Antonyms and linguistic quantifiers in fuzzy logic

Fuzzy Sets and Systems 124 (2001) 335–351

www.elsevier.com/locate/fss

Antonyms and linguistic quanti#ers in fuzzy logic  Vil%em Nov%aka; b; ∗ a University

b Institute

of Ostrava, IRAFM, 30. dubna 22, 701 03 Ostrava, Czech Republic of Information and Automation Theory, Academy of Sciences of the Czech Republic, Pod vod'arenskou v)ez)'* 4, 186 02 Praha 8, Czech Republic Received June 2000; received in revised form May 2001

Abstract The paper is a contribution to the theory of fuzzy logic in broader sense (FLb), namely the discussion of linguistic expressions fundamental for it—the evaluating linguistic predications, the pairs “nominal syntagm–antonym”, and the theory of linguistic quanti#ers. The aim is to develop a theory of natural human reasoning, whose characteristic feature is the use of natural language. Formalism of FLb is based on the theory of fuzzy logic in narrow sense with evaluated syntax, which provides us means for modelling of the concepts of intension, possible world and extension. Characterization of some of the main properties of the above expressions is provided. We also propose a modi#ed de#nition of the linguistic variable. c 2001 Elsevier Science B.V. All rights reserved.
Keywords: Fuzzy logic in narrow and broader sense; Antonyms; Generalized quanti#ers; Intension; Extension; Possible world

1. Introduction The paper is a discussion of two main topics raising in connection with the attempt to use formal theory of fuzzy logic in the model of semantics of limited part of natural language. The #rst topic is the theory of the, so called, evaluating linguistic predications—the expressions such as “temperature is high”, “pressure is very low”, etc. The second topic is the theory of  This paper has been supported by the project VS96037 of A MSMT of the Czech Republic. ∗ Correspondence address: University of Ostrava, IRAFM, 30. dubna 22, 701 03 Ostrava, Czech Republic. Tel.: +420-266052874; fax: +420-2-821227. E-mail address: [email protected] (V. Nov%ak).

linguistic quanti5ers, which are words such as “many, most, a lot of, a few, a little”, etc. and which are used together with the linguistic predications. Both kinds of expressions are fundamental for the theory of fuzzy logic in broader sense (FLb) whose aim is to develop a theory of natural human reasoning. Recall that the characteristic feature of the latter is the use of natural language. Therefore, a model of its semantics is necessary. The technical background is rendered by fuzzy logic in narrow sense with graded syntax (FLn) (presented, e.g., in [12] and elsewhere), namely by its many-sorted predicate version. This, among others, makes possible to formalize the concepts of intension, extension and possible world. Intension of the linguistic expression is a representation of the property denoted by it. One

c 2001 Elsevier Science B.V. All rights reserved. 0165-0114/01/$ - see front matter  PII: S 0 1 6 5 - 0 1 1 4 ( 0 1 ) 0 0 1 0 4 - X

336

V. Nov'ak / Fuzzy Sets and Systems 124 (2001) 335–351

intension then leads to a class of extensions, each of them being a grouping of objects having the property in concern in some possible world. In our formalism, the intension is a set of evaluated instances of some formula of FLn. This determines a fuzzy theory, which if being consistent, has models. Some speci#c ones are taken as possible worlds. The extension is then a satisfaction fuzzy set of the given formula in a model—a possible world. A part of our theory is the theory of pairs of simple evaluating syntagms “nominal adjective—antonym”. They are formalized using the concept of “horizon”, being mathematized using a predicate L(x) ful#lling speci#c axioms, special functional symbol  for order-reversing automorphism, and a set of additional unary connectives {/ |  ∈ }. The meaning of a nominal adjective and its antonym are multiformulas constructed from the simple formulas /(L(x)) and /(L((x))), respectively. The meaning of the middle term (i.e. “medium”) is then de#ned on the basis of the formula /(¬L(x)∧¬L((x))). We will demonstrate that under axiomatics stemming from linguistic considerations, we can prove some reasonable properties of fuzzy theories containing such formulas and, following from that, the linguistically formulated knowledge bases. The theory of linguistic quanti#ers roots to the theory of generalized quanti#ers as proposed by A. Mostowski. Though a nice theory, its drawback is that no vagueness is involved in it. There is a straightforward way how generalized quanti#ers can be introduced in fuzzy logic. However, they provide means for determination of the truth value (in a possible world) only and give no way how to rely on the above concept of intension. Therefore, we propose to model linguistic quanti#ers using the concept of formula-based fuzzy quanti#ers (FB-fuzzy quanti#ers), which are obtained formally using a universally (existentially) closed formula together with a certain simple formula. This enables us to characterize both extension as well as the truth of quanti#ed linguistic expressions. The obtained theory seems to comply with the intuition. The paper is organized as follows. In the next section, we brieNy overview the main concepts of fuzzy logic in narrow and broader sense. In Section 3 we introduce a special fuzzy theory of evaluating syntagms T EV , which contains also the theory of the pairs of

antonyms, as well as canonical model of T EV . Among others, we also introduce a modi#ed de#nition of the linguistic variable. Section 4 is devoted to the theory of linguistic quanti#ers and their formalization in FLb. The theory of FB-quanti#ers is included there. 2. Preliminaries 2.1. Mathematical preliminaries The set of truth values is denoted by L. In general the structure L of truth values on L is supposed to be a residuated lattice and more speci#cally, an MValgebra. In this paper, however, we will suppose that L is the Lukasiewicz algebra, i.e. that L = [0; 1] with the usual Lukasiewicz operations of conjunction ⊗, disjunction ⊕, implication → and negation ¬ (see, e.g. [12]). We will work with a many-sorted predicate language of fuzzy logic J . The set of sorts of J is denoted by #. We suppose it to be #nite. By Q we denote a type, which is some #nite sequence of sorts from #. A predicate P as well as a function f symbols from J may be of type Q, or untyped. By M we denote a set of closed terms of the sort . By FJ we denote the set of all the well formed formulas from J . If the sort of variable is not important then we will write a formula A(x) without stressing a speci#c sort of x. Recall that a=A is an evaluated formula where A ∈ FJ is a formula and a ∈ L is its evaluation. The other concepts, such as fuzzy theory, fuzzy sets of logical formulas, logical constants, etc. can be found in [12], and elsewhere. Let A(x) be a formula of the language J , x is a sequence of free variables of the sorts from the corresponding type Q = 1 ; : : : ; n . Then the multiformula is a set of instances of the evaluated formulas A
x = {at =Ax [t] | t ∈ M};

(1)

where M = M 1× · · · ×M n is a set of sequences of closed terms of the same sort as variables from x. A structure D for the language J is D = {D | ∈ #}; PD; Q; : : : ; fD; Q; : : : ; {u ; : : : | ∈ #} ; where PD; Q ⊂ D 1× · · · ×D n are fuzzy relations as∼

signed to the n-ary predicate symbols P ∈ J of the type

Q = 1 ; : : : ; n . If P is an untyped n-ary predicate

V. Nov'ak / Fuzzy Sets and Systems 124 (2001) 335–351

symbol then it is assigned a fuzzy relation PD ⊂ ∼


∈# D . Furthermore, if f is a functional symbol of the type

Q then fD; Q is a function fD; Q : D 2 × · · · × D n → D 1 : The constants from the language J are assigned elements from the support of D in accordance with their sorts. The (many-valued) interpretation of formulas from J in a structure D are de#ned in a usual manner. If A ∈ FJ is a formula then D(A) ∈ L is a truth value assigned to it in the structure D. Given a structure D, we extend (temporarily) the language J into the language J  =J ∪{d |d ∈D ; ∈#} where are d new constants being names of elements from the set D corresponding to the sort . The satisfaction fuzzy set of the formula A is the fuzzy set Q SatD (A; x) = {D(Ax [d])= d

1 ; : : : ; d n | d 1 ∈ D 1 ; : : : ; d n ∈ D n }

(2)

of elements satisfying the formula A in the correspondQ where dQ is a sequence of the ing degrees D(Ax [d]) constants d of various sorts corresponding to the sorts of the variables from x. Note that the formula A may in general have more free variables than are contained in x. We will alternatively also write D(A(x=d)) for the truth value of A(x) after the variable x is assigned the element d ∈ D (for some de#nite sort). In the sequel, we will often con#ne ourselves only to one variable of some sort. If there is no danger of misunderstanding, we will omit the subscript for its sort. Similarly, the symbol M will denote the set of closed terms of some sort. We will also work with (n-ary) additional connectives o. They are supposed to be logically #tting, i.e. if o is the operation interpreting them then there are m1 ; : : : ; mn such that (x1 ↔ y1 )m1 ⊗ · · · ⊗ (xn ↔ yn )mn 6 o(x1 ; : : : ; xn ) ↔ o(y1 ; : : : ; yn )

(3)

holds for all xi ; yi ∈ L; i = 1; : : : ; n. It can be proved that (3) means that o is Lipschitz continuous (for the Lukasiewicz algebra—see [5]).

337

By F(U ) we denote the set of all fuzzy sets in the universe U . If f : A → R is a real function de#ned on some subset A ⊆ R then f∗ denotes the function  f(x) if f(x) ∈ [0; 1];    ∗ if f(x) ¿ 1; f (x) = 1    0 if f(x) ¡ 0: 2.2. Linguistic preliminaries In this paper, we will work with the so called evaluating syntagms (see [12]). These are special expressions of natural language, which characterize sizes, distances, etc.—in general, they characterize a position on an ordered scale. Among them, we distinguish atomic evaluating syntagms and predications which consist of atomic evaluating syntagm being any of the adjectives “small”, “medium”, or “big”, or fuzzy quantity “approximately z”, which is a linguistic expression characterizing some quantity z from an ordered set, and simple evaluating syntagms, which are expressions of the form linguistic hedge atomic evaluating syntagm : Examples of fuzzy quantities are thirty two, the value z, etc. Simple evaluating syntagms are very small, more or less medium, roughly big, about twenty 5ve, approximately z, etc. Let us stress, that the atomic syntagms “small, medium, big” should be taken as canonical for a lot of other corresponding words, such as “short, average, long”, “deep, medium deep, shallow”, etc. Atomic evaluating syntagms usually form pairs of antonyms, i.e. the pairs nominal adjective – antonym : Of course, there are a lot of pairs of antonyms, for example “ young–old”, “ ugly–nice”, “stupid–clever”, etc. When completed by the middle term, such as “ medium”, “average”, etc., they form the so called basic linguistic trichotomy. A special case of syntagms are evaluating linguistic predications, which are linguistic expressions of the form noun is A

(4)

338

V. Nov'ak / Fuzzy Sets and Systems 124 (2001) 335–351

where A is an evaluating syntagm. If A is a simple evaluating syntagm then (4) is a simple evaluating predication. Examples of simple evaluating predications are, e.g. “temperature is very high” (here “high” is taken instead of “big”), “pressure is roughly small”, “income is roughly three million”, etc. Important syntagms is the conditional clause

Recall that a multiformula being an intension of a natural language expression is a set of evaluated formulas. But then it can be understood as a fuzzy set of special axioms of some fuzzy theory. Hence, each set T of natural language expressions

C := IF A THEN B

T = {Ai | i ∈ I }

(5)

and the compound evaluating predication C := A AND B;

(6)

where A, B are evaluating linguistic predications. All the above discussed expressions of natural language (evaluating syntagms, predications, etc.) form a certain part S of natural language, which will be formalized using the means of fuzzy logic. For this, we suppose existence of translation rules using which items of the syntax of fuzzy logic to the expressions from S and its constituents. More concretely, variables are assigned to the nouns and formulas to the adjectives and more complex syntagms. Let us assume that the basic assignment has already been done. Then we may introduce the concepts of intension, extension and possible world. The latter is taken as a speci#c structure D for J . We usually suppose that all the fuzzy relations assigned to predicate symbols of J in the possible world D have continuous membership function. Moreover, some further assumptions D can be made, for example the unimodality of some membership functions, speci#c topological structure de#ned on the support D, etc. Denition 1. Let A ∈ S be a natural language expression having been assigned a formula A(x). (i) The intension of A is the multiformula (1) Int(A) = A
x :

(7)

(ii) The extension of A in the possible world D is the satisfaction fuzzy set (2), i.e. Ext D (A) = SatD (A; x):

(8)

(iii) The meaning of A is the couple Mean(A) = Int(A); Ext(A) ;

(9)

where Ext(A) = {Ext D (A) | D is a possible world} is a class of all its extensions.

(10)

(I is some index set) will be called a theory of FLb. Each theory of FLb (10) is assigned a fuzzy theory T = {Ai | i ∈ I };

(11)

where Ai = Int(Ai ) are multiformulas being intensions of Ai ∈ T; i ∈ I , thus serving as fuzzy sets of special axioms of T . A speci#c expression of natural language is the “typical example”. Let us consider an evaluating predication A with the intension A
x . Then the linguistic expression TExm(A) := “typical example of A” has the intension TExm(A
x) = {1=Ax [u0 ]}

(12)

for some term u0 such that the evaluation of a0 =Ax [u0 ] ∈ A
x is either a0 = 1 (1 is the unit of MValgebra of truth values), or it can be in some sense “signi#cantly big” value. Determination of u0 , however, is out of logic. There are algorithms for solution of this task, implemented, e.g. in the software system LFLC (cf. [9]). We will also need the special syntagm “utmost” which has the following meaning: Mean(Utmost) = {1=Lx [u1 ]}; {{1="(D)} | D is a possible world} ;

(13)

where "(D) is a measure of the support D, usually set to be equal to 1. 3. Evaluating syntagms in FLb In this section, we will present a formalization of the meaning of evaluating syntagms in FLb. We start with de#nition of a many-sorted language of special fuzzy theory T EV . Then we will specify fuzzy sets of

V. Nov'ak / Fuzzy Sets and Systems 124 (2001) 335–351

339

its special axioms, and also its canonical model. This theory is further used for modeling of the meaning of the evaluating syntagms. In this paper, we focus especially on the simple evaluating syntagms.

The hedge /0 will be called central. Then the hedges /−k ; : : : ; /−1 will be called the hedges with narrowing e?ect and /1 ; : : : ; /p the hedges with widening e?ect.

3.1. Language, special axioms and canonical model of T EV

Denition 2. The fuzzy set of special axioms SAxEV of the fuzzy theory T EV consists of the following evaluated formulas:

The language J of the fuzzy theory of evaluating syntagms T EV is many-sorted predicate language J = {6; L; # = {/ |  ∈ }; {u ;a | a ∈ [0; 1]; ∈ #}; }; where 6 is an untyped predicate symbol for classical ordering, L is an untyped unary predicate called the left horizon predicate,  is a unary functional symbol, which will represent an order-reversing automorphism and u ; a are special constants using which we will specify intensions of the above considered linguistic syntagms. Finally, / ,  ∈  ( is a #nite set of subscripts) are unary connectives interpreted by logically #tting (i.e. Lipschitz continuous) functions. We will denote them by the corresponding non-bold symbols / . The classical equality = is supposed to be an untyped predicate ful#lling all the classical equality axioms in the degree 1, which is also crisp, i.e. it ful#lls the axiom 1=(∀x)(∀y)(x = y ∨ x = y): The connectives / will be assigned to linguistic hedges (“very, roughly, signi#cantly”, etc.). Recall that L. A. Zadeh proposed to model their meaning using certain transformations of the fuzzy sets, which represent the meaning of the words with linguistic hedges standing before them. From our point of view, the main principle of the model of hedges consists in shifting of the horizon. This is achieved by application of certain unary connective. In general, we speak about the linguistic hedges with narrowing effect (very, highly, etc.) and those with widening e?ect (more or less, roughly, etc.). We will suppose a more subtle structure of the set of connectives #, namely # = {/−k ; : : : ; /−1 ; /0 ; /1 ; : : : ; /p }:

(i) The classical axioms for crisp binary linear ordering 6 in the degree 1. Except for the ordinary axioms, the following axiom must also hold: (Ei1) 1=(∀x)(∀y)(x6y∨¬(x6y)). (ii) The axioms on constants u ; a : (Eii1) 1=(∀x)((u ; 0 6x)&(x6u ; 1 )), (Eii2) {1=u ; a 6u ; b | a6b; a; b ∈ [0; 1]}. Among the constants, we will distinguish three constants u ; 0 ; u ; c0 ; u ; 1 for each sort ∈ # where 0¡c0 ¡1 is a #xed element. The constant u ; c0 for each sort will be called the semantic center. (iii) The axioms specifying the left horizon predicate L: (Eiii1) 1=(∀x)(∀y)(x6y ⇒ (L(y) ⇒ L(x))), (Eiii2) 1=L(u ; 0 ), (Eiii3) 1=¬L(u ; c0 ), (Eiii4) {(1−(b−a)=c0 )∗ =(Lx [u ; a ] ⇒ Lx [u ; b ]) | a; b ∈ [0; 1]}. 1 (iv) Classical axioms to determine the function assigned to  to be the order reversing automorphism: (Eiv1) 1=(∀x)(∃y)((y = (x))∧(∀y)(∃x) (y = (x))), (Eiv2) 1=(∀x)(∀u)(∀v)((u = (x)∧v = (x)) ⇒ u = v), (Eiv3) 1=(∀x)(∀y)(x6y ⇒ (y)6(x)), (Eiv4) 1=(∀x)(x = ((x))), (Eiv5) 1=((u ; c0 ) = u ; c0 ). (v) The axiom of linguistic hedges: (Ev1) {1=(/ (¬A) ⇒ ¬/ (A)) | 60}. (vi) The axioms taken from the de#nition of intension of the basic linguistic trichotomy (see Definition 5). 1 It is clear that the axioms 1=(L [u ] ⇒ L [u ]) for b6a, x ; a x ; b can be derived from the #rst one of (iii). However, our aim is to characterize the fuzzy theory for simple evaluating syntagms disregarding the optimality of the set of speci#ed axioms.

340

V. Nov'ak / Fuzzy Sets and Systems 124 (2001) 335–351

Finally, we will introduce the inference rule of linguistic hedge introduction rLH :

a=A ; / (a)=/ (A)

 ∈ :

(14)

It is not diUcult to see that the rule rLH is sound. Before we will continue with speci#cation of the meaning of evaluating syntagms in FLb, we will present the completeness theorem for fuzzy logic extended by the above unary connectives. Theorem 1 (Completeness). Let J be a language extended by the logically 5tting unary connectives / ∈  and the inference rule rLH . Then T a A i? T |=a A: for every fuzzy theory T and a formula A ∈ FJ (T ) . Proof. This follows from the assumptions and [12, Corollary 4:6]. By this theorem, we can use freely the introduced unary connectives / without harming the completeness property. The following lemma assures a correct behaviour of the new connectives with respect to complete fuzzy theories. Lemma 1. Let T be a complete fuzzy theory in the language J containing the unary connectives / ∈ # and the rule rLH . Then there is a canonical model D0 such that T a A i? D0 (A) = a holds for every A ∈ FJ (T ) . Proof. For formulas not containing the unary connectives is the proof identical with the proof of Theorem 4:18 from [12]. Hence, it only suUces to check the induction step for A := / (B). By the induction assumption, we have T  B ⇔ b for T b B and D0 (B) = b. But then D0 (A) = / (D0 (B)) = / (b). At the same time, since T is complete then from the #tting axiom we conclude that T  A⇔ / (b) which implies a = / (b).

The following are special formulas using which we will later introduce mathematical model of the meaning of simple evaluating syntagms. Denition 3. (i) Basic nominal formula W (x) := / (L(x)): (ii) Antonym of basic nominal formula Ant W (x) := W ((x)) := / (L((x))): (iii) Middle member MW (x) := / (¬L(x)∧¬L((x))): The formulas de#ned due to items (i) – (iii) as well as their instances will be called simple. Let us now introduce the canonical structure E for the language of evaluating syntagms. Denition 4. (a) The canonical structure for the language J (T EV ) is E = {E | ∈ #}; 6; LE ; #E = {/−k ; : : : ; /−1 ; /0 ; /1 ; : : : ; /p }; E ; {0 ; c 0 ; 1 | ∈ #} ;

(15)

where the items of (15) are de#ned as follows: (i) E = [0; 1]; ∈ #. (ii) The 6 is an untyped ordinary
linear inequality relation, i.e. it is de#ned in ∈# E = [0; 1] in a classical way and assigned to the inequality symbol 6. Similarly, the = is assigned classical equality relation. (iii) The constants u ; a ; a ∈ [0; 1] are assigned elements e ∈ E = [0; 1] for every sort ∈ #. The distinguished constants u ; 0 ; u ; c0 ; u ; 1 are assigned the elements 0 ; c 0 ; 1 , respectively, where 0 ¡c 0 ¡1 . Since in the canonical model, they are taken from [0; 1] for each sort equally, we will usually omit the subscript for the sort.
(iv) The LE ⊂ [0; 1] is a fuzzy set LE ⊂ ∈# E de∼

#ned by

 x LE (x) = 0 ∨ 1 − 0 ; c

∼

x ∈ [0; 1]

V. Nov'ak / Fuzzy Sets and Systems 124 (2001) 335–351

for every sort and assigned to the untyped predicate symbol L. (v) The operations / : [0; 1] → [0; 1],  ∈  are Lipschitz continuous functions with the following properties: there are a ; b ∈ [0; 1] such that 0¡a ¡b ¡1: / (x) = 0;

x ∈ [0; a ];

/ (x) = 1;

x ∈ [b ; 1]

iV a/ 6 a 2 :

/ (x) = (−x2 + k1 x − k2 )∗ ;

(16)

where k1 ; k2 are constants set properly to ful#l the previous conditions (see below). (vi) The symbol  is assigned an order-reversing automorphism E on [0; 1] with the #xed point c0 . More speci#cally, we will take

E (x) =

0    c (1 − x) ; 1 − c0

Int(Ant W) := Ant W0;
x

  a−c /0 (Lx [(ua )]) | a ∈ [0; 1] : = /0 0 ∨ 1−c (iii) The middle syntagm is assigned a middle member formula MW0 (x). Then its intension is

More speci#cally, we will put

 0 c (x + 1) − x   ;  c0

Denition 5. (i) The nominal syntagm is assigned a basic nominal formula W0 (x). Then its intension is

  c−a Int(W) : W0;
x = /0 0 ∨ c /0 (Lx [ua ]) | a ∈ [0; 1] : (ii) The antonym of nominal syntagm is assigned an antonym of basic nominal formula AntW0 (x). Then its intension is

and / (x) is strictly increasing on [a ; b ]. Furthermore, we de#ne 6/

341

x ∈ [0; c0 ]; (17) 0

x ∈ [c ; 1]:

(b) The possible world is any structure isomorphic to the canonical one. 3.2. The meaning of evaluating syntagms and predications

Int(MW) := MW0;
x

  a 1−a = /0 1 ∧ ∧ c 1−c

/0 (¬Lx [ua ]∧¬Lx [(ua )]) | a ∈ [0; 1] :

(iv) Let C := ‘ linguistic hedge A’ be a simple evaluating syntagm where the linguistic hedge is assigned the hedge / ∈ #\{/0 } and the intension A of its atomic evaluating syntagm A is one of those given in (i) – (iii). Then the intension of C is a multiformula obtained from A by replacing the hedge /0 by / . The canonical linguistic trichotomy is W := “small”;

Ant W := “big”;

MW := “medium”:

3.2.1. Intension of the basic linguistic trichotomy Using the results from the previous section, we can now de#ne intensions of the atomic evaluating syntagms forming the basic linguistic trichotomy.

Of course, there are a lot of other linguistic trichotomies, which the reader may de#ne in addition to these three ones. The de#nition of the intension of its atomic syntagms should be similar to the above ones. The evaluated formulas from De#nition 5 complete fuzzy set of special axioms of the fuzzy theory T EV .

2 It might also be required that b 6b holds at the same time.  / Till now, however, it is not completely clear whether this condition is necessary or not. The functions interpreting the concrete linguistic hedges are set to ful#l both.

Lemma 2. (a) T EV  (u 0 )=u1 and T EV  (u1 )=u 0 . (b) T EV  (∀y)(uc0 6y ⇒ ¬L(y)). (c) T EV  (∀x)(x6uc0 ⇒ uc0 6(x)).

342

V. Nov'ak / Fuzzy Sets and Systems 124 (2001) 335–351

Proof. (a) From the axioms about constants, we have T  u0 6(u1 ), which implies T  ((u1 ))6(u0 ). But this gives T  u1 6(u0 ) and part (a) of lemma follows. (b) Using the axioms for the left horizon and contraposition, we have T  (∀x)(∀y)(x6y ⇒ (¬L(x) ⇒ ¬ L(y)). Then taking its instance for uc0 and the axiom (Eiii3) we obtain the lemma. (c) follows from the axioms (Eiv3) and (Eiv5) using the equality theorem. The following lemma will be used later. Lemma 3. Let T be an extension of the fuzzy theory T EV such that

Therefore, noun is assigned some sort of variables x; y; : : : occurring in the formulas. Denition 6. (i) Let C := ‘ noun is A’ be an evaluating predication, x be a variable of the sort assigned to noun and the intension of the evaluating syntagm A is the multiformula A
x . Then the intension of C is the multiformula C
x := A
x : (ii) The intension of the conditional clause (5) is C
x;y = A
x ⇒B
y = {at → bs =Ax [t]⇒By [s] | t ∈ M 1 ; s ∈ M 2 }

T b (∃x)(W (x) & Ant W (x)) for some 60 and b¿0. Then T is contradictory. Proof. Note that the assumption implies that T e1 / (L(x)) and T e2 / (L((x))) for some e1 ; e2 ¿0. Then the proof may be split into two possibilities. (a) Let T  x6uc0 . From Lemma 2(c) and (b) we obtain T  ¬L((x)). Using this, the rule rLH and axiom (Ev1) we conclude that T  ¬(/ (L((x))) and thus, T e2 / (L((x))) & ¬(/ (L(x))); i.e. T is contradictory. (b) Let T  uc0 6x. In the same way as above we prove T e1 / (L(x)) & ¬(/ (L(x)); i.e. T is again contradictory. Lemma 4. The structure E speci5ed in De5nition 4 is a model of the fuzzy theory T EV ; i.e. E |= T EV . Proof. The proof is a technical exercise where we must verify that all the axioms from De#nitions 2 and 5 are true in E, at least in the corresponding degree. This is left to the reader. 3.2.2. Intension and extension of evaluating predications The #rst constituent of the evaluating predications (4) is a noun. However, in the syntax of our logic, we have no means how to specify the objects concretely.

(18) (iii) The intension of the compound evaluating predication (6) is C
x;y = A
x ∧B
y = {at ∧ bs =Ax [t]∧By [s] | t ∈ M 1 ; s ∈ M 2 }; (19) where A
x and B
y are intensions of the corresponding linguistic predications A; B. Of course, we can also de#ne intension of other compound evaluating predications. We do not need them in this paper. The intension of some linguistic expression leads to an extension, which can be understood as its interpretation in a possible world. Recall that in our theory, we have identi#ed possible worlds with some speci#c structures for the formal language J and thus, the extension is, in general, a fuzzy relation (2). Having speci#ed the intensions of simple evaluating predications (syntagms), their extension in the above speci#ed canonical model is depicted on Fig. 1. The corresponding membership functions In this #gure, the curves have been obtained using the values of the constants k1 ; k2 in accordance with the linguistic modi#ers speci#ed in Table 1. For better orientation, membership functions of the three main evaluating expressions of each kind are pointed in the #gure.

V. Nov'ak / Fuzzy Sets and Systems 124 (2001) 335–351

343

Fig. 1. Shapes of the membership functions representing extensions of the simple evaluating predications in a canonical model. The curves on the left hand side correspond to extremely, signi5cantly and very small, small, more or less, roughly, quite roughly and very roughly small, similarly for big on the right hand side, and medium, more or less, roughly, quite roughly and very roughly medium in the middle.

Table 1 Assignments of values of constants to the concrete linguistic hedges

Denition 7. The linguistic variable is a tuple

Modi#er

k1

k2

Extremely Signi#cantly Very — More or less Roughly Quite roughly Very roughly

3.9 3.7 3.5 3.0 2.8 2.7 2.6 2.8

1.9 1.6 1.35 0.85 0.65 0.5 0.35 0.2

where T (X) ⊂ S is a set of evaluating predications (4) containing the same noun , which is the name X of the linguistic variable, G is a syntactic rule according to which the evaluating syntagms are constructed, M is a set of closed terms of the sort assigned to noun ,

3.2.3. Linguistic variable The above de#nitions enable us to present a slightly modi#ed concept of the linguistic variable in comparison with the de#nition originally given by L. A. Zadeh (see [14]). Our de#nition takes the concept of intension and possible world into consideration. The signi#cant diVerence lays in the semantical part since the original de#nition is purely extensional considering some universe U .

X := noun ; T (X); G; M ; P; M ;

P = {D | D is a possible world}: is a class of possible worlds and M is a semantical rule assigning to each evaluating predication A ∈ T (X) its meaning (10). Recall that the possible world is understood to be a speci#c structure D for the language J (cf. our discussion above). 3.3. Generalized inference 3.3.1. Linguistic description An important question studied in fuzzy logic is, how some kind of dependence can be linguistically

344

V. Nov'ak / Fuzzy Sets and Systems 124 (2001) 335–351

described. We have (at least) two possibilities. First, it may be described using a set of conditional clauses (5), which are implications characterized using natural language. In this case, also cases when the antecedent predications may not be true are taken into account. However, we may consider only positive cases of the dependence and then, the dependence is described using a set of linguistically characterized conjunctions (compound evaluating predications). We will speak about linguistic description of the dependence and put forth the following de#nition. Denition 8. The linguistic description in FLb is one of the following sets: (i) A #nite set LDI = {R1I ; : : : ; RmI } of conditional clauses (5) with the intensions (18). (ii) A #nite set LDA = {R1A ; : : : ; RmA } of compound evaluating predications (6) with the intensions (19). Let us remark that both kinds of the linguistic expressions occurring in the linguistic descriptions LDI as well as LDA are usually called fuzzy IF–THEN rules. Therefore, the left part of each rule (the evaluating predication Ai ) is called antecedent and the right part (the evaluating predication Bi ) succedent. A linguistic description in which IF–THEN rules consist of simple evaluating predications is called simple. In the next section, we focus only on the simple linguistic description LDI and reasoning based on it. For the discussion of reasoning based on LDA —see [12, Chapter 6]. 3.3.2. Inference based on linguistic description We assume that a simple linguistic description LDI consists of m IF–THEN rules. Furthermore, let a simple evaluating predication A be given. This may not be completely arbitrary since otherwise, the inference might hardly have any sense. In general, A may be a modi#cation of the antecedent Ak of some rule Rk from LDI . More precisely, the intension A = Int(A ) may diVer from the intension Ak = Int(Ak ) in the evaluations but not in the corresponding formulas, i.e. if at =Ax [t] ∈ A and akt =Ak; x [t] ∈ Ak then Ak; x [t] = Ax [t]. As a special case, A = TExm(Ak ) with the intension (7).

The following theorem is a slight modi#cation of Theorem 6:1 from [12]. Theorem 2. Let LDI be a simple linguistic description and A be the above speci5ed evaluating predication. Let T = {A ; LDI } be a theory of FLb. Then we may derive a conclusion B with the intension    

 = bs = (at ⊗ cts )=By [s]|s ∈ M 2 ; (20) B
y   t∈M 1

where cts = akt → bks and all bs for s ∈ M 2 in the mul tiformula B
y are the provability degrees of By [s] in the fuzzy theory T adjoint to T. Proof. The proof proceeds analogously as the proof of Theorem 6:1 from [12] since the evaluating predications lead to sets of independent formulas in the sense assumed there. By this theorem, if the IF–THEN rules are formed of the simple evaluating predications then the deduction based on them leads to the conclusion, which is the best possible one—in the sense of maximization of the provability degrees within the adjoint fuzzy theory T . Theorem 3. Let T = {A ; LDI } be a theory of FLb and T be its adjoint fuzzy theory where A := Ak ; or A := TExm(Ak ). Let LDI contain rules of the form IF Ak THEN W;k IF Ak THEN Ant W;k for 60. Then the fuzzy theory T adjoint to T is contradictory. Proof. Due to de#nition of the intension of the evaluating predications, there are terms t; s such that

V. Nov'ak / Fuzzy Sets and Systems 124 (2001) 335–351

1=Ax [t] ∈ A and the evaluations of cts =(Ak;x [t]⇒W;k;y [s]) ∈ Ak ⇒W;k ; ets =(Ak;x [t]⇒Ant W;k;y [s]) ∈ Ak ⇒Ant W;k are such that cts ⊗ ets ¿0 (as a special case, both cts ; ets can eventually be equal to 1). But then we can construct a proof w of the formula (∃x)(W;k (x) & Ant W;k (x)) with the value Val(w) = cts ⊗ ets ¿0. The theorem then follows from Lemma 3. This theorem explicitly states that IF–THEN rules which are in contradiction lead to a contradictory fuzzy theory—a degenerated fuzzy theory equal to the set of all the well-formed formulas FJ . Note, however, that the contradiction raises only with the basic antonyms and their narrowing modi#ers. 4. Linguistic quantiers 4.1. Linguistic and generalized quanti5ers in classical and fuzzy logic In this section, we will focus on the concept of generalized (linguistic) quanti#ers and the ability of fuzzy logic to contribute to their theory. Generalized quanti#ers have been introduced by Mostowski [8]. The main idea was to extend the de#nition of classical universal and existential quanti#ers so that quanti#ers such as “most”, “some” and others could be introduced in logical theory. Their investigation in connection with natural language has been set in motion by Montague [7] who has demonstrated that a great deal of natural language expression can be logically analysed and interpreted using the generalized quanti#ers. This direction has further been developed by J. Van Benthem, D. WesterstZahl, J. Barwise, E.I. Keenan, and others [4,13]. Example 1. The following is an example of quanti#ed linguistic expressions of various kinds: (a) Peter laughs; Either Rex or Sophie barks. (b) All (some) students cheat; No healthy woman smokes; Most spoons lay on the table; and the expression containing the words each; several;

345

few; a few; more than enough; too many; nearly a hundred; no more than; most male and all female; all but 5nitely many. (c) More students than teachers work hard; Not as many boys as girls passed the exam; almost as many students as teachers; At least three times as many as. (d) Most (many) critics reviewed just four (few) 5lms; At least three girls gave more roses than lilies to John. In linguistics, quanti#ers provide a unifying point of view to interpretation of a large class of natural language expressions. It is generally argued that they are determiners. For example, the sentence “Most spoons lay on the table” can be analysed as Most lay on the table    spoons       Det N VP    NP

where NP denote noun phrase and VP verb phrase. In logic, quanti#ers can be analysed as follows. Recall that given a formula A(x) ∈ FJ of the language J and D a structure for J , its satisfaction set SatD (A; x) = {d ∈ D | D(Ax [d]) = 1}

(21)

is a set of all elements from D on which the formula A(x) is satis#ed (cf. the formula (2)). Then quanti#ers are taken as functions determined by subsets of the power set P(D) of D. Since classical quanti#ers are included in these considerations, we will speak about generalized quanti5ers (cf., e.g. [13]). Denition 9. A generalized quanti#er of type k1 ; : : : ; kn is a function Q together with the symbol Q which to each set D assigns a relation QD ⊆ P(Dk1 ) × · · · × P(Dkn )

(22)

so that D((Qx1 · · · xn )(A1 ; : : : ; An )) = 1

iV

SatD (A1 ; x1 ); : : : ; Sat D (An ; xn ) ∈ QD ;

(23)

where xi = xi1 ; : : : ; xiki , i = 1; : : : ; n are sequences of variables. Example 2. (a) On the basis of De#nition 9, the universal quanti#er ∀ is of type 1 and it is a function

346

V. Nov'ak / Fuzzy Sets and Systems 124 (2001) 335–351

which to each set D assigns a relation ∀D = {D} so that D((∀x)A) = 1

iV

SatD (A; x) ∈ ∀D :

(b) Another example of the quanti#er of the same type is given in the sentence from Example 1(a): D((xPeter )(Laugh)) = 1 iV

generalized quanti#ers fuzzy ones and proposed to interpret them as fuzzy numbers characterizing cardinality of fuzzy sets in a #nite universe U . He introduced two kinds of fuzzy quanti#ers: (i) Fuzzy quanti#ers of the #rst kind (several, few, many, etc.), which characterize cardinality of the fuzzy set A ⊂ U ∼

Sat D (Laugh; x)

8 Count(A) =

= {dPeter } ⊆ D

A(x):

x∈U

where xPeter is the quanti#er and Laugh is a formula formalizing the property “to laugh”. Furthermore, the satisfaction set Sat D (Laugh; x) consists of the elements “who laugh”, i.e. it is the element dPeter representing “Peter”. 3 (c) The following example of the quanti#er of the type 1; 1 is a formalization of two sentences from Example 1(b). The support D in the second sentence is assumed to be a #nite set. D((Some x; y)(Stud; Cheat)) = 1



iV

Sat D (Stud; x) ⊆ SatD (Cheat; y); D((Most x; y)(Spoon; Lay)) = 1 iV |Sat D (Spoon; x) ∩ Sat D (Lay; y)| ¿ |SatD (Spoon; x)\Sat D (Lay; y)|; where | · | denotes the number of elements. It can be seen from these examples that there is no model of vagueness involved in this theory. However, vagueness is one of the most distinguished features of the semantics of natural language. Thus, fuzzy logic should be able to provide tools for making theory of linguistic quanti#ers better corresponding to human way of understanding to them. In fuzzy set theory, the #rst who came with the idea of introducing of generalized quanti#ers was L.A. Zadeh. He published the paper [15] where he calls 3 To be precise, we must realize that everything what we do in mathematics is a model formulated in its language, which essentially is based on the language of the set theory. Consequently, all elements that we deal with are sets. Thus, dPeter is a mathematical representation of “Peter” being again some chosen set taken from the universe of sets and not “Peter” himself.

(ii) Fuzzy quanti#ers of the second kind (most, a large fraction, much of, etc.), which characterize relative cardinality of the fuzzy set B ⊂ U ∼

relative to A ⊂ U ∼

8 Count(B|A) =



x∈U (A

∩ B)(x) x∈U A(x)



(recall that U is in both cases #nite). Then, the fuzzy quanti#er is set to be a fuzzy number Q ⊂ R and given a linguistic quanti#er Q, we may ∼

interpret simple quanti#ed sentences as follows: (a) “There are QA’s” = Q(8 Count(A)), (b) “QB’s are A” = Q(8 Count(B|A)), where Q(x) is a truth degree of x ∈ R and A; B are predicates representing properties of elements from U , which are interpreted by the corresponding fuzzy sets. 4 H%ajek in [3] has introduced the interpretation of the quanti#er “many” in the formal system of fuzzy logic as the fuzzy quanti#er of the #rst kind (i). He extended the  language of predicate fuzzy logic by a new symbol . If D is a #nite model then the interpretation of the  quanti#ed formula A dx is 

1 D A dx = D(Ax [d]); (24) n d∈D

where n is the number of elements of D. He proved that fuzzy logic extended by such de#ned fuzzy quanti#er keeps its completeness. Mesiar and Thiele in [6] have discussed mathematical background of the cardinal quanti#ers generalizing the idea that a quanti#ed formula (∀x)A(x) can 4 The symbols for predicates and fuzzy sets are not distinguished.

V. Nov'ak / Fuzzy Sets and Systems 124 (2001) 335–351

be understood as a conjunction (possibly in#nite) of sentences of the form Ax [d] for all d ∈ D (and analogously, (∃ x)A(x) means disjunction of them). If the conjunction (disjunction) is interpreted by the operation of minimum (maximum) then both these quanti#ed formulas are interpreted using the respective operations of in#mum and supremum taken over arbitrary subsets of [0; 1]. Since, t-norms (t-conorms) generalize interpretation of the logical conjunction (disjunction), the authors solve the problem, how the interpretation of the above quanti#ers can be de#ned when replacing minimum (maximum) by arbitrary t-norm (t-conorm).

347

so that D((∀x)A) = a

iV

∀D (Sat D (A; x)) = a:

(27)

It is clear that (27) is equivalent with the usual de#nition of universally quanti#ed formula in fuzzy logic (see [12])  D((∀x)A) = D(Ax [d]): d∈D

(b) The fuzzy interpretation of the second sentence of Example 1(b) can be obtained as follows. Let us consider a fuzzy relation “greater than” between fuzzy sets

4.2. Generalized quanti5ers in FLb

GT ⊂ F(D) × F(D):

4.2.1. Direct generalization There are at least two possibilities how generalized quanti#ers can be introduced into the theory of fuzzy logic in broader sense: a direct generalization of the above outlined theory and a more radical departure using the means of fuzzy logic theory. The truth value of a formula with a generalized quanti#er may be any value from the set of truth values L. Direct generalization of De#nition 9 can be obtained when replacing the relation of subsethood in (22) by some general fuzzy relation.

For example, we can consider some measure " : F(D) → [0; 1] and take  "(E) − "(F) if "(E) ¿ "(F); GT(E; F) = 0 otherwise

∼

Denition 10. A generalized fuzzy quanti#er of type k1 ; : : : ; kn is a function Q together with the symbol Q which to each set D assigns a fuzzy relation QD ⊂ F(Dk1 ) × · · · × F(Dkn )

(25)

∼

so that

for all (measurable) E; F ⊂ D. As a special case, if D ∼

is #nite then we may de#ne "(E) to be the measure on the right-hand side of (24). Then D((Most x; y)(Spoon; Lay)) = a

GT("(SatD (Spoon; x) ∩ Sat D (Lay; y)) "(SatD (Spoon; x)\Sat D (Lay; y))) = a: Let us now consider linguistic expressions of the form QA

D((Qx1 · · · xn )(A1 ; : : : ; An )) = a

iV

QD ( Sat D (A1 ; x1 ); : : : ; Sat D (An ; xn ) ) = a:

(26)

(recall that in (26), fuzzy sets (2) are considered). Example 3. (a) The universal quanti#er ∀ in fuzzy logic is a function which to each set D assigns a fuzzy relation       ∀D = E(d)=E  E ⊂ D  ∼ d∈D

iV

(28)

where Q is a linguistic quanti#er and A is a linguistic predication. Our aim is to characterize intension and extension of (28). De#nition 10, however, does not oVer satisfactory solution since only truth value of the quanti#ed proposition can be determined. This leads us to diVerent concept of linguistic quanti#ers presented in the next subsection. 4.2.2. FB-fuzzy quanti5ers Except for the problems with de#nition of intension and extension of the quanti#ed linguistic expressions, the previous theory has also the disadvantage

348

V. Nov'ak / Fuzzy Sets and Systems 124 (2001) 335–351

that it requires a special de#nition for each quanti#er. However, when analyzing the linguistic meaning of the quanti#ers we may come to the conclusion that a great deal of them has similar nature and diVer only in the extent of their scope. Therefore, we will attempt to establish a uni#ed theory of linguistic quanti#ers in FLb. This leads us to the concept of the fuzzy quanti#ers which we will call formula-based quanti5ers (in short FB-quanti#ers). We, in essence, rely on the above mentioned idea of L.A. Zadeh. We will consider the many-sorted language J speci#ed above but extended by a quanti#er symbol Q. Further, we extend the set of sorts # by a special sort 0 and use the letter m to denote the corresponding variable. The de#nition of formula is extended as follows: • Let B(m) be a simple formula, m variable of the sort 0 and (∀x)A a formula where the sort of x is diVerent from 0. Then (QB )((∀x)A)

(29)

is a formula. For short, we will write (QB∀ x)A instead of (29). Since, as argued, linguistic quanti#ers imprecisely characterize cardinality we will extend the de#nition of the structure for the (many-sorted) language J by considering a measure " : S" → D0 for each sort ∈ # where S" ⊆ P(D ) is some set (9-)algebra on D . Then the structure D for J is the tuple D = { D ; " | ∈ #}; PD; Q; : : : ; fD; Q; : : : ; {u ; : : : | ∈ #} ;

(30)

where we set D0 = [0; 1]. The symbols from J are interpreted in the same way as in the preceding section, except for the quanti#er symbol Q. The truth value of the new formula (QB∀ x )A is de#ned by

D((QB∀ x )A) = (D(B(m=" (X ))) X ∈S"

Card(X )¿1

∧ D|X ((∀x)A(x )));

The measures " strongly inNuence properties of FB-fuzzy quanti#ers. A problem may occur when considering one-element subsets of D . If we take their measure to be 0 then, if using the evaluating syntagm “small” in the de#nition of the quanti#er and the interpretation of the formula A is a normal fuzzy set then we may always obtain the truth value (31) to be equal to 1. Therefore, we have assumed that the cardinality of X in (31) is greater than 1. Moreover, we assume that " (D ) = 1 and " (X ) = 0 if Card(X )62. For determinacy, we also set " (X ) =

Card(X ) − 2 Card(D) − 2

(32)

for every #nite D and at least two-element subset X ⊆ D. We can also introduce the dual FB-fuzzy quanti#er (QB )((∃x)A) = (QB )(¬(∀x)¬A) (we will write (QB∃ x)A instead). Lemma 5. Let a formula C does not contain a variable of the sort and B be a simple formula. Then (FB1) |= (((QB∀ x )A) & C) ⇒ (QB∀ x )(A & C); (FB2) |= (QB∀ x )(C ⇒ A) ⇒ (C ⇒ (QB∀ x )A); (FB3) |= (∀x)A ⇒ (QB∀ x)A. Proof. Since C does not contain variable of the sort

, we immediately obtain that D|X (C) = D(C). The proof then follows from the properties of residuated lattices. The following inference rule should be introduced: rFQ :

a=A : a=(QB∀ x)A

(33)

Lemma 6. The inference rule (33) is sound for every simple formula B. Proof. It is suUcient to prove that D((QB∀ x )A) ¿ D(A)

(31)

where D|X (A) is a truth value of the formula A in the structure D in which the support set D is replaced by X.

holds for every structure D. Since B(m) is a simple formula, by its de#nition and the de#nition of " there exists a set X ⊂ D such that D(B(m=" (X ))) = 1. But then D|X ((∀x )A(x ))¿D((∀x )A(x )) which gives the required inequality.

V. Nov'ak / Fuzzy Sets and Systems 124 (2001) 335–351

349

When extending fuzzy logic by FB-quanti#ers, we will take the formulas (FB2) and (FB3) as additional logical axioms. Since by Lemma 6, the rule rFQ is sound, we obtain the following theorem.

The truth value of QA in the given possible world D is the value (31)

Theorem 4. Fuzzy logic calculus with the FB-fuzzy quanti5ers of the type QB∀ is sound.

Let us stress that we should always clearly distinguish between the meaning of the quanti#ed linguistic predication from its truth value.

A wide class of linguistic quanti#ers can be modelled using FB-fuzzy quanti#ers. Their interpretation is essentially determined by some simple evaluating expression B. An example of the possible assignments is the following

TruthD (QA) = D((QB∀ x )A):

Example 4. Let us consider the quanti#ed linguistic predication B := “All x are A”. Then its meaning is Mean(B) = (∀x)Int(A); Ext(All x are A) ; (∀x)Int(A) = A
x = {at =Ax [t]|t ∈ M };

∀ Many ≈ QBig ; ∀ Most ≈ QVery

ExtD (All A) = ExtD (A)

big ; ∀ Several ≈ QSmall ; ∀ A few ≈ QVery small ; ∃ Some ≈ QSmall ; ∀ All ≈ QUtmost ;

because by (13), D(B(m=" (X ))) = 0 for all X = D (for some, non-speci#ed sort, ). The truth value of the linguistic predication B with respect to any possible world is determined within the corresponding logical system (cf. (31)) and is equal to

∃ Exists ≈ QUtmost :

Quanti#ed evaluating predications seem to have no speci#c intension. Instead, they refer to some possible world (model) where their meaning becomes senseful. On the other hand, they always give a truth value with respect to a possible world. Denition 11. Let Q be a linguistic quanti#er assigned a fuzzy quanti#er QB∀ and A be a linguistic predication with the intension Int(A) = A
x . Then the meaning of the quanti#ed linguistic predication QA is Mean(QA) = Int(QA); Ext(QA)

TruthD (All x are A)  = D((∀x)A) = D(Ax [d]): d∈D

Example 5. The following example demonstrates speci#c FB-fuzzy quanti#ers. Let us consider a twosorted language and a model D = D0 = [0; 1]; "0 ; D1 = {1; : : : ; 12}; "1 ; : : : : Furthermore, let A(x) be a formula with the satis#ability fuzzy set SatD(A;x) = {1=1; 0:9=2; 0:8=3; 0:75=4; 0:7=5; 0:6=6; 0:5=7; 0:4=8; 0:35=9; 0:3=10; 0:25=11; 0:2=12}

= {ExtD (QA)|D is a possible world} ;

(35)

where the intension of QA we write as Int(QA) = (QB∀ )A
x and the extension Ext(QA)D is the fuzzy set

being extension of some linguistic expression A. Obviously,

Ext(QA)D = {D(B(m=" (X )))=Ext D|X (A)|X ∈

holds true. Furthermore, if a subset X ⊂ D1 contains n elements then we will write X (n) . The interpretation of the simple evaluating syntagms will be those

S" ; Card(X ) ¿ 1}:

(34)

D((∀x)A)) = 0:2

and

D((∃x)A)) = 1

350

V. Nov'ak / Fuzzy Sets and Systems 124 (2001) 335–351

from Fig. 1. If we take the measure (32), then, for example, we get D(WBig (m="1 (0:9)) = 0:96; D (WBig ( m="1 (0:7)) = 0:4; D (WSmall ( m="1 (0:1)) = 0.84, etc. and we obtain the following extensions: ExtD (Most x is A) = {1=Ext D (A); 0:87=Ext D|X (11) (A); : : : ; 0:54=ExtD|X (10) (A); : : : ; 0:15=Ext D|X (9) (A); : : :}; ExtD (Many x is A) = {1=Ext D (A); 0:96=Ext D|X (11) (A); : : : ; 0:71=ExtD|X (10) (A); : : : ; 0:4=Ext D|X (9) (A); : : : ; 0:04=Ext D|X (8) (A); : : :}; ExtD (Several x is A) = {1=Ext D|X (2) (A); : : : ; 0:84=Ext D|X (3) (A); : : : ; 0:4=Ext D|X (4) (A); : : :}: The truth values are ∀ TruthD (Most x is A) = ((QVery

big )A)

= 0:3;

(36)

∀ TruthD (Many x is A) = ((QBig )A) = 0:35;

(37)

∀ TruthD (Several x is A) = ((QSmall )A) = 0:9:

(38)

We argue that our assumption to consider at least two element sets X is not in contradiction with the intuition. We may indeed hardly agree that “several” or “a few” should mean one element only. This is supported also by our example since the fact that the truth of (38) is smaller than that of (∃ x)A seems to be intuitively satisfactory. On the other hand, this example also demonstrates that our model of the meaning of quanti#ed linguistic expressions reNects the concept of linguistic context (think of the support set D1 having 10 000 elements; how many elements would correspond to “several”?). 5. Conclusion In this paper, we have focused on linguistic expressions, which are worked out in FLb, namely the simple evaluating predications, pairs of their antonyms, and

linguistic quanti#ers. The theory of antonyms, which includes also the third member forming thus an evaluating linguistic trichotomy seems to be now satisfactorily elaborated. The philosophical background of the presented formalism is extensively discussed, e.g. in [11]. The second important class of linguistic expressions are those based on using some linguistic quanti#er. We have outlined a slightly diVerent theory which conforms with the general ideas of construction of the meaning in FLb. From the formal point of view, this raised a series of questions. In the #rst place completeness of the obtained formal calculus should be studied. This seems to be somewhat dubious since in our presented theory, we provided no means for modi#cation of the provability degree in the syntax with FBquanti#ers. The question, whether a more elaborated syntax is possible (and necessary) should be clari#ed in the future. Other questions, such as characterization of interconnection among FB-quanti#ers, their relation to cardinal quanti#ers, strength for modelling of natural language sentences, etc. should be further investigated. References [1] M. Black, Vagueness: An Exercise in logical analysis, Philos. Sci. 4 (1937) 427– 455, Reprinted in Int. J. General Systems 17 (1990) 107–128. [2] S. Gottwald, A Treatise on Many-Valued Logics, Research Studies Press Ltd., Baldock, Herfordshire, UK, 2001. [3] P. H%ajek, Metamathematics of Fuzzy Logic, Kluwer, Dordrecht, 1998. [4] L.E. Keenan, D. WesterstZahl, Generalized Quanti#ers in Linguistics and Logic., in: J. van Benthem, A. ter Meulen (Eds.), Handbook of Logic and Language, Elsevier, Amsterdam, 1997, pp. 837–893. [5] R. Mesiar, V. Nov%ak, Operations Fitting Triangular-NormBased Biresiduation, Fuzzy Sets and Systems 104 (1999) 77–84. [6] R. Mesiar, H. Thiele, On T-quanti#ers and S-quanti#ers, in: V. Nov%ak, I. Per#lieva (Eds.), Discovering the World with Fuzzy Logic. Springer, Heidelberg, 2000, pp. 310–326. [7] R. Montague, The proper treatment of quanti#cation in ordinary english, in: R. Thomason (Ed.), Formal Philosophy: Selected Papers of Richard Montague, Yale Univ. Press, New Haven, 1974, pp. 247–270. [8] A. Mostowski, On a generalization of quanti#ers, Fund. Math. 44 (1957) 12–36. [9] V. Nov%ak, The Alternative Mathematical Model of Linguistic Semantics and Pragmatics., Plenum, New York, 1992.

V. Nov'ak / Fuzzy Sets and Systems 124 (2001) 335–351 [10] V. Nov%ak, Linguistically Oriented Fuzzy Logic Controller and Its Design, Int. J. of Approx. Reasoning 12 (1995) 263–277. [11] V. Nov%ak, I. Per#lieva, Evaluating linguistic expressions and functional fuzzy theories in fuzzy logic, in: L.A. Zadeh, J. Kacprzyk (Eds.), Computing with Words in Information=Intelligent Systems 1., Springer-Verlag, Heidelberg, 1999, pp. 383–406. [12] V. Nov%ak, I. Per#lieva, J. MoAckoAr, Mathematical Principles of Fuzzy Logic., Kluwer, Boston, 1999.

351

[13] D. WesterstZahl, Quanti#ers in formal and natural languages, in: D. Gabbay, F. Guenthner (Eds.), Handbook of Philosophical Logic, Vol. IV, D. Reidel, Amsterdam, 1989, pp. 1–131. [14] L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning I, II, III, Inform. Sci. 8, 199 –257 (1975) 301–357; 9 (1975) 43–80. [15] L.A. Zadeh, A computational approach to fuzzy quanti#ers in natural languages, Comp. Math. Appl. 9 (1983) 149–184.