attention to content words. Topological models for the antonymy relation will not be considered.</Paragraph>
    <Paragraph position="1"> 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~,)\[y ~ s x ~&gt; ~y ~ ~1 Axiom ~: (Vx)(Vy)(~x) (aNy) Ix ~ y ------&gt; Nx n Ny -- ~\]  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.</Paragraph>
    <Paragraph position="2"> First, neighborhood Axiom 1 is satisfied.</Paragraph>
    <Paragraph position="3">  implies s(x) = s(y) since y e n(x) c_ s(x) = {z : zSx\] implies ySx and ySx implies s(y) = s(x)deg 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))\[y ~ n(x) ~ n(y) ~- nCx)\].</Paragraph>
    <Paragraph position="4"> In fact, the neighborhood topology satisfies Axiom 4, which is a separation axiom: TheorT: (Vx)(Yy)(~n(x))(~n(y))\[x ~ y =&gt; n(x) n n(y) = ~\] Proof. Assume x ~ y. Let nCx) = (x} and n(y) = {y}.</Paragraph>
    <Paragraph position="5"> 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.</Paragraph>
    <Paragraph position="6"> Therefore, with respect to synonymy, words have a neighborhood  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 ~&amp;quot;  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:  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..</Paragraph>
    <Paragraph position="7"> 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.</Paragraph>
  </Section>
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