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<Paper uid="P84-1064">
  <Title>A COMPUTATIONAL THEORY OF DISPOSITIONS</Title>
  <Section position="3" start_page="312" end_page="313" type="metho">
    <SectionTitle>
EDF A WHITE \[Sample;p\] + USUALLY\[Proportion;p\],
</SectionTitle>
    <Paragraph position="0"> in which + should be read as and. The ith row in WHITE is a tuple (Si,ri), i = 1,...,m, in which S i is the ith sample of snow, and ri is is the degree to which the color of S i matches white. Thus, r i may be interpreted as the test score for the constraint on the color of Si induced by the elastic constraint WHITE. Similarly, the relation USUALLY may be interpreted as an elastic constraint on the variable Proportion, with p representing the test score associated with a numerical value of Proportion.</Paragraph>
    <Paragraph position="1"> The steps in the procedure which represents the meaning of p may be described as follows:  1. Find the proportion of samples whose color is white:</Paragraph>
    <Paragraph position="3"> in which the proportion is expressed as the arithmetic average of the test scores.</Paragraph>
    <Paragraph position="4"> 2. Compute the degree to which C/ satisfies the constraint induced by USUALL Y: r ~ ~ USUALLY\[Proportion ~ p\] , in which r is the overall test score, i.e., the degree of compatibility of p with ED, and the notation ~R\[X = a\] means: Set the variable X in the relation R equal to a and read the value of the variable p.</Paragraph>
    <Paragraph position="5"> More generally, to represent the meaning of a disposition it is necessary to define the cardinality of a fuzzy set. Specifically, if A is a subset of a finite universe of discourse U ---- {ul,...,u,}, then the sigma-count of A is defined as ~Count(A ) = I:~pA(U~), (2.1) in which pA(Ui), i ---- l,...,n, is the grade of membership of u/ in A (Zadeh, 1983a), and it is understood that the sum may be rounded, if need be, to the nearest integer. Furthermore, one may stipulate that the terms whose grade of membership falls below a specified threshold be excluded from the summation. The purpose of such an exclusion is to avoid a situation in which a large number of terms with low grades of membership become count-equivalent to a small number of terms with high membership.</Paragraph>
    <Paragraph position="6"> The relative sigma-count, denoted by ~ Count( B / A ), may be interpreted as the proportion of elements of B in A. More explicitly,</Paragraph>
    <Paragraph position="8"> where B D A, the intersection of B and A, is defined by  itBnA(U)fUS/U) ^ US(U), U e U , where A denotes the sin operator in infix form. Thus, in terms of the membership functions of B and A, the relative slgma-count of B and A is given by ~,#B(u,) A tin(u,) Z Count( B / A } = (2.3} ~,tJa(u,) As an illustration, consider the disposition d A overating causes obesity (2.4) which after restoration is assumed to read 2 p A most of those who overeat are obese . (2.5) To represent the meaning of p, we shall employ an explanatory database whose constituent relations are:</Paragraph>
    <Paragraph position="10"> The relation POPULA TION is a list of names of individuals, with the variables Overeat and Obese representing, respectively, the degrees to which Name overeats and is obese. In MOST, p is the degree to which a numerical value of Proportion fits the intended meaning of MOST.</Paragraph>
    <Paragraph position="11"> To test procedure which represents the meaning of p involves the following steps.</Paragraph>
    <Paragraph position="12">  1. Let Name~, i -- 1 ..... m, be the name of ith individual in POPULATION. For each Name, find the degrees to which Namei overeats and is obese: ai A POVEREA r(Namei) A 0 ..... t POPULA T/ON(Name = Namei\] #, A ItonEsE( Namei} ~ o6,, POPULA TlON\[Name ~ Namei\] . 2. Compute the relative sigma-count of OBESE in OVEREAT: =iai A #i p @ ~Count(OBESE/OVEREAT)= E,ai 3. Compute the test score for the constraint induced by MOST: r-~ ~MOST\[Proportion --~ p\] .</Paragraph>
    <Paragraph position="13"> This test score represents the compatibility of p with the explanatory database.</Paragraph>
    <Paragraph position="14"> 3. The Scope of a Fuzzy Quantifier  In dealing with the conventional quantifiers all and some in flint-order logic, the scope of a quantifier plays an essential role in defining its meaning. In the case of a fuzzy quantifier which is characterized by a relative sigma-count, what matters is the identity of the sets which enter into the relative count. Thus, if the sigma-count is of the form ECount(B/A ), which should be read as the proportion of BIs in A Is, then B and A will be referred to as the n-set \[with n standing for numerator) and b-set (with b standing for base), respectively. The ordered pair {n-set, b-set}, then, may be viewed a~ a generalization of the concept of the scope of a quantifier. Note, however, that, in this sense, the scope of a fuzzy quantifier is a semantic rather than syntactic concept.</Paragraph>
    <Paragraph position="15"> As a simple illustration, consider the proposition  p A most students are undergraduates. In this case, the n-set of most is undergraduates, the b-set is students, and the scope of most is the pair { undergraduates, students}. 2. It should be understood that (2.5) is just one of many possible in null terpret~.tions of (2.4), with no implicat;on that is constitutes a prescriptive interpretation of causality. See Suppes (1970}. As an additional illustration of the interaction between scope and meaning, consider the disposition d A young men like young women . (3.1) Among the possible interpretations of this disposition, we shall focus our attention on the following (the symbol rd denotes a restoration of a disposition): rd I A most young men like most young women rd 2 A most young men like mostly young women .</Paragraph>
    <Paragraph position="16"> To place in evidence the difference between rd I and rdz, it is expedient to express them in the form rdl -~- most young men PI</Paragraph>
    <Paragraph position="18"> in which Name is the name of a male person and # is the degree to which the person in question satisfies the predicate.</Paragraph>
    <Paragraph position="19"> \[Equivalently, p is the grade of membership of the person in the fuzzy set which represents the denotation or, equivalently, the extension of the predicate.) To represent the meaning of PI and P2 through the use of test-score semantics, we assume that the explanatory data-base consists of the following relations (gadeh, 1983b):</Paragraph>
    <Paragraph position="21"> In LIKE, it is the degree to which Namel likes Name9 ; and in YOUNG, it is the degree to which a person whose age is Age is young.</Paragraph>
    <Paragraph position="22"> First, we shall represent the meaning of PI by the follow- null ing test procedure.</Paragraph>
    <Paragraph position="23"> 1. Divide POPULATION into the population of males, M.POPULATION, and the population of females,</Paragraph>
  </Section>
  <Section position="4" start_page="313" end_page="315" type="metho">
    <SectionTitle>
F.POPULA TION:
M.POPULA TION A N .... Ag, POPULA TION\[Sez---Male\]
F.POPULA TON A Ne,,,,age POPULA TION\[Sez---Female\] ,
</SectionTitle>
    <Paragraph position="0"> where N~mc,AocPOPULATION denotes the projection of POPULATION on the attributes Name and Age.</Paragraph>
    <Paragraph position="1">  2. For each Name:,j ~ 1 ..... L, in F.POPULATION, find the age of Namei: Ai A Age F.POPULA TION\[Name~Namei\] . 3. For each Namei, find the degree to which Name i is young: ai A ~YOUNG\[Age=Ai \] ,  where a i may be interpreted as the grade of  membership of Name i in the fuzzy set, YW, of young women.</Paragraph>
    <Paragraph position="2"> 4. For each Namei, i=l,...,K, in M.POPULATION, find the age of Namei: Bi A Age M.POPULA TlON\[Name---Namei\] . 5. For each Namei, find the degree to which Namei likes Name i : ~ii ~- ~LIKE\[Namel = Namel;Name2 = Namei\] , with the understanding that ~i/ may be interpreted as the grade of membership of Name i in the fuzzy set, WLi, of women whom Name, likes. 6. For each Name/ find the degree to which Name, likes Name i and Name i is young: &amp;quot;Tii A ai A #ii * Note: As in previous examples, we employ the aggregation operator rain (A) to represent the meaning of conjunction. In effect, 70 is the grade of membership of Name i in the intersection of the fuzzy sets WLI and YW.</Paragraph>
    <Paragraph position="3"> 7. Compute the relative sigma-count of women whom</Paragraph>
    <Paragraph position="5"> F. i a i  8. Compute the test score for the constraint induced by MOST:</Paragraph>
    <Paragraph position="7"> This test-score way be interpreted as the degree to which Name i satisfies PI, i.e., ri = p PI \[Name = Namei\] The test procedure described above represents the meaning of P,. In effect, it tests the constraint  expressed by the proposition E Count ( Y W/WL i ) is MOST and implies that the n-set and the b-set for the quantifier most in PI are given by:</Paragraph>
    <Paragraph position="9"> which implies that the constraint which defines P2 is  Thus, whereas the scope of the quantifier most in PI is {WLi, YW}, the scope of mostly in P2 is { YW, WL~}. Having represented the meaning of P1 and P~, it becomes a simple matter to represent the meaning of rd, and rd~. Taking rd D for example, we have to add the following steps to the test procedure which</Paragraph>
    <Section position="1" start_page="314" end_page="315" type="sub_section">
      <SectionTitle>
defines Pr
</SectionTitle>
      <Paragraph position="0"> For each Namei, find the degree to which Name i is young:</Paragraph>
      <Paragraph position="2"> where /f i may be interpreted as the grade of membership of Name i in the fuzzy set, YM, of young men.</Paragraph>
      <Paragraph position="3"> Compute the relative sigma-count of men who have property P* among young men:</Paragraph>
      <Paragraph position="5"> Test the constraint induced by MOST:</Paragraph>
      <Paragraph position="7"> The test score expressed by (3.6) represents the overall test score for the disposition d A young men like young women if d is interpreted as rd 1. If d is interpreted as rd2, which is a more likely interpretation, then the procedure is unchanged except that r i in (3.5) should he  The approach described in the preceding sections can be applied not only to the representation of the meaning of dispositions and dispositional predicates, but, more generally, to various types of semantic entities as well as dispositional concepts. null As an illustration of its application to the representation of the meaning of dispositional commands, consider dc A stay away from bald men , (4.1) whose explicit representation will be assumed to be the comm and c A stay away from most bald men . (4.2) The meaning of c is defined by its compliance criterion (gadeh, 1982) or, equivalently, its propositional content (Searle, 1979), which may be expressed as ee A staying away from most bald men .</Paragraph>
      <Paragraph position="8"> To represent the meaning of ce through the use of test-score semantics, we shall employ the explanatory database</Paragraph>
      <Paragraph position="10"> The relation RECORD may be interpreted as a diary -kept during the period of interest -- in which Name is the name of a man; pBald is the degree to which he is bald; and Action describes whether the man in question was stayed away from (Action~l) or not (Action=0).</Paragraph>
      <Paragraph position="11"> The test procedure which defines the meaning of dc may be described as follows:  1. For each Name i, i~I ..... n, find (a) the degree to which Namel is bald; and (b) the action taken: #Baldi A ,B~IdRECORD\[Name --. Namei\] Action i A a~tionRECORO\[Nam e --. Namei\] .</Paragraph>
      <Paragraph position="12"> 2. Compute the relative sigma-count of compliance: 1 \[~i pBaldl A Actidegni}&amp;quot; (4.3) p=--# 3. Test the constraint induced by MOST:</Paragraph>
      <Paragraph position="14"> The computed test score expressed by (4.4) represents the degree of compliance with c, while the procedure which leads to r represents the meaning of de.</Paragraph>
      <Paragraph position="15"> The concept of dispositionality applies not only to semantic entities such as propositions, predicates, commands, etc., but, more generally, to concepts and their definitions. As an illustration, we shall consider the concept of typicality -- a concept which plays a basic role in human reasoning, especially in default reasoning '(Reiter, 1983), concept formation (Smith and Media, 1981), and pattern recognition (Zadeh, 1977}.</Paragraph>
      <Paragraph position="16"> Let U be a universe of discourse and let A be a fuzzy set in A (e.g., U A cars and A ~ station wagons). The definition of a typical element of A may be expressed in verbal terms as follows:  t is a typical element of A if and only if (4.5) (a) t has a high grade of membership in A, and (b) most dements of ,4 are similar to t.</Paragraph>
      <Paragraph position="17">  it should be remarked that this definition should be viewed as a dispositional definition, that is, as a definition which may fail, in some cases, to reflect our intuitive perception of the meaning of typicality.</Paragraph>
      <Paragraph position="18"> To put the verbal definition expressed by (4.5) into a more precise form, we can employ test-score semantics to represent the meaning of (a) and (h). Specifically, let S be a similarity relation defined on U which associates wi~h each element u in U the degree to which u is similar to t ~. Furthermore, let S(t) be the Mmilarity clas~ of t, i.e., the fuzzy set of elements of U which are similar to t. ~Vhat this means is that the grade of membership of u in S(t) is equal to #s(t,u), the degree to which u is similar to t (Zadeh, 1971).</Paragraph>
      <Paragraph position="19"> Let HIGH denote the fuzzy subset of the unit interval which is the extension of the fuzzy predicate high. Then, the verbal definition (4.5) may be expressed more precisely in the form:  t is a typical element of A if and only if (4.6) 3. For consistency with the definition of A, S must be such that if u and u I have a high degree of similarity, then their grades of membership in A should be close in magnitude.</Paragraph>
      <Paragraph position="20"> (a) Pa(t) is HIGH (b) ECount(S(t)/A ) is MOST.</Paragraph>
      <Paragraph position="21">  The fuzzy predicate high may be characterized by its membership function PHtCH or, equivalently, as the fuzzy reinton IIIGfI \[Grade; PL in which Grade is a number in the interval \[0,1\] and p. is the degree to which the value of Grade fits the intended meaning of high.</Paragraph>
      <Paragraph position="22"> An important implication of this definition is that typicality is a matter of degree. Thus, it follows at once from (4.6) that the degree, r, to which t is typical or, equivalently, the grade of membership of t in the fuzzy set of typical elements of A, is given by</Paragraph>
      <Paragraph position="24"> In terms of the membe~hip functions of HIGH, MOST,S and A, (4.7} may be written as \[ ~, Pstt, u) A PA( u) I r A V.-LF. J' (4.8) where tHIGH, PMOSr, PS and PA are the membership functions of HIGH, MOST, S and A, respectively, and the summation Zu extends over the elements of U.</Paragraph>
      <Paragraph position="25"> It is of interest to observe that if pa(t) ----- 1 and .s(t,n) = ~a(u), (4.9) that is, the grade of membership of u in A is equal to the degree of similarity of u to t, then the degree of typicality of t is unity. This is reminiscent of definitions of prototypicality (Rosch, 1978) in which the grade of membership of an object in a category is assumed to be inversely related to its &amp;quot;distance&amp;quot; from the prototype.</Paragraph>
      <Paragraph position="26"> In a definition of prototypicality which we gave in gadeh (1982), a prototype is interpreted as a so-called a-summary. In relation to the definition of typicality expressed by (4.5), we may say that a prototype is a a -summary of typical elements of A. In this sense, a prototype is not, in general, an element of U whereas a typical element of A is, by definition, an clement of U. As a simple illustration of this difference, assume that U is a collection of movies, and A is the fuzzy set of Western movies. A prototype of A is a summary of the summaries {i.e., plots) of Western movies, and thus is not a movie. A typical Western movie, on the other hand, is a movie and thus is an element of U.</Paragraph>
    </Section>
  </Section>
  <Section position="5" start_page="315" end_page="316" type="metho">
    <SectionTitle>
5. Fuzzy Syllogisms
</SectionTitle>
    <Paragraph position="0"> A concept which plays an essential role in reasoning with dispositions is that of a fuzzy syllogism (Zadeh, 1983c). As a general inference schema, a fuzzy syllogism may be expressed in the form QIA'a are Bin (5.1) Q2 CI8 are DIs fQs E' a are F~ a where Ql and Q2 are given fuzzy quantifiers, Q3 is fuzzy quantifier which is to be determined, and A, /3, C, D, E and F are interrelated fuzzy predicates.</Paragraph>
    <Paragraph position="1"> In what follows, we shall present a brief discussion of two basic types of fuzzy syllogisms. A more detailed description of these and other fuzzy syllogisms may be found in Zadeh (1983c, 1984).</Paragraph>
    <Paragraph position="2"> The intersection~product syllogism may be viewed as an instance of (5.1) in which  fuzzy arithmetic. Thus, we have as the statement of the syllogism: null Q1A's are B' s (5.2) QT(A and B)' s arc CI s</Paragraph>
    <Paragraph position="4"> In particular, if B is contained in A, i.e., PB --&lt; PA, where PA and P8 are the membership functions of A and B, respectively, then A and B = B, and (5.2) becomes  where (~ , ~ ~ and @ are the operations of addition, subtraction, rain and max in fuzzy arithmetic.</Paragraph>
    <Paragraph position="5"> As a simple illustration, consider the dispositions dl A students are young d 2 ~-- students are single.</Paragraph>
    <Paragraph position="6"> Upon restoration, these dispositions become the propositions Pl A most students are young P2 A most students are single Then, applying the consequent conjunction syllogism to Pl and P2, we can infer that Q students are single and young where 2 most 01 &lt;_ Q &lt;_ most . (5.8) Thus, from the dispositions in question we can infer the disposition null d A students are ,ingle and young on the understanding that the implicit fuzzy quantifier in d is expressed by (5.8).</Paragraph>
  </Section>
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