COLING 82, J. Horeck.f, (ed.) 
North-Holland Publishing Company 
© Academia, 1982 
TEST-SCORE SEMANTICS 
FOR NATURAL LANGUAGES 
Lotfi A. Zadeh 
Computer Science Division 
University of California 
Berkeley, California 
U.S.A. 
Test-score semantics is based on the premise that 
almost everything that relates to natural languages 
is a matter of degree. Viewed from this perspective, 
any semantic entity in a natural language, e.g., a 
predicate, predicate-modifier, proposition, quanti- 
fier, command, question, etc. may be represented as 
a system of elastic constraints on a collection of 
objects or derived objects in a universe of dis- 
course. In this sense, test-score semantics may be 
viewed as a generalization of truth-conditional, 
possible-world and model-theoretic semantics, but 
its expressive power is substantially greater. 
INTRODUCTION 
Test-score semantics represents a break with the traditional approaches to seman- 
tics in that it is based on the premise that almost everything that relates to 
natural languages is a matter of degree. The acceptance of this premise entails 
an abandonment of bivalent logical systems as a basis for the analysis of natural 
languages and suggests the adoption of fuzzy logic (Zadeh (1975), Bellman and 
Zadeh (1977), Zadeh (1979)) as the basic conceptual framework for dealing with 
natural languages. 
In fuzzy logic, as in natural languages, almost everything is a matter of degree. 
To put it metaphorically, the use of fuzTy logic may be likened to writing with 
a spray-can, rather than with a ball-point pen. The spray-can, however, has an 
adjustable orifice, so that one may write, if need be, as finely as with a ball- 
point pen. Thus, a commitment to fuzzy logic does not preclude the use of a biva- 
lent logic when it is appropriate to do so. In effect, such a con~itment merely 
provides a language theorist with a much more flexible framework for dealing with 
natural languages and, especially, for representing meaning, knowledge and strength 
of belief. 
An acid test of the effectiveness of a meaning-representation system is its ability 
to provide a basis for inference from premises expressed in a natural language. 
In this regard, an indication of the capability of test-score semantics is pro- 
vided by the following examples, in which the premises appear above the line and 
the question which may be answered is stated below it. 
(a) During much.of the past decade Pat earned far more than all of his close 
friends put together 
How much did Pat earn during the past decade? 
(b) Most tall men are not fat 
Many fat men are bald 
Big is tall and fat 
425 
426 L.A. ZADEH 
How many big men are bald? 
(c) If X is large then it is not likely that Y is small 
If X is not very large" then it is very likely that Y is large 
X is not large 
How likely is it that Y is more or less small? 
In fuzzy logic, the answer to a question is, in general, a possibility distribution 
(Zadeh (1978)). For example, in the case of (a) the answer would be a possibility 
distribution in the universe of real numbers which associates with each number u 
the possibility, ~(u), o ~(u) ~ I, that u could be the cumulative income of 
Pat given (i) the premise, and (ii) the information resident in a database. 
In test-score semantics, a semantic entity such as a proposition, predicate, predi- 
cate-modifier, quantifier, qualifier, command, question, etc., is represented as a 
system of elastic constraints on a collection of objects or derived objects in a 
universe of discourse. Simple examples of semantic entities whose meaning can be 
represented in this manner are the following: 
I. Anca has a young son. (Proposition.) 
2. When Dan is tired or tense, he smokes a lot. (Conditional proposition.) 
3. It is not quite true that John has very few close friends. (Truth-qualified 
proposition.) 
4. It is very likely that Marie will become well-known. (Probability-qualified proposition.) 
5. It is almost impossible for Manuel to be unkind. (Possibility-qualified 
proposition.) 
6. Expensive car. (Fuzzy predicate.) 
7. Very. (Modifier) 
8. Several large apples. (Second-order fuzzy predicate.) 
9. More or less.(Modifier/Fuzzifier.) 
I0. Not very true. (Qualifier.) 
II. Very unlikely. (Qualifier) 
12. Much taller than most. (Fuzzy predicate.) 
13. Bring me several large apples. (Fuzzy command.) 
14. /Who are Edie's close friends. (Question.) 
Although test-score semantics has a much greater expressive power than the meaning- 
representation systems based on predicate, modal and intensional logics, its ex- 
pressiveness is attained at the cost of downplaying, if not entirely severing, the 
connection between syntax and semantics. In particular, the homomorphic connection 
between syntax and semantics which plays a central role in Montague semantics 
(Montague (1974), Partee (1976) and attributed grammars for programming languages 
(Knuth (1968)), plays a much lesser role in test-score semantics-a role represented 
in the main by a collection of local translation rules governing the use of modi- 
fiers, qualifiers, quantifiers and connectives. In effect, the downplaying of the 
connection between syntax and semantics in test-score semantics reflects our belief 
that, in the case of natural languages, the connection is far too complex and far 
too fuzzy to be amenable to an elegant mathematical formulation in the style of 
Montague semantics, except for very small fragments of natural languages in which 
the connection can be formulated and exploited. 
The conceptual framework of test-score semantics is closely related to that of 
PRUF (Zadeh (1978)), which is a meaning-representation system in which an essential 
use is made of possiblity theory (Zadeh (1978))- a theory which is distinct from 
the bivalent theories of possibility related to modal logic and possible-world 
semantics (Cresswell (1973), Rescher (1975)). 
In effect, the basic idea underlying both PRUF and test-score semantics is that 
most of the imprecision and lack of specificity which is intrinsic in natural lan- 
guages is possibilistic rather than probabilistic in nature, and hence that possi- 
TEST-SCORE SEMANTICS FOR NATURAL LANGUAGES 421 
bility theory and fuzzy logic provide a more appropriate framework for dealing with 
natural languages than the traditional logical systems in which there are no gra- 
dations for truth, membership and belief, and no tools for coming to grips with 
vagueness, fuzziness and randomness. 
In what follows, we shall sketch some of the main ideas underlying test-score seman- 
tics and illustrate them with simple examples. A more detailed exposition and ad- 
ditional examples may be found in Zadeh (1981). 
BASIC ASPECTS OF TEST-SCORE SEMANTICS 
As was stated earlier, the point of departure in test-score semantics is the assump- 
tion that any semantic entity may be represented as a system of elastic constraints 
on a collection of objects or derived objects in a universe of discourse. 
Assuming that each object may be characterized by one or more fuzzy relations, the 
collection of objects in a universe of discourse may be identified with a collection 
of relations which constitute a fuzzv relational database or. equivalently, a state 
description (Carnap (1952)). In this database, then, a derived'object would be char- 
acterized by one or more fuzzy relations which are derived from other relations in 
the database by operations expressed in an appropriate relation-manipulating lan- 
guage. 
In more concrete terms, let SE denote a semantic entity, e.g., the proposition 
p 4 During much of the past decade Pat earned far more than 
all of his close friends put together, 
whose meaning we wish to represent. To this end, we must (a) identify the con- 
straints which are implicit or explicit in SE; (b) describe the tests which must 
be performed to ascertain the degree to which each constraint is satisfied; and 
(c) specify the manner in which the degrees in question or, equivalently, the par- 
tial test scores are to be aggregated to yield an overall test score. In general, 
the overall test score would be represented as a vector whose components are num- 
bers in the unit interval or, more generally, possibility/probability distributions 
over this interval. 
Spelled out in greater detail, the process of meaning-representation in test-score 
semantics involves three distinct phases. In Phase 1, an explanatory database 
frame or EDF, for short, is constructed. EDF consists of a collection of rela- 
tional frames each of which specifies the name of a relation, the names of its at- 
tributes and their respective domains,with the understanding that the meaning of 
each relation in EDF is known to the addressee of the meaning-representation pro- 
cess. Thus, the choice of EDF is not unique and is strongly influenced by the de- 
sideratum of explanatory effectiveness as well as by the assumption made regarding 
the knowledge profile of the addressee of the meaning-representation process. For 
example, in the case of the proposition p 4 During much of the past decade Pat --- -- 
earned far more than all of his close friends put together, the EDF might consist ---7----- ----- 
of the relational frames 
FRIEND [Namel; Name2; p], where p is the degree.to which Name1 is a friend of 
Name2; INCOME [Name; Income; Year], where Income is the income of Name in year 
Year, counting backward from the present; MUCH [Proportion; ~1, where in is the de- 
gree to which a numerical value of Proportion fits the meaning of much in the con- 
text of p; and FAR.MORE [Numberl; Number2; P]. in which p is the degree to which 
Number1 fits the description far more in relation to NumberP. In effect, the com- 
position of EDF is determined by the information that is needed for an assessment 
of the compatibility of the given SE with any designated object or, more generally, 
a specified state of affairs in the universe of discourse. 
In Phase 2, a test procedure is constructed which upon application to an explan- 
atory database -that is, an instantiation of EDF - yields the test scores, 
428 L.A. ZADEH 
Tl,... 
bv the 
3Tn, which represent the degrees to which the elastic constraints induced 
constituents of SE are satisfied. For example, in the case of p', the test 
ppocedure would yield the test scores for the constraints induced by close friend, 
@, far more, etc. -- 
In Phase 3, the partial test scores obtained in Phase 2 are aggregated into an 
overall test score, T, which serves as a measure of the compatibility of SE with 
ED, the explanatory database. As was stated earlier, the components of T are num- 
bers in the unit interval or, more generally, possibility/probability distributions 
over this interval. In particular, when the semantic entity is a proposition, p, 
and the overall test score, T, is a scalar, 'I may be interpreted as the truth of 
p relative to ED or, equivalently, as the possibility of ED given p. In this in- 
terpretation, then, the classical truth-conditional semantics may be viewed as a 
special case of test-score semantics which results when the constraints induced 
by p are inelastic and the overall test score is allowed to be only pass or fail -- 
The test procedure which yields the overall test score T is interpreted as the 
meaning of SE. 
To illustrate the phases in question, we shall consider a few simple examples 
(a) SE 4 Ellen resides in a'small city near Oslo. 
In this case, EDF is assumed to comprise the following relational 
frames ( + stands for union ): 
EDF 4 RESIDENCE [Name; City.Name]+ 
POPULATION [City.Name; Population]+ 
SMALL [Population; PI+ 
NEAR [City.Namel; City.Name2; lo] 
In RESIDENCE, City.Name is the name of the city in which Name resides; in POPULA- 
TION, Population is the number of residents in City.Name; in SMALL, P is the de- 
gree to which a city with a population equal to the value of Population is small; 
and in NEAR, P is the degree to which City.Namel is near City.NameZ. 
The test procedure which leads to the overall test score T -- and thus represents 
the meaning of SE - is described below. In this procedure, Steps 1 and 2 involve 
the determination of the value of an attribute given the values of other attri- 
butes; Steps 3 and 4 involve the testing of constraints; and Step 5 involves an 
aggregation of the partial test scores into the overall test score T. 
1. Find the name of the residence of Ellen: 
RE! c,ty RameRESIDENCEIName=Ellen] 
which means that the value of Name is set to Ellen and the value of City.Name is 
read, yielding RE, the residence of Ellen. 
2. Find the population of the residence of Ellen: 
n 
PRE = Population PDPULATION[City.Name=RE] 
3. Test the constraint induced by SMALL: 
r,guSMALLIPopulation=RE] 
where ~~ denotes the resulting test score. 
4. Test the constraint induced by NEAR: 
T2=uNEAR[City.Name=Oslo; City.Name2=RE] 
5. Aggregate ~~ and T2: 
T = T1 ,. T2 
where A stands for min in infix position, and T is the overall test score. This 
'TEST-SCORESEMANTICSFORNATURALLANGUAGES 429 
mode of aggregation implies that, in SE, the denotation of conjunction is taken to 
be the Cartesian product of the denotations of the conjuncts (Zadeh (1981)). 
(b) SEA During much of the past decade Pat earned far more than all of 
his close friends put together. 
In this case, we shall employ the EDF described earlier, that is: 
EDF 2 INCOME[Name; Year; Amount]+ 
FRIEND[Namel; Name2; u]+ 
FAR.MORE[Numberl; NumberP; n]+ 
MUCH[Proportion; u] 
The test procedure comprises the following steps: 
1. Find the fuzzy set of Pat's friends: 
FP4 NamelxnFRIEND [Name2 = Pat] 
in which the left subscript Namelxu signifies that the relation FRIEND [NameP=Pat] 
is projected on the domain of the attributes Name1 and u, yielding the fuzzy set of 
friends of Pat. 
2. Intensify FP to account for the modifier.*: 
CFP 4 FP2 
in which FP* denotes the fuzzy set which results from squaring the grade of member- 
ship of each component of FP. The assumption underlying this step is that the 
fuzzy set of close friends of Pat may be derived from that of friends of Pat by in- 
tensification. 
3. Find the fuzzy multiset of incomes of close friends of Pat in year 
Year; , i=l,...,lO: 
ICFP. 4 , AmountINCOMEIName = CFP; Year=Yeari] 
In stipulating that the right-hand member be treated as a fuzzy multiset, we imply 
that the identical elements should not be combined, as they would be in the case of 
a fuzzy set. With this understanding, ICFPi will be of the general form 
ICFPi = 6,/e1+6,/e+...+6,/em . 
where el,..., e m are the incomes of Name ..., Name 1' m, 
respectively, in Year., andsl,...,,m 6 are the grades of membership of Name,;.-, 
Namem in the fuzzy se t of close friends of Pat. 
4. Find the total income of close friends of Pat in Yeari , i=l;.., 10: 
TICFPi =6,el+...+ 6 e mm 
which represents a weighted arithmetic 
in Yeari. 
5. Find Pat's income in Yeari: 
IPi 4AmountINCOME[Name=Pat; 
6. Test the constraint induced 
sum of the incomes of close friends of Pat 
Year=Yeari]. 
by FAR.MORE: 
ri$pFAR.MOREINumberl=IPi; Nurnber2= TICFPi] 
7. Find the sigma-count (Zadeh (1981)) of years during which Pat's income 
was far.greater than the total income of all of his close friends: 
c iq'.; 
, 
430 L.A. ZADEH 
a. Test the constraint induced by MUCH: 
TePMUCH[Proportion=C] 
'16 
where T represents the overall test score. 
The two examples described above are intended merely to provide a rough outline 
of the meaning-representation process in test-score semantics. A more detailed ex- 
position of some of the related issues may be found in Zadeh (1978) and Zadeh 
(1981) 
'Research supported in part by the NSF Grants MCS79-06543 and IST-801896. 

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