Quantified propositions in a linguistic logic

Int. J. Man-Machine Studies (1983) 19, 195-227

Quantified propositions in a linguistic Iogict RONALD R. YAGER

Machine Intelligence Institute, Iona College, New Rochelle, New York 10801, U.S.A. (Received 20 January 1981, and in revised form 14 October 1982) We introduce two methodologies for interpreting quantifiers in binary logic. We then extend these interpretations to the case where the quantifiers arc linguistic. We use the formalism of fuzzy subset theory to provide a framework in which to interpret linguistic quantifiers. We discuss various methodologies for measuring the cardinality of a fuzzy set including the concept of a fuzzy cardinality. Among the important questions we study in the paper is the problem of making quantified statements based upon the observation of a sample from a set of objects.

Introduction In e v e r y d a y language, quantifiers serve the p u r p o s e of enabling us to m a k e s u m m a r y type statements a b o u t properties of a class of objects. T o build a logical system capable of representing h u m a n discourse we must include the ability to interpret quantified statements in a c o h e r e n t m a n n e r . O u r p u r p o s e here is to investigate and extend the p r o c e d u r e s used in binary logic to represent quantified propositions to, a m o r e natural and useful, linguistically based logic.

Quantifiers in logic A s s u m e D is s o m e class of objects or elements generally called the universe of discourse. A term or predicate is an expression which results in a description or an assertion a b o u t any object when the variable in the predicate is replaced by some d ~ D. F u r t h e r m o r e , it is assumed that the truth of the assertion can be meaningfully d e t e r m i n e d for each d ~ D. A s s u m e Px is s o m e predicate, the x implies the variable, then by Pxl for xi ~ D, we m e a n the predicate evaluated at x --xi. W e shall d e n o t e the truth value of the predicate at x = x i a s ]Pxil.:~ F o r example, if D is a set of students and Px the term or predicate, "x is a f e m a l e " , then for each xl e D, Pxi b e c o m e s a tangible assertion whose truth value can be meaningfully determined. It is i m p o r t a n t to note that it is only w h e n the variable associated with the predicate takes on some tangible value f r o m D that we can d e t e r m i n e the truth of an assertion. Quantified statements provide a c o n v e n i e n t m e t h o d of summarizing the information a b o u t the elements in a universe of discourse D with respect to a predicate P. F o r example, the statements "all students are f e m a l e s " or " t h e r e exist s o m e female s t u d e n t s " are quantified statements a b o u t o u r predicate female which summarizes the information we k n o w a b o u t the elements in D. In the a b o v e statements "all" and " t h e r e exist s o m e " are called quantifiers. -t This paper is dedicated to L. A. Zadeh whose intuition in the attempt to make precise the imprecise never ceases to amaze and inspire. ~:If the context is clear we may also use P(xt) or Pxi to indicate the truth value of P evaluated at xl. 195 0020-7373/83/080195 + 33503.00/0

9 1983 Academic Press Inc. (London) Limited

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T h e r e appears to be at least two interesting practical questions with respect to quantifiers and quantified statements. T h e first of these pertains to the situation in which the value IPxi[ is known for each xt e D and we ask what kinds of quantified statements can be m a d e as summaries of the situation. M o r e specifically, this involves the question of assigning a truth value to a quantified statement based upon the known truth values of the elements in D. T h e second question pertains to the situation in which we have some asserted quantified statement as being true and we are concerned about the possible values which the individual IPxils can assume to be consistent with the known quantified statement. For example, if I know that all students are females, I can easily deduce the fact that the only possible assignment of [Pxl is such is IPxl true for all x E D. W e see that the first problem is one of induction in the sense that we are trying to m a k e a m o r e general statement from some elementary facts we are trying to summarize. T h e second situation is one of deduction in that we are trying to deduce possible worlds based upon a general fact. In order to investigate coherently these two types of p r o b l e m we must provide a f r a m e w o r k in which to interpret and process the meaning of quantified statements in terms of the information available a b o u t the truths of the individual elements of D. One c o m m o n l y used a p p r o a c h of accomplishing this is called the substitution approach (Suppes, 1957). In this approach, we try to represent a quantified statement by an equivalent logical sentence involving atoms which are instances of the predicate evaluated at the elements in D. W e shall briefly describe this methodology. Assume that D is a finite set of n elements and P is some predicate which has a ~ruth value IPx, I for each x~ ~ D. F u r t h e r m o r e , we shall assume that IPx,I ~ {0, 1}, where 0 is false and 1 is true. T h e universal quantifier will be denoted (Vx)Px, is read as "for all x e D the predicate P is true". In terms of the c o m m o n l y used language quantified statements of this type take m a n y forms: for example, "all Americans are peaceful", "Chinese like rice" or "all even numbers are divisible by 2". Using the substitution approach to defining quantified statements the usual definition of the universal quantifier is (Vx)Px = Pxl and Px2 and P x 3 . . . and Pxn, and hence the truth of the statement (Vx)Px, denoted [(Vx)Pxl is IPxl ^ Px2 A Px3 ^ ' ' " ^ Px,[ which is seen f r o m ordinary binary logic to be Minx,~D IPxil. Thus, given the value of IPx~[ for all x ~ D , we can use the above definition to determine the truth of the statement (Vx)Px and hence validate its truthfulness as a summarizer of the known information about the elements of D with respect to P. On the other hand, if we know I(Vx)Pxl = 1, then the only possible set of truth values for the elements [Px, lare IPx,I -- a for all xl ~ D. T h a t is, any association of truth values with the elements of D having at least one IPx,I = 0 is impossible. The existential quantifier will be denoted (3x)Px. It can be read as "for at least one x e D the predicate P is true". In terms of c o m m o n l y used language existentially quantified statements take such forms as "there exists a positive n u m b e r " or " I have an acceptable alternative". The c o m m o n interpretation of the existential quantifier is (3x)Px = Pxx or Px2 or Pxa o r . . . or Px,, and hence the truth of the statement (3x)(Px) denoted ](::lx)(Px)] is IPxl v Px2 v .

9

9

V

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Pxn I, which from ordinary binary logic is MaXx~D IPxi I. Thus, given the values of IPxi] for all xi ~ D, we can determine the truth of the statement (3x)Px and hence validate its truthfulness as a summarizer of our knowledge. On the other hand, if we know 1(3x)Pxl = 1, then any association of truth values with the elements of the universe of discourse which has at least one element having a truth value of one is a possible deduction from this information. We can observe certain facts about these quantifiers: (1) I(Vx)Pxl = l ~ l ( 3 x ) p x l = 1; (2) 1(3x)(not P)l = 1 ~ I(Vx)Pxl -- 0; (3) if P and Q are two predicates, then I(u

^ Q)I = 1 ~ I(Vx)P] = 1 I(Vx)QI = 1.

An alternative framework in which to investigate quantified propositions is possible. We shall call this the algebraic approach. Again assuming D to be our universe of discourse, a finite set of size n, P our predicate and [Px~l~{0, 1} the truth of P for xg e D. Letting Q indicate a general quantifier we can interpret the quantified statement (Qx)Px by associating with Q (1) a subset So c R (reals), and (2) a function Fo(lPxl[, I P x = l , 9 9

9 Ir'x

f)

R such

that

(Qx)Px is true, I(Qx)Px I -- 1, if Fo c So. For example, the universal quantifier, Q = V, if we define So={n}

and

F o = ~ IPx~r, i=1

we will obtain the same results as using the substitution approach. For the existential quantifier, Q = 3 if we define So = {1, 2, 3 . . . . , n}

and

Fo= ~ IPxil, i=1

then we again get the same results as using the substitution approach. Having this alternative framework we can easily introduce more sophisticated quantifiers by appropriately defining So and Fo. For example, the quantified statement "a majority of D are P" can be expressed by letting Q = M (majority), then defining SM = {all integers in (n + 1)/2, n}, FM = ~ IVx, I. i=1

We note that we could provide an equivalent definition for the quantifier "a majority" using the substitution approach; however, this would involve a fairly complex logical sentence. Thus, in many cases, it appears that this new algebraic formulation gives us a easier way of defining quantifiers than does the substitution approach. We could also easily define the quantifier, "a vast majority of D are P" by appropriately selecting some nl, such that nl >> (n + 1)/2 and setting So = In1, n]. Note 1. In the terminology of fuzzy subset theory I(Qx)Px] = So(Fo), is the grade of

membership of Fo in So.

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Note 2. An alternative approach to these quantifiers could involve allowing So c [0, 1] and d e f n i n g FQ = ( l / n ) Y.I'=I IPxll in this case: (1) (2) (3) (4)

"for all" has So={1}, "there exists" has So = (0, 1] "a majority" has So = (0.5, 1] "a vast majority" has SQ = [ n l / n , 1], perhaps n l / n = 0.8.

Note 3. An equivalent methodology has been proposed by Mostowski (1957). In this methodology, we let C1 equal the n u m b e r of true elements in D and C2 equal the number of false elements in D. Then a quantifier is defined by assigning appropriate sets Scl and Scz and determining the truth of a quantified proposition by m e m b e r s h i p in Scl and Sc2. Let us again consider the substitution approach to defining quantified statements. As we noted we can also use this approach to represent the new quantifiers we introduced. For example, the quantifier, M, a majority can be represented in this form as-

(Mx)Px = U Bl, /eL

where Bt = Pxi 1 ^ Pxt2 ^" 9 9 A Px~, where {xt1, x~. . . . . . xt~} = l is a subset of D consisting of k elements, k > (n + 1)/2. L indicates the index set of all such subsets of D consisting of at least (n + 1)/2 elements. We note in this case !(Mx)Pxl = t~.JLB L

,

Without loss of generality, assume the xs are ordered with respect to their truth in P, that is i > ] if Pxj-> Pxg. Since [ ( M x ) P x [ = Maxt e L in~l and In~l = Minxj~l IPxj], it can be easily seen that

](Mx)Pxl

= =

Min

IPx, I = IPxr

if n is odd

1Vfin

IPx, l=lPxr

ifn iseven.

i=1,2,...,(n+1)/2

i=1,2,...,(n/2)+1

It should be noted that both the substitution and algebraic approach give the same result. Furthermore, if we are given J(Mx)Pxl = 1 than any allocation of truths for the indivudal elements which has at least a majority of elements as true is possible. Let us consider the quantifier " A t least n~ D are P". Using our algebraic approach we let

FQ-- ~ IPx~l i=1

and So={n~,n~+l,n~+2

. . . . . n}.

Thus given a particular allocation IPxxl, IPx21. . . . . IPx.I for the truths of the elements of D we can determine the truth of the assertion, "at least n~ D are P" by determining So(FQ), that is if FQ c So. Using the substitution approach the assertion becomes equivalent to I._JtCLB~, where B again is of the form Pxt, A Pxz~ ^ 9 9 9A Pxt~, where {xt,, xz. . . . . . x~} = l is a subset of

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D having k -> na elements and L is the index over all such sets. Again, if the elements are ordered in descending truths then we can easily show:

IU THEOREM. Assuming D = {xi,

IPx,I = IPx,,,I. X2,..

9 , Xn}

is a set of elements such that

i > j ~ Pxj ---Pxi,

then using the substitution interpretation of quantifiers where "and" and " o r " are defined as Min and Max, respectively, the truth of the assertion "at least nl D are P" is [PXn, I. Proof. T h e interpretation of the assertion is I...JBL as described above. Since

U B, =Max IB,I and IB,I= 1~1. L Max

Inzl occurs

Min

xi~l

IPxJ,

for the case when l ={xl, x2 . . . . . x,,} and thus [UBt[=

Man

XC{XlsX2,.-,,Xnl}

[Px] = [Px~,[.

Note. While we are primarily at this point concerned with binary logic the above t h e o r e m holds in any multivalued logic where " a n d " and " o r " are defined by min and max. Note. In the case where our logic is binary then both approaches, substitution and algebraic, yield the same results. Consider the quantified statement "at most n l D approach we get

Fo=

i=l

are P". Using the algebraic

IPx, I

and So = {0, 1 . . . . . n 1}. Using the substitution approach the above statement becomes equivalent to the assertion "not at least n l + 1 elements of D are P". Considering (_.J~. B~ as before, however, in this case k = nl + 1, we get tcCJLgt = ]Px,,+l], as the truth of the statement "at least n l + 1 elements of D are P", however, in this case we want the truth of the statement "not at least n l + l elements of D are P" which is (1 - I P x . , + d ) . If we consider the quantifier "exactly n~ D are P", then the algebraic approach has Fo = ~ IPxil

and

So={nl}.

l=1

A formulation based upon the substitution approach can also be obtained. Without loss of generality assume the elements of D are indexed in order of decreasing truth,

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i > j ~ Pxi >-Pxi. Then the representation would be S = Pxl

A Px2

^ Px3

^

9 9 9 ^

Px.1 ^ Px.I+I

A

Pxnl+2

A 9 " " ^

Px,

then [SI = IPxll ^ IPx2l A Vx3[ A ' . . ^ [Px.II ^ IPx-~-~+~l^ " "

^ IP-~.l.

Since Pxi >-"Pxi for i > / , then IPx,I A ]Px2l ^ IPx.,] = I P x J

and

for i > i , then IPx.,, 11^ IPx--~,+21^ IPx.,+31 ^ .... ^

lP~x.I = IPx.~+xl;

furthermore, since ]Px-~+~[ = 1 -IPx.,~-al we get

N = IPx.,I ^ (1 -lex.l+lf). W e note that both formulations lead to the same results when our logic is binary. If we start with a given true quantified proposition (Qx)Px, where Q is defined and the D is known, we can use this to give us some information about the possible allocations of truth values to the individual elements in D. By an allocation we shall m e a n an assignment of truth values with respect to P to each of the elements in D. Thus an assignment or allocation is a mapping A : D--> {0, 1}, where A(xi) = IPxil. Thus, if we have a quantifier Q, we have the set So and a function Fo, if the quantifier is known to be true then if A is the underlying allocation of truths associated with P for the elements, then Fo(A) ~ So, where Fo(A) indicates the function evaluated for this allocation A. However, if we do not know A, but just know that the quantified proposition is true, then we can infer at the very m i n i m u m that any assignment A such that Fo(A) c S is a possibility for the allocation of truth values for the elements of D. Thus, we can associate with a given quantified proposition a set II of possible allocations for the elements of D based upon the known truth of the proposition. Note that II is a subset of the set of all possible allocations for the elements of D.

Example. A s s u m e D = {Xl, x2, x3} and we have the proposition " a majority of the Ds are P". T h e n any allocation of truths to the xs such that their sum is greater than or equal to 2 is possible. Consider the set {A1, A2 . . . . . As} of all allocations.

A1 A2 A3 A4 A5 A6 A7 As

Pxl

Px2

Px3

0 0 0 1 1 1 0 1

0 0 1 0 1 0 1 1

0 1 0 0 0 1 1 1

QUANTIFIED PROPOSITIONS OF A LINGUISTIC LOGIC

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Thus, of the above allocations, only As, A6, A7 and A8 are possible assignments of truths to the individual elements of D. As a result of the above exercise we can see that a quantified proposition can be used to extract information about the truth values of the elements of D with respect to P. We note that this information is not very specific in the sense that many allocations may be d e e m e d possible. It should be noted that if we have some additional information, such as IPxl] -> IPx21, we may further reduce the possiblities. It should also be noted that we can use the substitution approach in an analogous manner to obtain the possiblity allocations. A second form of information which may be extracted from the given quantified proposition is probabilistic type information. In the above example, if I ask what is the probability of selecting an element x ~ D, such that IPxl= 1, I can easily say that it is at least 2/3. If I ask, selecting two elements from D without replacement, what is the probability that at least one will have property P, the answer is obviously 1. More generally, if from a quantified proposition (Qx)Px, I obtain the set II of possible allocations and if ProbA(E) is the probability associated with a particular event E for a given A ~ II, then the set B, of probabilities of E for all elements in II give me the possible probabilities associated with this experiment based upon the knowledge of (Qx)Px. A third type of information available from a quantified proposition pertains to the validation of the truth of another quantified proposition. Two types of situations may be of interest. Assume Q1 and Q2 are two quantifiers with their associated sets $1 and $2 and their associated functions F1 and F2. Assume (Qlx)Px is true. If F1 =F2 then if $1 c $2, then (Qlx)Px implies (Q2x)Px is true. For example, knowing that a majority of students are P implies that at least one student is P. In the second type of situation assume we know that (Q~x)Plx is true and we are interested in determining whether (Q2x)P2x is true. Note that P1 and P2 are not the same. In some cases, when there exists some relationship between P~ and P2, we may be able to make quantified statements about P2 based upon the statement (Q~x)PlX. For example, if P2 = not PI, then IPEXI = 1 - I P I X t. Consider then the situation in which IPEXl = G(IP1x]). Assume (Qlx)Plx is true, then this supplies us with a set II of possible allocations of truth values for elements. For a given allocation A ~ H, there exists a corresponding allocation for P2, such that IPEXll = G(IPlxil), let us denote this as G(A). Let G(H) be the set of all possible allocations associated with P2 from those possible under (Qlx)(PlXl). If for all G(A) ~ G(II), FE(G(A)) ~ $2 then (Qlx)P1x implies (Q2x) (P2x) is true. On the other hand, if for all G ( A ) e G(II)F2(G(A))~ So2 then (Q2x)PEX is false.

Multiple quantified predicates Assume D and H are two classes of objects not necessarily distinct. A two valued predicate is a term which results in some verifiable statement for each pair (x, y) D x H. Assume D is a set of males and H is a set of females. An example of a two valued predicate could be "x loves y". In general, a two valued predicate indicates some relationship between x and y. Thus, if D = H = Reals, then we can say x -> y is a two valued predicate.

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If P(x, y) is a two valued predicate by P(xu yi) we shall mean the valuation of P at the values x = x~ and y = yj. W e shall use IP(xi, Yi)I to indicate the truth of the assertion implied by the predicate for these two values. Assume Q1 is a quantifier over the set D and Q2 is a quantifier over the set H. We can consider the quantified statement (Qlx)[(QEy)P(x, y)], for example, for all males there exists m a n y females he loves. We shall use our methodology to evaluate the truth of such a statement. First we have associated with Qx a set $1 and function FI: similarly for Q 2 w e have $2 and Fz. Consider an arbitrary element x i ~ D and the simply quantified statement (Q2y)P(xi, y). Note that with xi fixed P(x~, y) becomes a simple predicate in H. Using the methods developed, we can evaluate the truth of assertion (Q2y)P(x~, y) for each x~ e D. In this manner, we obtain a new simple predicate P' over set D such that IP'x~] = [(QEy)P(x~, Y)I. Now if we consider the simply quantified statement (Q~x)P'x, we can use our definition of Qx to obtain the truth of this quantified statement which is equivalent to I(QIx)((Q2y)P(x, y))l. More generally consider the class of n sets D1, D2 . . . . . D~, the quantifiers Q1, Q2, 99 9 Q , and the n valued predicate P. Consider the multiply quantified proposition (QID1)[(Q2D2)[(Q3D3)... [(Q,Dn)P]]]. Let U~ = D 1 x D 2 x D s x Un-I=DIXDzxD3x

999 x D . , ''

. x D , , ~,

61 =D1. We note that P is a function of Un. for each y e U~_I we can determine the truth of the statement (Qnx)(P(yi, x)), over the set D,. This results in a new predicate P' defined over U , 1. for each zi c U , - 2 , we can determine the truth of the assertion ( Q , - l x ) P ' ( z i , x) over the set D~_I. This results in a new predicate P" defined over Un-2. We can continue in this m a n n e r until we find the truth of the whole statement. QUANTIFIERS IN M U L T I V A L U E D LOGICS

W e have thus far restricted ourselves to the case where our truth values were drawn from the binary set {0, 1}. W e shall now consider the situation when the truth values associated with a predicate are drawn f r o m the set I = [0, 1]. We can associate the concept of a predicate verifiable over a universe of discourse (or base set) D with a fuzzy subset defined over D (Zadeh, 1965). If P is a predicate the fuzzy subset associated with this predicate, denoted P, is such that P ( x ) = IPx 1, where P(x) indicates the degree of m e m b e r s h i p of x in the fuzzy subset P and IPxl, as before, indicates the truth of the assertion that object x ~ D satisfied the predicate P. In the following we shall drop the bold face and simply use P to indicate the fuzzy subset associated with the predicate. Because of the above relationship we shall use the concepts of fuzzy subsets and predicate interchangeably.

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In extending the concepts associated with quantified propositions to multivalued logics certain problems arise. The most glaring is the divergence between the two methodologies for interpreting quantifiers previously discussed, i.e. substitution and algebraic. When using a binary logic the substitution and algebraic approach always lead to the same result whereas when the truth values are drawn from the set I these two approaches do not necessarily give the same results. In the following we shall develop both of these approaches to defining quantifiers when the logic is multivalued. It will be seen that an aspect related to the extension of quantifiers under multivalued logics will be based upon two differing definitions associated with the idea of the cardinality of a fuzzy subset. We recall that the cardinality of a crisp set B is simply the number of elements in B. Assume U is a set of n elements and F is a fuzzy subset of U. One possible way of defining the cardinality of F is to use the power of the fuzzy subset F (DeLuca & Termini, 1972), lEt

= ~ F(u,), i=l

where F(ui) is the membership grade of ui in F. A second approach to defining the cardinality of a fuzzy subset F is to use the fuzzy cardinality of R, denoted IFIr (Yager, 1979; Zadeh, 1979c). Let F , indicate the a-level set of F, that is F~ = {/zl/z ~U, F ( u ) - a } . Since F~ is a crisp subset of U we can unambiguously define the cardinality of the number of elements in F~. Using these level sets we can than define the fuzzy cardinality of the fuzzy set F as

F=, IF=l, as

It should be pointed out that IFIr is a fuzzy subset of {0, 1, 2 . . . . . n}. It is a fuzzy integer number.

Example.

Assume F=

{11 0.7 0.5 0.2 0.2} , , , , . U2

/23

124

Then IFI = 2 . 6

while o., o, {Fir=

Example.

' 2'

3'

Assume

F=Iu~,u~,u~}.

-

}

"

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Then

IFI-- 3 while IFlr = {]} = 3.

Thus, for crisp sets both definitions for cardinality lead to the same results. Recalling that quantifiers supply us with a means of summarizing the known information about the elements of some set with respect to a predicate P, we shall see that the introduction of multivalued logic affords us a means of representing very sophisticated natural language based quantifiers.

Linguistic quantifiers based upon power cardinality In a number of papers (Zadeh & Bellman, 1977; Zadeh, 1979a, b, c, 1981), Zadeh has developed a procedure for interpreting linguistic quantifiers in his theory of approximate reasoning. This approach is based upon an extension of the algebraic method, with the underlying truths drawn from the unit interval. The method uses the power of a fuzzy subset as a measure of cardinality. This method has the significant advantage of allowing, via the use of fuzzy subset theory, the introduction of linguistic quantifiers to help in the summarization of data. Thus, by using such linguistic quantifiers as few, many, most, approximately, some, several, in addition to the usual quantifiers of all and there exists, we are able to obtain a theory to represent better the imprecision and complexity of common language usage. The linguistic quantifiers are handled in this theory as fuzzy subsets, similarly to the handling of any linguistic value (Zadeh, 1965). In some cases the quantifiers are represented as fuzzy subsets of the unit interval, such as many, few, etc., and other cases we use fuzzy subsets of the sets R § (the non-negative reals). Assume O is a linguistic quantifier, with representation both linguistically and as a fuzzy subset. Let X be some set of objects, called the base set, and let F be a concept definable as a fuzzy subset of X, i.e. tall, sick, fat, rich, etc. A statement of the form " Q Xs are F " is called a quantified fuzzy proposition. The purpose of such a statement is to summarize the information known about the elements of X with respect to the concept F, for example, " m a n y Americans are tall" or " a b o u t half the students are passing". According to Zadeh (1977, 1979a, b, c, 1981), the information contained within or the effect of such a proposition is to supply a restriction on the definition of F as a fuzzy subset of X. In its general form Q Xs are F ~ R ( F / X , Q) = Q(r). In particular, if F is a particular manifestiation of the concept in terms of a specific fuzzy subset of X, if r is an appropriate means of evaluating the information in F, Q(r), the membership grade of Q at r is a measure of the truth of the proposition given F and Q. Furthermore, R ( F / X , Q), the restriction on possiblity of this F given Q, can be seen to be equal to Q(r).

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In this methodology, there are two possible ways of summing to obtain the information r contained in F. The choice depends upon the base set over which the linguistic quantifier is defined. Assume X is a finite set of n elements. Then (1) if the linguistic quantifier is represented as a fuzzy subset of [0, 1], then for a fuzzy subset F, r =-

1

E (Fxi)

ni=l

and the truth or consistency of this F with our proposition is R(F) = Q(r); (2) if the linguistic quantifier is represented as a fuzzy subset of R +, then for a fuzzy subset F r = ~ F(xl) t=1

and the truth or consistency of this F with our proposition is again R(F) = Q(r). As we noted in previous sections, there are two situations which are of interest when using quantifiers. Case I In the first case we are given a quantified proposition. For example, " m a n y alternatives are good", in which the linguistic variable many is defined as a known fuzzy subset of either I or R § The problem here is to find which definitions for the fuzzy subset F, good alternatives, are compatible with the given proposition. We shall refer to this situation as the deductive, or possibility extraction, problem. Thus, if F is a hypothesized definition of the concept good as a fuzzy subset of X, the set of alternatives, then R(F) would measure how consistent or truthful this definition of F is in light of the given proposition. Alternatively, the proposition can be seen to express a restriction on the possible definitions for good as a fuzzy subset of X. If ~: is the set of all fuzzy subsets of X, understood to be the meaning of good alternative, then our quantified proposition can be seen to induce a possibility distribution,

I-I:F-~ I, where for each F c H(F) = R(F) indicates the possiblity of F being the true meaning for good, Case 2 In this case we are given a data base in terms of a fuzzy subset F of X and are interested in summarizing the information about X with respect to X. This problem

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will be called the summarization, or inductive, problem. In this case given a prospective quantifier Q, to summarize the data, Q(r) would measure the truth of this summarization. Let us consider the first case, where Q is known and F is just specified as some concept and we are trying to deduce the possible definitions of F in terms of a fuzzy subset of the base set X. Let U be some other base set on which F has already been measured. For example, if we have the statement " m a n y Swedes are tall" let U be a set of heights on which "tall", F, has already been calibrated, F : U -->I. Assume G maps each element of X into an appropriate element in U, G:X-~U. For example, it would m a p each Swede into his height. For a given xi ~ X, we get G(xl) ~ U and then F(G(xl)) ~ I can be obtained from the known mapping F. Using the methodology developed for interpreting O X is F, we get r=-

F(G(xi))

or

r=

n i=l

F(G(x/)). i=l

If we set R ( G ) = Q(r), we can obtain a restriction on the rule G as being the rule mapping the Xs into the Us. Thus, the given quantified proposition induces a possibility distribution II over the set of all mappings G from X into U.

Example. Consider

a set X. Assume we have the statement " m a n y Xs are tall" where m a n y is defined by the fuzzy subset Q : I ~ I such that

Q(v)=vl/2eI. (a) Consider as a definition of F over X F = I0"2 0-6 0.7 0.___5~ t-'x1

X2

' X3 ' x , J

Then r=~ ~ F(xl)=0.5 i=1

and hence Q(0.5) = (0.5) x/2 = 0.7. Thus the consistency of this particular definition of F with our proposition " m a n y Xs are tall" is 0.7. (b) Assume we have defined "tall" as a fuzzy subset over a set of heights, in inches, U = [60, 84] such that Tall (u) = 0,

u --- 70,

Tall ( u ) = 112.5((u - 7 0 ) / 7 5 ) 2, Tall (u) = 1 - 112.5((u - 8 0 ) / 7 5 ) 2, Tall (u) = 1,

u>-80.

7 0 - > u ---75, 75 -< u -<80,

QUANTIFIED PROPOSITIONS OF A LINGUISTIC LOGIC

207

Assume X -- (xl, x2, x3, X4). Let G be expressed as G(xl) = 75,

G(x2) =

80,

G(x3) = 67,

G(x4) = 77.

Thus F(xl) = Tall (G(Xl)) = Tall (75) = 112.5((75 - 70)/75) 2 = 0.5, F(x2) = Tall (80) = l, F(x3) = Tall (67) = 0, F(x4) = Tall ( 7 7 ) = 1 - 1 1 2 . 5 ( ( 7 7 - 8 0 ) / 7 5 ) 2 = 1 - 112.5(3/75) 2= 1 - 0 . 1 8 = 0.82. Therefore r=

0.5+1+0+82 4

=

2.32 =0.58, 4

Q(r) = (0.58) 1/2 = 0.76. Thus II(G) = 0.76, expresses the possibility of this G. Note that in the above deviations associated with G no restriction was placed on the cardinality of U. Again assume we have quantified fuzzy proposition " Q X are F", where F is only known as a fuzzy subset of U. Further, assume that Q is a fuzzy subset of I and U is an interval of the real numbers. Let p (u) be defined such that p (u) du indicates the proportion of the Xs lying in the interval [u, u + d u ] c U. In this case the average m e m b e r s h i p grade of U, r, is r = fu p (u)F(u) du. Then we can associate with this particular manifestation, p, a consistency with our quantified fuzzy proposition such that Q(r) = R(p). M o r e generally, if P is the class of all such proportional functions of the Xs along U, then our proposition induces a possibility distribution H over P such that H:P-*I and for each p c P H ( p ) = R(p). Thus, II indicates the possible of each p being the distribution of the Xs along U based on our proposition. W e note that this is valid for X being finite or infinite. Recapitulating, f r o m a statement " Q Xs are F " we obtain Q(r). If F is known, then Q(r) determines the truth of the use of the quantifier Q. If Q is known then Q(r) determines the possibility of F.

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R.R. YAGER

FURTHER OPERATIONS WITH QUANTIFIED PROPOSITIONS

We can use the framework suggested by Zadeh to interpret more complicated forms of quantified propositions. Consider the statement " m a n y tall Swedes are fat". We are interested in using the preceding methodology to determine the truth of such a statement. Let Q be a linguistic quantifier, let X be a set of elements, called the base set, in this case the set of Swedes, let T and F be two fuzzy subsets definable over X, in this case tall and fat, respectively. Thus our statement reads " Q (TXs) are F". Furthermore, this is equivalent to " Q (TXs) are F c~T", where we let H = F c~T. There are two cases to consider. (i) If Q, the linguistic quantifier is a fuzzy subset of R*, then all we are saying is that the number of tall and fat Swedes is Q. Thus, if r = ~. H(xi), the cardinality of H, i

then the truth of the assertion is Q(r). (ii) If Q, the linguistic quantifier is a fuzzy subset of I, then we are saying the proportion of tall Swedes that are tall and fat is Q. If ~i H(xi), measures the number of tall and fat Swedes, the cardinality of H, and if ~iT(xi) measures the number of tall Swedes, the cardinality of T, then

r=~H(x,)/~T(x,) and Q(r) measures the truth of the assertion. Let us consider the situation with multiple quantifiers. Consider the statement "for most Swedes there are many acceptable cars". If we let X be the set of Swedes, Y be the set of cars, Q1 be the linguistic quantifier "most", Q2 be the linguistic quantifier " m a n y " and A is a fuzzy relationship on X x Y measuring the acceptability to Swede x of car y as A(x, y). We can express our proposition as (Q1 Xs)[(Q2 Ys)A]. In order to evaluate the truth of this proposition for given values of Q1, Q2 and A we can use the methodology suggested earlier. 1. Evaluate the truth of simply quantified proposition (Q2 Y)A(xi, y), for a given x~X.

2. Repeat step 1 for all x ~X, this gives us a new fuzzy subset of X, A*(x). 3. Evaluate the truth of (Q1 Xs)A*(x) to give us the truth of the whole assertion.

Example.Let X = {xl, x2}--Swedes, Y = {yl, y2, ya}--cars. Q1 :I~,I,

Ql(r) = r2--most,

Q2: I --*I,

Q2(r) = r - - m a n y ,

209

QUANTIFIED PROPOSITIONS OF A LINGUISTIC LOGIC

and 1 0.4 0-5 0.9 0.3 0.8 \ A(x, y) = (xl,yl) (x2, Yl) (xl, Y2)' (x2, y2)' (xl, ya)' (x2, y3) A(xl, y ) = rl = 1.8/3 = 0.6, A(x2,

{ (xll-yl)' 1 (xl,0,y2)' (x~,ya 0,i} Q2(0.6) = 0.6, 0.4

,

yl)-- t(x ,

r2 = 2.1/3 = 0.7,

0.9

(x2,

'

A*(xa) = 0.6, 0.8

1 (x?, Lfl'

Q2(0"7) = 0.7,

A*(x2) = 0.7,

n,=Io'6 o' t, I.

XI -'~'-21

r* = 1.3/2 = lY65,

Q1(0.65) = 0.4225.

Thus 0.4225 is the truth of the above assertion. Consider the statement "for most tall Swedes there are many acceptable small cars". Reformulate this as (for most tall Swedes)(there are many small cars) that are acceptable, or (Qx T Xs)[(Q2 SYs)A]. To evaluate this we would proceed as follows. 1. For each Swede we can consider the truth of the statement: for Swede X, "many small cars are acceptable". This gives us a new fuzzy subset A 1 over the set of Swedes, called "many small cars acceptable". 2. Using these new predicates we can evaluate the quantified proposition "most Swedes find many small cars acceptable". Example.

Using the data from the above with the following added information: I0-7, 1 1 , T = Tall Swede = t-~-i ~22J S=Smallcar=

{10"601 y l ' y"2~' ~ J '

for xl, we get Q2 SY are A ~ Q2 SY are A~ 1:~ S,

AXlAS = { 1 ,0.5y2, ~~3 0 . }3 (. {11 ,0.6y2' O3} = {~1 ,0-5y2' O3} = Bxl" Thus rxl = Y~ B X l ( y , ) / Y. S ( y , ) = 1 - 5 / 1 - 6 = 0 - 9 4 yI~/Y

yt~Y

210

R . R . YAGER

and hence Q(rxl) = 0.94. For

X2,

0.0.08 rx2= Y.

o.6 o I

0., o.6 o}

Bx2(yi)/ ~ S(y,)= 1 / 1 " 6 = 0 . 6 2 5

yieY

Yi Y

and hence Q(Fx2) = 0.625. Thus A , = ( 0 ' 9 4 , 0"6__ 25 ] L

X1

X2

J "

Next we proceed as follows. Let C=A*c~T=t r=

I0"94,0'625/ I0"7, 1 I0"7,0'625.l ~ x2 J r ~ t x , XE}=tx-ll x2 J '

Y. C ( x i ) / Y. T ( x i ) = l . 3 2 5 / 1 . 7 = 0 . 7 7 9 , xtEX

IJ~iEX

and Ql(r) = 0.607. Thus, the statment is deemed true to degree 0.607. POSSIBILITIES, CERTAINTIES AND TRUTHS OF RELATED PROPOSITIONS

Assume we have a proposition P1 involving a variable V which can take values in a base set U. Under Zadeh's translation rules in P R U F (Zadeh, 1978) this proposition generates a possibility distribution P1, a mapping from U to I, associated with the values the variable V can assume. Let P2 be a second proposition associated with V, which generates a possibility distribution P2 o v e r U. There exists a number of measures of compatibility between these two propositions. I. The possibility of P2 given P1 (Zadeh, 1977) is Poss [P2 ] Pl] = M a x [Pl(u) ^ P~(u)] ueU

This measures how easy it is for P2 to be in effect given a world in which PI is in effect. Note that Poss [P2 [ P1] = Poss [Px [ P2]. II. The certainty of P2 given Px (Zadeh, 1979b) is C E R T [P21Pt] = 1 - Poss [not P2] P1] = 1 - M a x [P~(u) ^ (1 - P2(u))] u~U

= Min [(1 - Pl(U)) v P2(u)]. u~:U

This measures how certain we are for P2 to be in effect given a world in which P1 is in effect. III. The truth of P2 given P1 (Zadeh, 1968) is

fP,(u)]

Truth [P2 IP1] = {P-~u)] '

u c U.

QUANTIFIED PROPOSITIONS OF A LINGUISTIC LOGIC

211

This measures the truth of the assertion P2 given that assertion P~ is assumed true. Note that Truth [P2IP1] is a fuzzy subset of the unit interval.

Example. 1,0.8,0.4, {0"5 2 P2=

~

1 0"_~47 }

' u2 u3

Poss [P2IP,] = 0.5,

C E R T [P2IPd = 0.2,

08 04 0o} '

Truth [P2]P1]= Consider

0}

' 0.2' 1 '

P3{ } Poss [P3[P,] = 0.4, Truth [P3IPI] = { ~ ,

1},

C E R T [Ps[P1] = 0, Truth [P1 [P3] = {0~4} = 0'4.

More generally, if we consider any fuzzy subset A of U, where m(u3) is the grade of membership of u3 in A. Let u3 ={1/ua}, the fuzzy subset corresponding to u3, Truth [AI u3] = A(u3) and Poss [u3l A] = A(u3). Thus, the grade of membership of u3 in A can be interpreted as either the possibility of us occurring when A is in effect or the truth of the assertion A when u3 has happened. Assume we have a linguistically quantified proposition " Q t Xs are F". As we noted, if Q1 is known this can be used to generate a possibility distribution II0, over the set ~r of fuzzy subsets of X such that for any K E ~ I l o , ( K ) indicates the possibility of K being an acceptable definition for the concept F based upon the statement. We recall that IIQI(K) = QI(FK). We may be interested in inquiring which other quantifier may be used to specify the information contained in the above proposition. Assume L is the set of all fuzzy subsets representing linguistic quantifiers over X. Let G ~ L, for each K ~ #r we can calculate G(rK). This would then generate a possibility distribution IIo(K) over ~" indicating the possiblity associated with quantifier G. Then Max [IlcffK) ^ IIo,(K)] would indicate the possibility that " G X are F" given that "Q1 X are F". Furthermore, Min [II~(K) v (1 -l-lo,(K))] Ke,~

would indicate the certainly that " G X are F" given that "Q1 X are F". Finally, IIQ,(K)

n---g J would indicate the truth of the assertion " G Xs are F " given that " Q Xs are F".

212

R.R. YAGER

In other instances, we may want to use the information contained in our original assertion to tell us some information about some other concept F* which may be related to F. For example, if we know that most students are honest we may want to know how many students are dishonest. Assume F~ and F2 are two concepts which can be represented by fuzzy subsets of X. Furthermore, assume there exists a mapping T: I-~ I such that T(FI(x)) = F2(x). For example, if Fx is " y o u n g " and F2 is "not young" then T(FI(X)) = F2(x) = 1 - F l ( x ) . Let us also denote the inverse of this map as T-~. As we noted the proposition P1 : Q1 X are F1 generates a possiblity distribution rio, over the set of all fuzzy subsets ~ of X. Thus t'or any K e ~, HoI(K) indicates the possibility of K being the generating function for F1. Consider some other proposition P2 : Q2 X are F2. This also generates a possibility distribution over the set ~ of fuzzy subsets of X, which we shall indicate as H02, where for any K e ~, 1-102(K) indicates the possibility of F2 being generated by the realization K. We also note I/o~(K) indicates the possibility of F1 being represented by the fuzzy subset K. However, if "Q2 X are F2" induces II02 since we know that T-I(F2) = F~ (because of the relationship between F1 and F2) we can say that P2 implies that the possibility of T-I(K) being the generating function of F~ is IIo~(K). Thus, we have possibility distributions associated with the generating functions for F1. The first II01 obtained from P~, the second which we shall denote as H* generated from P2, in which for K ~ ~', the possibility of K being the generating function is II*(K) where II*(K) = IIo~(T(K)). Note Ho2(T 1G) =Q2 [~x T-aG(x) ] . With these two induced possibility distributions we can now apply our measures of compatibility. Thus Poss [Q2 Xs are FEIQI Xs are F1] = Max [H*(K) AHo2(K)]. K~

The certainty that Q2 Xs are f2 given Q1 Xs are F1 is Min [II*(K) v (1 -IIol(K))]. Ke,~

The truth of the statement Q2 Xs are F2 given Q~ Xs are F1 is {Hol(K)/II*(K)} for all Ke~. With respect to this truth value there are some cases of special interest. If IIol(K) -II*(K) for all K we say that Q2 XS are F2 is semantically equivalent to Q1 Xs are F1. If I I * ( K ) - H o l ( K ) for all K ~ then we say Q1 Xs are F semantically implies (or entails) Q2 Xs are F2.

213

QUANTIFIED PROPOSITIONS OF A LINGUISTIC LOGIC

Example. Assume Q1 and Q2 are defined on [0, 1] and Q: is the antonym of QI, i.e. Q2(u) = QI(1 - u ) . Let F2 be the negation of F1, i.e. F2(s) = 1 - F l ( x ) . The proposition Q1 Xs are F1 induces the possibility distribution such that for K e :~,

\ n xeX

The proposition Qz Xs are F induces the possible distribution Qz such that for K 9 4,

z

\ n x~X

Since TK = 1 - K , then

(

II*(K) = IIo2(TK ) = Q2 \ ~ -

(1

=

g

However, since Q 2 ( 1 - u ) = Q l(U), we see that H*(K)= IIol(K). Thus, the two statements are semantically equivalent. SPECIFICITY OF QUANTIFIED STATEMENTS

Assume A and B are two normal fuzzy subsets of U indicating the value of some variable V. A is said to be a more specific version of B, if A(u)-
P2 :"Q2 Xs are F".

Each one induces a possibility distribution over :~, indicating the possibility of K 9 ~: being the possibility function for F. For P1 we get 1-I1, and for P2 we get [I2. First we note that there exists some K* e :~ such that Hi(K*) = 1 and necessarily II2(K*) = 1. For any K e 4, Hi(K) = Q1 ( 1 ~ K(x)) -< l'I2(K). Thus P1 induces a more specific distribution than P2 induces. Let Q be a monotonically non-decreasing linguistic quantifier, i.e. Q(u,)>-Q(u2) if//1>/,/2 . Consider the propositions P~ : Q Xs are F~,

P2 : Q Xs are F2,

where F~ is a more specific concept than F2, that is F,(x)-
(E Hi(K) = Qxx~x

K(x))

214

R.R.

YAGER

indicates its possibility distribution for being the generating function of F1 and

indicates its possibility distribution for being the generating function of F2. Let T : F2 -~ F1 such that T(F2(x)) = Fl(X). Since F1 is m o r e specific than F2, then T(F2(x))-~Fl(x). As we noted in the previous section I I * ( K ) = II2(TK) induces from P2 a possibility distribution on FI:

Thus Px semantically entails P2. PROBABILISTIC INFORMATION FROM QUANTIFIED STATEMENTS A quantified statement can be used to generate probabilistic information about the elements in the base set. However, first we must introduce some ideas about probabilities of fuzzy events. Assume X = {Xl, x2 . . . . . x~} is a set of elements. Let P1, P~ . . . . . P~ be a probability distribution over X, Y~i Pi = 1, and let A be a fuzzy subset of X with m e m b e r s h i p function A(x). If we select an element from X, denoted e, based upon our probability distribution, then Z a d e h (1968) has suggested that Prob {e 9 A} = ~ P i A ( x i ) . i

The special case when the element e is selected randomly occurs when Pi = 1 / n . Thus Prob {random element ~ A} = ~ 1 A ( K , ) . i=In

We also note that if A and 13 are two fuzzy events, then P ( A / B ) = P(AB)/P(B). Two cases are of interest. If A and B come f r o m the same trial, for example, if A and B are "tall" and "fat", then P(AB) is Prob (e is tall and fat), then A ~ B = C , C ( x ) = Min [A(x), B(x)] and Prob (AB) = Y, PgC(x~). i

If A and B come from different trials, thcn if we let Yli = (xi, xi), xi being the outcome of the first trial and xj being the outcome of the second trial, then Mill [A(xi), B(xi)] = C(yij) and we must calculate P0" = Prob (x,., xi) then P(AB) = Z P,i c (Y/i). As we discussed previously a quantified statement such as " Q Xs are F " induces a possibility distribution I I o over the set ~ of all fuzzy subsets of X such that for any K 9 .~', H(K), indicates the possibility of k being the definition of the concept F. Assume we randomly select an element from X and ask what is the probability that x E F.

215

Q U A N T I F I E D P R O P O S I T I O N S OF A L I N G U I S T I C L O G I C

Since F is defined by the possibility distribution II over ~, then for each K 9 ~, Prob {x r K} =- Z 1K(x,.), i=ln

and therefore

rio(K) Prob (x e ~r)= KV~ {1/n

~K(xi))'

which is a linguistic probability, a fuzzy subset of the unit interval. In the special case when the linguistic quantifier 0 is defined as a fuzzy subset of I, then I-IQ(K) = Q (nl--/~x K(xi)). Assume we selected two people randomly from X, without replacement, and asked for the probability that they both have property F. Then Pr~ {x'e F and x 2 9 F}=

1 / ( n 2 - n ) Y.j=t F,=1 [K(xl) ^ K(xi)] n

n

9

Assume G is some property related to F, in particular assume there exists a mapping T: I -~ I, such that T(F(x)) = G(x). We shall say T F = G. For example, if F is "tall" then G could be '"not tall", "very tall", etc. Assume we know " 0 Xs are F" and we want Prob {x 9 G}, where x is a random element in X. First we note our proposition induces the possibility distribution IIo, indicating the possibility of any K 9 ~ of being the distribution for F. From this we can deduce the fact that the possiblity distribution for G induced from this is t~., (Ho(K) ] and hence

tJ /

Prob {x 9 G} = K'~'~t

Uo(K) / 1/n ~ii-~(xi)J"

S U M M A R I E S B A S E D ON S A M P L E S

As we noted, linguistic quantifiers allow us to make summary statements about properties of the elements in some base set. In many instances we may want to make such a statement based upon only seeing a sample of the base set, Assume X is a finite set of objects, called the base set. Let Y be some crisp subset of X and let F be some property associated with the elements of X which can be represented as a fuzzy subset. Assume we know the values of F(x) for only those x ~ Y. We are interested in determining the validity of a quantified statement about the elements of X with respect to the property F based upon our sample of F over the set Y. Let ~- be the class of all fuzzy subsets of X such that G ~ ~" if for all x ~ Y G(x) = F(x). Thus, ..~ is the set of possible values for F based upon the known information. For any G 9 ~ we can calculate the truth of the statement " Q Xs are F", where F is defined by G. Let us denote this as IG].

216

R.R. YAGER

Thus we have a mapping

M(G) = IG[.

M: ~ - ~ I, Let a = Min Ge~-

IGI

and

B =Max G~

IGI.

From this we can conclude that based upon our sample, the statement " Q Xs are F" is at least a true and at most fl true. The interval [a, fl] becomes a confidence like interval for the truth of our assertion based upon our data, the observations about the property F. If Q is a linguistic quantifier, represented as a fuzzy subset of the unit interval, then for a particular G ~ ~ the truth of the assertion " Q Xs are G " is Q(ra), the membership grade of Q at r~, where re = -

1 " Y~ G(Xl).

ni=l

Assume that X={xl, xz .....

and

xm,...,x,,}

Y = { x l , x2 . . . . . x,,}.

Then

1

r,=-

n i=1

G(x,>= /

O(x,>+

n i=1

G(x,> n i=m+l

If we let our known portion sum to C we get

II i = m + l

We shall consider the case where Q is a monotonically non-decreasing type of quantifier, that is rx > r2 implies Q(r,) -> Q(r2). Examples of this type of quantifier are " m a n y " , "most", "at least 5 0 % " , etc. For this type of quantifier it is obvious that B = M a x o ~ IGI, occurs for G* e .~ such that G*(x) =

I for x~ Y.

Thus

Ic+n -m tl

n

Also, in this case a = M i n c ~ IG] occurs for the (~ ~ ~r such that G(x) = 0 for x ~ Y , and therefore 1 rd = - C . n

QUANTIFIED

PROPOSITIONS

OF A LINGUISTIC

217

LOGIC

T h e r e f o r e for any monotonically non-decreasing quantifier c~ = Q ( r o ) =

o(1 )

C ,

/~/=Q(re.) =

o(1_

C+ n n

First we note that when m = 0, a necessarily assumes its minimal value, Q(0) and /3 assumes its maximal possible value Q(1). When m = n, a =ft. F u r t h e r m o r e c~ is always less than or equal to/3. Consider aa

am

_

aQ(r) Or

1 OC Or am = Q ( r ) n a-m"

Since Q is monotonically non-decreasing and C is also a monotonically non-decreasing function of m, then Oot/am is also monotonically non-decreasing. Consider

013 OO(r) Or O---m= a~ am

OQ(r)[10C

Or L;z am

1]

"

its obvious that ac/am < 1, thus OB/am is monotonically increasing. F r o m this we can conclude the following for monotonically non-decreasing type quantifiers Q, with respect to samples. 1. O u r lower confidence is minimal when our sample size is zero and increases as the sample size increases 9 2. Our upper confidence is maximal when our sample size is zero and decreases as the sample size increases 9 3. O u r interval p = fl - a, decreases as the sample size increases until m = n, when p =0, and//=a. The above conclusion is also valid in the case when the quantifier Q uses r = E, F(x,) instead of 1/n Y. F(xi). Let us consider the case when Q is a monotonically non-increasing. In this o~, our lower confidence occurs when G = G* and our upper confidence occurs when G = (~. It can easily be shown that also in this case ot increases as m increases a n d / ~ decreases as m increases and p = f l - a monotonically approaches zero. Thus, for any monotonic type quantifiers, our lower truth level continuously increases as our sample size increases and the distance between fl and a decreases as m increases. Let us now consider a quantifier which is n o n - m o n o t o n i c over I. Examples of this would be " a p p r o x i m a t e l y half", " a b o u t 7 5 % " , etc. Again assume we have a sample of size m and let ~- be the set of possible definitions of F based on this sample. First we note that there exists some G* e ~" such that

O(ro*) =/3 = Max

G a .'~-

IGI

and some G such that

= ML.Iol.

218

R.R. YAGER

However, it is not all that obvious what the values for G* and G are. It requires considerable calculation to find these. Secondly, it it quite possible that as m increases a, our lower confidence, may decrease. Thus these non-monotonic qualifiers are very difficult to use with confidence given some sample. OTHER MEASURES OF CARDINALITY

To evaluate the truth of a linguistically quantified proposition, such as " O Xs are F" for a particular manifestation of F as a fuzzy subset of X, we have assumed this truth to be the m e m b e r s h i p grade of the cardinality of F in the fuzzy subset representing the linguistic quantifier O. This approach is basically an extension of the algebraic type approach where we allow the elements of F(x) to be drawn from the multivalued logic I and we allow Q to be a fuzzy subset with grades also drawn from I. In this extension of our approach we have used as our definition for the cardinality of the fuzzy set F the sum of the m e m b e r s h i p grades of the elements of X in F. The question naturally arises as to whether this m e t h o d of measuring the cardinality of a fuzzy subset is a valid extension of the concept of cardinality from crisp sets, recalling that the cardinality of finite crisp set is the n u m b e r of elements in the set. A problem with the above extension hinges on the question of whether, for example, two elements with grade of m e m b e r s h i p 0.5 corresponds to 1 total membership. T h e r e exists other possible ways of extending the concept of cardinality from crisp subsets to fuzzy subsets. In extending this definition there appears to be at least certain minimal properties that any extension of the definition must satisfy. Assume X is a finite set of elements, let X~ be the class of all fuzzy subsets of X, note ~ also includes all the crisp sets as a special class, and let C a r d : g - * R be a measure of the cardinality of any element of g . Any definition of Card should satisfy at least the following properties. (1) For any F s ~7~',Card (F) is non-negative. (2) For any F ~ X~ which is a crisp subset of X, Card F - - # of elements in F. (3) If

for any x, y c X and if ~ -~ B, then Card F1 -> Card F2. (4) Card (F1 ~ F2) - Card (F1 c~ F2) = Card (F1) + Card (F2). I f a measure of cardinality satisfies conditions (1)-(4) then if A c B, Card A <- Card B.

THEOREM.

Proof. Let )..., I. X l

,

,

X2

l-X1

F r o m our hypothesis, b~ - ai for all i. Let

,od

B

,..., X2

.

Xn/

QUANTIFIED PROPOSITIONS OF A LINGUISTIC LOGIC

219

F r o m p r o p e r t y (3) Card B~ -> Card Ai. F u r t h e r m o r e ,

AinAi=BinBi=

0

i#/'.

Since A=AluAzuA3u"

9 "wAn,

B = BI~BauB3~"

9 9u B n ,

from property (4) Card A = Card A1 + Card A2 +" 9 9+ Card An, Card B = Card B1 + Card

B2+"

" "+Card

Bn,

Since Card B~ -> Card A~, then Card B _>Card A. THEOREM.

If a measure of cardinality satisfies conditions (2) and (4), then if A

[XI,X2

,

Xn

Carda= i=t ~ Card{~ Proof.

Let A~={a~}

A=AlwAzw...uA,

and

AinAi=Q5

F r o m condition (2), Card ~ = 0. Then our theorem follows from condition (4). Thus the cardinality is determined by how we measure the cardinality of each molecule Ai. A general class of cardinality measures could be the following. L e t F be the m e m b e r s h i p function of a fuzzy subset of X, and let H(x) = Fa(x),

F(x)-> a,

H(x) = F~(x),

F(x)
where a , b --> 0 , b --- a a n d a ~ [ 0 , 1], then Card F = ~ H(xi). i--1

When a = b = 1 and any a, we get Card F = ~. F(xi). i=l

220

R.R. YAGER

W h e n b = oo and a = 0, we get H(x) = 1,

F(x) - a ,

H ( x ) = 0,

F(x) < a,

C a r d F = n u m b e r of x ~ X such that F(x) ~ a. when b = oo and a = 1 we get H ( x ) = F(x),

F(x) ->t~,

H(0) = 0,

F(x) < a,

C a r d F = Z F(x ) o v e r all x such that F(x) -> a. W h e n a --- b, we get C a r d F = ~ F a (x). x

T h e r e exists an alternative a p p r o a c h to m e a s u r i n g the cardinality of a fuzzy subset, which is b a s e d u p o n the c o n c e p t of level sets associated with a fuzzy subset. This a p p r o a c h leads to the c o n c e p t of a fuzzy cardinality (Yager, 1979). A s s u m e F is a fuzzy subset of X the a - l e v e l set associated with F is the crisp set F defined as F~ = { x IF(x) ->a, x ~X}. Since F~ is an o r d i n a r y set we can define the cardinality of F~, d e n o t e d C a r d Fa or [Fa [, as the n u m b e r of e l e m e n t s in Fa. Since F=

U ~F~,, at

we can use the extension principle to define the fuzzy cardinality of F, d e n o t e d IFI: or Card F as

Example. A s s u m e X ={xl, x2, x3, x4, xs}. L e t F={11,0.7

0.5 0-3

0},

X 2 ' X 3 ' X 4 ~ XS'

then a =0,

F,, =(xl, xz, x~,x4, x~},

IF~I = 5,

0
F,~ ={xl, x2, x3, x4},

IF,,I = 4,

0"3
F~,={xl, x2, x3},

[F,,t = 3,

0"5 < a -<0"7,

F~ = {Xl, x2},

]F~]=2,

F~ - - { x d ,

IF~I = 1.

a >0.7,

221

QUANTIFIED PROPOSITIONS OF A LINGUISTIC LOGIC

Therefore L1'2'3'

"

We note that [Fit is a fuzzy subset of the real line. We can use this fuzzy measure of cardinality of F to determine the truth of a linguistically quantified proposition " Q Xs are F " as follows. Realizing that for a crisp numerical cardinality of F equal to r, Q(r) measures the truth of " Q Xs are F " then from the extension principle, for fuzzy cardinality we obtain Q(IFIr) where if , . . . ,

,

/'2

where rl E R, then ,)' O(r2)

.

.

.

.

.

.

which gives us a linguistic truth value for the truth of quantified proposition.

Example. Consider the proposition " Q Xs are F", where F is defined as in the preceding example and Q is the linguistic quantifier defined by the membership function Q : R-~ I such that Q(r)=r2/16,

r<_4,

Q(r) = 1,

r > 4,

then the truth of our assertion is

T=

[ 1716' 1 / 4 ' 9 / 1 6 ' 07/ - "

Using a m e t h o d suggested by Yager (1981) we can use the fuzzy cardinality to obtain an alternative measure of cardinality, called the equivalent cardinality, denoted IFIE. Assume F is a fuzzy subset of X and [Fir = B is the associated fuzzy cardinality. Let B,~ be the a-level set of the fuzzy cardinality, amax be the maximal membership grade in B and let M(B~) be the mean of the elements in B~. Then Equiv. Card =

Example.

~0~max

M(B~) da.

For our previous example F={1

,

0.7 0.5 __0"3 O } ,

X2

{Fir=

,

X3

,

X4

07 05 YI '

' 2 ' 3 ' -'

,

222

R.R.

YAGER

The level sets for F are

{1,2, 3,4},

M=2.5,

0 . 3 < a --0.5,

11, 2, 3},

M=2,

0.5 < a <-0.7,

{1, 2},

M = 1.5,

0.7
{1},

M=I,

O < a -<0.3,

/* 1

IVlr

[

,Ic}

M d a = (2.5)(0.3) + (0.2)(2) + (0.2)(1.5) + (0.3)(1) =0.75+0.4+0.3+0.3=

1.75.

We note that this measure of cardinality gives us a number. Let us consider the same procedure except in this case for each level set B~ of the fuzzy cardinality instead of calculating the mean of elements in that set we use the Max of the elements in B~. Thus our measure of cardinality would be fo~"~ Max

da.

Assume tXI

X2

where al->a2>-a3--- 9 9 " ~ a , . Then

'2 .....

=B.

The level sets of B are {1, 2 , . . . , n},

Max B~ = n,

a n "~ o~ "~ a n - 1

{1, 2 , . . . , n - 1},

MaxB~ = n - l ,

a 2 *~ o~ ~-0~1,

{i},

Max i3,, = 1.

0<~a

~an,

,

Then the cardinality of F in this case is

Io'~ Max [13 ,~] da

n

,] + ( n - 1) (an_t

+(n-2)(a,,-l-~,

e~n) 2)+'--+ l(a~-d2)

=nan+ f ( i - l)(Oq-l--ai) = f a,, i=2

i =1

which is the original concept of cardinality suggested by Zadeh (1979c). FUZZY

CARDINALITY

AND

THE

SUBSTITUTION

METHOD

In binary logic we noted that there exist two possible ways of interpreting quantified propositions, the first of these methods is based on the algebraic cardinality approach

OUANTIFIED

PROPOSITIONS

OF A LINGUISTIC

LOGIC

223

and the second is based upon the substitution approach. Thus far we have just extended the algebraic approach to our multivalued situation. We shall now extend the substitution approach. We shall see that in m a n y cases the substitution approach is easily interpretable in terms of the concept of fuzzy cardinality. Furthermore, while in a binary logic both approaches give the same result in multivalued logic they m a y give different results. Assume " Q Xs are F " is a quantified proposition, in which F is a predicate (a fuzzy subset), X is a base set of n elements and Q is a quantifier. The substitution approach involves the writing of an equivalent formulation of the quantified statement in terms of logical sentences consisting of the disjunction, conjunction and negation of the truth of the predicate for the class of elements in X. (The truth of the predicate F for x ~ X is the grade of m e m b e r s h i p F(x) of x in F.) Let V be the family of all logical sentences whose atomic propositions consist of the predicate F evaluated over the base set F. We can represent a linguistic quantifier Q by a fuzzy subset So of V, such that for each v ~ V, So(v) indicates the degree of possibility of the sentence v as a meaning for the quantifier Q. For example, if Q is "all", then if v = F ( x 0 and F(x2) and F(x3) and F(x~), So(v) = 1, whereas for all other v ~ V, So(v) :- 0. If Q is "at least one", then v = F(xl) has S o ( v ) = 1, as does v = F(x2) and v = F(x3), as does v = F(xl) or F(xz) or F(xs) or . . . . To evaluate the truth of the proposition " Q Xs are F" we need to find an equivalent sentence which is true. Let So indicate the fuzzy subset of its equivalence as a substitute, thus So(v), for v ~ V, indicates the degree to which v is satisfied by the concept Q. Furthermore, for each v ~ V, let T(v) indicate the truth of the sentence v based upon the structure of the sentence, the truth of the predicate for the elements of X and our rules of logic. For example, if v = F(Xl) and F(xz) then (T(v) = Min (F(xl), F(xz)). Let T be the fuzzy subset of these truths of V. Then we want a sentence v that is T and So. From this it follows that we can calculate the truth of our quantified proposition P : Q Xs are F as

Truth of P = Max [So(v) ^ T(v)]. v~W

Using this approach we can express linguistic quantifiers as fuzzy subsets of the possible substitutions. In m a n y cases the fuzzy cardinality becomes a very useful tool to help in the evaluation of the truth of an individual sentence, T(v) and, in turn, in the determination of the truth of the whole proposition. If Q our quantifier is a crisp concept, such as all, then So becomes a crisp set and our truth of P becomes Max [T(v)]. v~SQ

224

R.R. YAGER

Consider the quantifier "at least m Xs are F", for example "at least 10 students are tall". In this case So = M Xs are F or M + 1 Xs are F o r . . . or N Xs are F, So = {M}u {M + 1}u { M + 2 } w . 99 u{N}. Furthermore {K} = K Xs are F = L_.I Bt, leLk

where Bi is the form F(x/,) AF(xl2) A" 99AF(xl~), where l = {xt,, xl2. . . . . xl~} is a subset of X having k elements and Lk is the index of all such sets. Thus the set So is made up of sentences v of the form of the B, where 1 E Lk. The truth of such a sentence is T(v) = Min [F(xtl), F(xt2) . . . . . F(xtk)]. Furthermore, the truth of our proposition is Truth P = Max T(v). V~SQ

Assume the elements in X are ordered such that i > / ~ seen that for this ease, at least M Xs are F,

F(xi)_> F(xi). It can easily be

Max T(v) = Min [F(xx), F(x2) . . . . . F(x,.)] = F(x,,). VESQ

If we consider the quantified statement "at most M Xs are F", as we have done in our section on binary logic, if we again consider the elements of X ordered as above the substitution approach yields as the truth of our proposition 1 -F(x,,+x). Finally if we consider the proposition "exactly M Xs are F", which is equivalent to "at most M Xs are F" and "at least M Xs are F" our substitution yields as the truth of this IF(x,,) A (1-F(xm+l))]. The concept of fuzzy cardinality provides a concise and useful way for evaluating the truth of the preceding types of quantifiers. Assume F is a fuzzy subset of the finite set X, assume X has n elements. Then let IFIt, which is a fuzzy subset of the integers, be the fuzzy cardinality of F. We note that the truth of the statement "at least m Xs are F" is equal to IFIr(m), the degree of membership of m in IF}t. Following Zadeh's (1979b) notation we shall define the fuzzy subset of {1, 2 , . . . , n}, F G count (F) as F G Count (F) =-[Fir. Furthermore we note that IFIt can be easily obtained from F by ordering the elements in F according to decreasing membership grade and replacing the element by the appropriate number we shall denote the ordinal set as F~.

Example.

/06 0.7 F=t Xl '

x2

0.1} ~ x3

~x4 ~ x5

"

225

QUANTIFIED PROPOSITIONS OF A LINGUISTIC LOGIC

Then I0_'l 0"7 0"6 0"3,0"1} F~=tx4' x2' xl'x3 x5

and

{Fit

={1,0"7,0"6,0"3,051 } 2 3 4

and

]Fir, {1 ' (F)=

F G Count (F) = F G Count

06 03

2 ' 3 ' 4 '

?1

"

Thus the truth of the statement "at least 3 Xs are F" is 0-6. We define the fuzzy subset FL Count (F) of the integers as follows: FL Count (F) : Integers ~ [0, 1] such that for any y E Integers FL Count (F)y = (1 -IFIr(y + 1)). Thus, for our example, F L C o u n t (F) =

{0~3 0.4 0.7 0.9 1~ ' 2 ' 3 ' 4 '5J'

It can be easily seen that the truth of the proposition "at most m Xs are F " is the degree of membership of m in FL Count (F). Finally, we can define F E Count (F) as FE Count (F) = F G Count (F) n FL Count (F). Thus, for our example, FECount(F)={0i3

0.4 0.6 0.3 051 } '2'3'4'

Furthermore, the truth of the proposition "exactly M Xs are F " is equal to the degree of membership of M in F E Count (F). Thus we see that the use of fuzzy cardinality provides a useful tool. Let us consider the following proposition "nearly all Xs are F". If X is a crsip set consisting of n elements then we can define "nearly all" by the fuzzy subset SQ where SQ = t - " ~ / J

'

where v~ = at least i elements, i.e. vi = F(xi) A F(x2) ^" " 9 ^ F(xi) SO(Vi) = i/n. Then the truth of the statement "nearly all X are F " is Max

i~{1,2,...,m}

[So(v/) ^ FG Count F(i)].

For our example

so /0i2 04 06 08 '2'3'4'

'

226

R . R . WAGER

O ount( t:{X

' 2'

06 03 0;1 3'4'

S o c ~ F G C o u n t ( F ) = { 0 i 2 , 0 ~ 4__ 0.6 0.3 0~1} ' 3'4' Therefore we get 0.6 as our truth. Assume we have two fuzzy subsets G and F of X. Consider the statement " Q (GXs) are F", where Q is a linguistic quantifier. An example of the above would be "several tall students are blond". To evaluate the truth of the above statement first realize that the above statement is equivalent to " Q Xs are G and F". Then, if we let G c~ F = H, we can apply our developed methodology to the proposition Q Xs are H. Assume we are given a quantified statement " Q Xs are F", where F is just specified as a concept but not defined as a fuzzy subset of X. Let ~ be the set of all fuzzy subsets of X. If we are interested in determining the possible elements of ~ which can be meaningful definitions of the concept of F we proceed as follows. Let FI be a possibility distribution II:~-~I such that for any G c ~-, I1(G) indicates the possiblity of G being the definition of F based upon the above quantified proposition. Let So be the definition of Q as a fuzzy subset of V, all the possible logical sentences. For each v ~ V, let TG(v) indicate the truth of sentence v with G as the assumed definition of F. Then II(G) = Max [So(v) ^ TG(v)]. u~G

Conclusion We have tried to present the necessary groundwork for a theory to help represent linguistic quantifiers in a manner useful for the building of intelligent machines.

References DELt.rCA, A. & TERMINI, S. (1972). A definition of nonprobabilistic entropy in the setting of fuzzy set theory. Information and Control, 20, 301-312. MOSTOWSKI, A. (1957). On a generalization of quantifiers. Fundamenta Mathematicae, 44, 17-36. RESCItER, N. (1969). Many-Valued Logic. New York: McGraw-Hill. SUPVES, P. (1957). Introduction to Logic. Princeton, New Jersey: Van Nostrand. YAGER, R. R. (1979). A note on probabilities of fuzzy events. Information Science, 18, 113-123. WAGER, R. R. (1981). A procedure for ordering fuzzy subsets of the unit interval. Information Science, 34, 143-161. ZADEH, L. A. (1965). Fuzzy sets. Information and Control, 8, 338-353. ZADEH, L. h. (1968). Probability measures of fuzzy events. Journal of Mathematical Analysis and its Applications, 23, 421-427. ZADEH, L. A. (1977). Fuzzy sets and information granularity. In GUPTA, M. M., RAGADE, R. K. & WAGER, R. R., Eds, Advances in Fuzzy Sets: Theory and Applications, pp. 3-18. Amsterdam: North-Holland. ZADEH, L. h. (1978). PRUF--a meaning representation language for natural languages. International Journal of Man-Machine Studies, 10, 395-460.

QUANTIFIED PROPOSITIONS OF A LINGUISTIC LOGIC

227

ZADEH, L, A. (1979a). Approximate reasoning based on fuzzy logic. Memo # E R L M79/32. Berkeley: University of California.

ZADEH, L. A. (1979b). Precisation of human communication via translation into PRUF. Memo # E R L M79/73. Berkeley: University of California. ZADEH, L. A. (1979c). A theory of approximate reasoning. In HAYES, J. E., MICItIE, D. 8r M1KULICH, L. 1., Eds, Machine Intelligence, vol. 9, pp. 149-194. New York: Wiley. ZADEH, L. m. (1981). Possibility theory and soft data analysis. In COBB, L. & THRAI.I., R. M., Eds, MathematicalFront&rs of SociaI Policy Sc&nces, pp. 69-129. Boulder: Westview Press. ZADEH, L. m. & BELLMAN, R. E. (1977). Local and fu2zy logics. In DUNN, J. M. & EPSTEIN, G., Eds, Modern Uses of Multiple Valued Logic, pp, 103-165. Dordrecht: Reidel,