fuzzy logics as families of bivaluated logics

Fuzzy Sets and Systems 64 (1994) 321-332 321 North-Holland

Fuzzy logics as families of bivaluated logics

J.L. Castro Depto de Ciencias de la Computaci6n e Inteligencia Artificial, Facultad de Ciencias. Universidad de Granada, 18071-Granada, Spain

Received June 1992 Revised January 1993

Abstract: In this paper we will prove the equivalence between L-valued logics and L-indexed families of bivaluated logics. It is also proved that this equivalence can be used as a tool for checking compactness and decidability properties of a L-valued logic.

Keywords: Fuzzy logic; consequence operator; decidability and compactness.

Introduction

In recent years a conspicuous interest in fuzzy logic has arisen and many interesting applications have been developed within Artificial Intelligence. The present paper tries to study those logics f rom an abstract point of view, in order to obtain general propert ies that can be useful for applications. This paper owes its inspiration to D.J. Brown and R. Suszko works on abstract logics [2]. Taking the generalization of a consequence opera tor introduced by Pavelka [8-10] as a general concept of fuzzy logic, it is shown that a family of Brown-Suszko ' s abstract logics can be associated to each fuzzy logic. Thus some propert ies exhibited by this abstracts logic are extended to L-fuzzy logics. In Section 1, a brief review of the concepts of Brown-Suszko ' s abstract logics is presented. In Section 2, after introducing the concept of L-closure system, we extend the results to Pavelka 's L-consequence operators. The associated classical abstract logics are presented in Section 3. Finally, in Section 4, that result is used in order to study multivalued logic f rom the propert ies of their associated bivaluated logic.

1. Consequence operators and closure systems

Definition 1. A consequence operator on S is defined as a mapping f rom the power set 2 s into itself, c : 2 S ~ 2 s, such that the three following conditions

(CI ) inclusion: A ~_ C(A) , (C2) monotony: If A ~_ B ~_ S, then C ( A ) ~_ C(B) , (C3) idempotence: C ( C ( A ) ) = C ( A ) ,

holds for each A, B ~_ S. Usually, a compactness proper ty is added. A consequence opera tor is said to be compact if (C4) compactness: for any X ~_S and any x ~ C ( X ) there exist a finite subset G ~_ X such that

x c C ( G ) , holds.

Definition 2. A family ~ of subsets of a non-empty set S closed under arbitrary intersections and such that S e c¢, is said to be a closure system on S.

Correspondence to: Dr. J.L. Castro, Depto de Ciencias de la Computacion, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain.

0165-0114/94/$07.00 © 1994---Elsevier Science B.V. All rights reserved SSDI 0165-0114(94)00063-D

322 J.L. Castro / Fuzzy logics as families o f bivaluated logics

Theorem 1. (i) Every consequence operator C on S defined the closure system

c~( C) = {A ~ S I C(A ) = A},

i.e. the family o f all C-closed subset o f S. (ii) Every closure system c~ on S defines the consequence operator

C(C~)(m) = n {T e ~ I A ~_ T}.

Theorem 2. The family o f all closure systems on S is a complete lattice under

inf cg~ = n cg~ = {s ~_ S I s E ~ for all i}, sup ~ = inf {cg I c~ ~ cg, for all i}. i i i i

and set inclusion.

Theorem 3. The family o f all closure operators on S is a complete lattice under the ordering

for each A ~_ F. C1<~C2 iff C1(A)<~C2(A)

Here,

and

inf C i ( A ) = n C i ( A ) i i

sup Ci = inf {C [ C i ~ C, for all i}. i i

Theorem 4. The complete lattice o f all consequence operators on S and the complete lattice o f all closure systems on S are dually isomorphic under the correspondences ~ ~ C(C¢), and C--* c¢(C).

The above theorems are due to Moore, Birkhoff, Tarski and Ore (see [2]).

Remark 1. The elements of q¢(C) are called the theories of the logic C. Thus, C(C~) is the logic whose theories are c~, and the previous theorem shows the equivalence between the theories of a logic and the consequence operator defining this logic.

2. L-closure systems and L-consequence operations

In this section we will extend the previous results to the framework of fuzzy set theory. Pavelka [8-10] introduced the notion of L-consequence operator in order to study the L-valued logic from a general point of view. In 1991, Murali [7] introduced the concept of L-closure system in fuzzy set theory and proved that both concepts are in a bijective correspondence. We will prove that the correspondence is a lattice anti-isomorphism.

Let L be a complete lattice with the order ~ and the operations infimum and supremum A and v respectively (inf and sup for the infinite case), minimum 0 and maximum 1. L will be considered as the lattice of truth values in L-valued logics. If S is a non-empty set, an L-subset s of S is defined as a mapping from S into L. Thus, the class of L-fuzzy sets of S is defined as L s. Defined intersection N, union U and inclusion _~ over L s as follows:

( n ~--)(a) = inf{s(a) Is ~ ~}, ( U ~ ) (a ) = sup{s(a) Is E ~},

sl -~s2 if and only if sl(a)<~s2(a) for each a in S.

Each non-fuzzy (crisp) subset s ~_ S can be seen as the L-subset defined by s(a) = 1 if a E S and s(a) = 0 i f a ~ts.

J.L. Castro / Fuzzy logics as families of bivaluated logics 323

Definition 3 [7]. A family ~ of L-subsets of a non-empty set S is said to be a L-closure system on S if is closed under arbitrary intersections, i.e. ['-) ~ is in q~ if ~ _ ~, and S • ~, i.e., the L-subset S(a) = 1 for each a • S is in c~.

If s is an L-subset of S, s , will denote the a-cut of s, by

s~ ={a • S I s (a )>a} .

Theorem 5. I f ~ is an L-closure system on S, for each a • L the family ~ defined by % = {s~ I s • ~} is a classical closure system.

Proof. Let ~ _ ~ . Let us denote ~ = {s • c~ I s~ • ~ ~_ cC Then O ~ is in ~ and ((--) ~)~ = (-') ~ is in ~ . Finally, obviously S, = S • qg~. []

Theorem 6. The family of all L-closure system on S is a complete lattice under

inf c~i = O ~i = {s • L s [ s E ~ for all i}, sup c~i = inf {c~ [ c~ ~ c~, for all i}. i i i i

L-set inclusion, minimum {s} and maximum L s.

Proof. See Murali's paper [7]. []

Example 1 [7]. (i) Let (X, r) be an L-fuzzy topological space; ~- be the collection of v-closed L-fuzzy subsets of X. Then v is an L-closure system.

(ii) The collection of all L-fuzzy subalgebras of an algebra is an L-closure system.

Remark 2. This lattice is an extension of the complete lattice of all closure systems on S, since a closure system is also an L-closure system, infimum and supremum operations being the same in both lattices.

Pavelka [10] introduced the notion of an L-consequence operator in order to give an abstract notion of L-multivalued logic.

Definition 4. A L-consequence operator on S is defined as a mapping from the power set L s into itself, : L s ~ L s, such that (CI ' ) fuzzy inclusion: A ~_ C(A), (C2') fuzzy monotony: If A g B ~_ S, then C(A) ~_ C(B), (C3') fuzzy idempotence: C(C(A)) = C(A),

hold for each A, B • L s. Usually, a compactness property is added. An L-consequence operator is said to be compact if, and

only if, it satisfies the condition (C4') fuzzy compactness: for any A E L s and any x • S there exist a finite subset G ~_ S such that

C(A)(x ) = C(A I G))(x), where X I G represents the L-subset

~ X ( y ) i f y • G , (X I G) (y ) = [0 otherwise.

Proposition 1. Let C be an L-consequence operation on S. Then, (i) The restriction o f C to 2 s, Ca, defined by

a e Cl(A) if and only if C ( A ) ( a ) = I , Va e S, VA ~_S,

is a classical consequence operator. (ii) If C satisfies the compactness axiom, C1 also satisfies the classical compactness axiom.

324 J.L. Castro / Fuzzy logics as families of bivaluated logics

Proof . (i) F rom ( C I ' ) and (C2') follows immediately (C1) and (C2). If a E CI(CI(A)) then C(CI(A))(a) = 1 and as C1(X)~-C(X) for each X, f rom (C2') and (C3') follows 1 = C(CI(A))(a)<~ C(C(A)(a) = C(A)(a), hence C(A)(a) = 1 and a ~ CI(A).

(ii) If C is compact , then for any A E L s and any x E S there exists a finite G~_S such that C(A)(x) = C(A [ G))(x). In particular, if A ~_ S and a E S, then there exists a finite G ~_ S such that C(A)(a) C(A [ G)(a) = C(G)(a), hence if a E q ( A ) , a c CI(G). []

Remark 3. Given a ~ 1, the restriction of (~ to 2 s, Ca, defined by

a ~ C a ( A ) if and only if C(A)(a)>la, V a ~ S , VA~_S,

is not in general a classical consequence operator , since idempotence is not always followed.

Proposition 2. 0 ( A ) = C(CI(A)), Vii ~_ S.

Proof. From A ~_ CI(A), follows C(A)~_ C(CI(A)) . On the other hand, from CI(A)~-O(A), follows ~' (CI(A)) ~ C'((~(A)) = (?(n). []

Proposition 3. For any a <~ 1, C~ satisfies inclusion (C1) and monotony (C2).

Proof. Let us suppose A ~_B. If a ~ C~(A) then C(B)(a)>~O(A)(a)>-a and a E Ca(B), hence Ca(A) ~- Ca(B). On the other hand, if a e A, then C(A)(a) = 1 and a e Ca(A), hence A ~_ Ca(A). []

Lemma 1. I f Ca is a classical consequence operator for every a E L, then

~C#(A) Ct3( C~(A ) ) = (Ca(A)

i f /3 ~ ~, V/3, VA ~_ S.

if a<~/3,

Proof. If fl < a then

Ca(A) ~_ C~(A) and C~(A) ~_ C~(Ca(A)) ~_ C~(C~(A)) = C~(A).

On the other hand, if a ~</3 then

[0(C~(A))]~ ~_ [0(Ca(A))]~ = C~(Ca(A)) = Ca(A),

hence Ca(A) ~- C~(C,(A)) ~_ Ca(A). []

Proposition 4. Ca is a classical consequence operator for every a E L if, and only if,

1 O(C,(A))(a) = { O(A)(a)

if a E Ca(a),

ira ~ Ca(a), V a ~ S , VA~_S.

Proof. This is immediate f rom the previous lemma. []

Corollary 1. The condition

l f A is a classical set then C(A) is a classical set

holds if and only if Ca = C1 for every a <<- 1.

Theorem 7. The family of all L-consequence operator on S is a complete lattice under the ordering

0~ <~02 iff O,(A)<~O2(A) for eachA ~ L s.


Here,

infC~(A)=(~(7~(A), sup(7g=inf{(TIc~c_(7, f o r a l l i }. i i i i

Proof. See [4]. []

Theorem 8. (i) Every L-consequence operator C on S defines the L-closure system

~((7) = {A e L s I O(A) = A},

i.e., the family o f all (7-closed L-subsets o f S. Clearly, the following are verified: (a) (-'l ~((7) = (7(0) where O(a) = 0 for each a e S, and (b) S e ~((7). (ii) Every L-closure system ~ on S defines the L-consequence operator

(7(~)(A) : (-') {T E @ ] A c_ T}.

Proof. See [7]. []

Theorem 9. The complete lattice o f all L-consequence operations on S and the complete lattice of all L-closure systems on S are dually isomorphic under the correspondences ~--~ (7( ~), and (7 ~ ~((7).

Proof. Let us consider two L-clsoure systems ~ and ~2 and denote by (7,, (72, CL2, ~t2 the L-consequence operators (7(~,), (7(~2), C(~1N ~2) and C ( ~ 1 0 ~2), respectively. We must prove

(i) If ~, c ~2, then (72 c (7~, (ii) (7,2 ~ sup((71, C2),

(iii) (7,2 = inf((71, (72). (i) This is evident since

{T E ~1 IA~--T}~--f-'I{T ~ ~2 I A ~ - T }

and so

(-'){T ~ ~2 I A c- T}~-("){T ~ ~, ] A C T } .

(ii) From (i) follows (7, c_ (7,2 and (72 ~- (7~2. If (7 is a consequence operator verifying (7~ c_ (7 and (72 c_ (7, then from

(7(A) ~_ C~(C(A)) c_ (7((7(A)) = (7(A), hence C~(C(A)) = (7(A) E ~ ,

C(A) ~_ C2(C(A)) c_ (7((7(A)) = (7(A), hence C2(C(A)) = (7(A) e ~2,

follows

Ac_(7(A) e ~ , N ~ 2 and

~7,2(A) = (~{T e ~, A ~2 I A c_ T } c C(A).

(iii) From (i) follows (7 ' 2 c _ (71 and (712~ (72- If (7 is a consequence operator verifying (7 c_ (7, and (7 c (72, then

(7(A) ~ T1 for every Tl E @x such that A ~ TI and

(7(A) ~_ T2 and every 7"2 ~ c~2 such that A c T2. Hence

(7(A)~_T, for every T E @ l O ~ 2 s u c h t h a t A c _ T and

(7(A) c (7 t2(A).


Table 1

T 0 T~ T 2 T 3 T 4

a 0 1 1 0.5 0.5 b 0 1 0.5 1 0.5 c 0 1 1 1 1

Conversely, let us consider two L-consequence operators C1 and C2 and denote by (~1, 6~2, (~12, (~12 with L-closure systems ~(C1), ~(C2), ~ (C l n C2) and ~(sup(C1, C2)) respectively. We must prove

(i') If C1 ~- C2, then ~2 =- ~1, (ii') ~12 = sup(~l, ~2),

(iii') @12 =_ ~1 fq ~2. (i') This is evident since if T c ~2 then T =_ (71(T) _ C a ( T ) = T, hence Ca(T) = T and T e @l.

(ii') From (i) follows ~1 =_ (~12 and c~2 c_ ~2. Let @ be an L-closure system verifying @1 _= c~ and c~ 2 c ~. If T e qg~2, then (C, f3 C 2 ) ( T ) = C,(T) n C 2 ( T ) = T. F r o m C I ( T ) E ~1 ~ c~ a n d C 2 ( T ) E q~2

follows T = (?~(T) n C 2 ( T ) ~ q~. Therefore (~12 C7 ~. (iii') From (i) follows ~12~ @~ and ~12~_ ~2. Let @ be an L-closure system verifying ~ =_ ~ and -= @2. If T ~ ~, then from T e @1 follows CI(T) = T and from T e ~2 f o l l o w s C 2 ( T ) = T. We will

prove that in these cases sup((7~, C 2 ) ( T ) - - - T and thus ~=_ ~12. We only need to consider the L-consequence operator ~(@12), which contains C~ and C2 and C(@~2)(T) = T.

Finally, we need to prove that both mapping are self inverses. That is, (a) C ( ~ ( C ) ) = C, ( b ) =

but this can be found in the Murali's paper [7]. []

Example 2. The following is an example of an L-consequence operator C (L = [0, 1]) such that the consequences of a crisp set is not crisp and such that C~ is a consequence operator for every a ~< 1. will be defined by the closure system ~ = {To, T~, T3, T3} (see Table 1).

If 0 < a ~< 0.5, then

C~({a}) = C~({b}) : C,({c}) : C,d{a, b}) = C,~({b, c}) : C,~({a, c}) = C,~({a, b, c}) : {a, b, c}.

If 0.5 < a ~< 1, then

C~({a}) = {a, c}, C~({b}) = {b, c}, C,~({c}) = {c},

C,~({a, b}) = {a, b, c}, C~,({a, c}) -- {a, c}, C,~({b, c}) = {b, c},

C,({a, b, c}) = {a, b, c}, C~(0) = 0.

Remark 4. Analogously to the bivaluated case, the elements of c~(~) are called the theories of the L-valued logic C. Thus, ~(c~) is the L-valued logic whose theories are ~, and the previous theorem shows the equivalence between the theories of an L-valued logic, and the L-consequence operator defining this logic.

3. L-valued logics as indexed families o f bivaluated logics

Given an L-consequence operator C on S, we have the associated L-closure system ~ ( C ) on S, which has associated the family q¢(C)~, Va E L of classical closure systems on S; and so, the family C '~= C(~(C)~) , Va e L of classical consequence operators on S associated to everyone of these


classical closure systems. Thus, we can associate a family of classical consequences opera to r C '~ for every L-fuzzy consequence opera tor .

Definition 5. For each a E L, every L-consequence opera to r C on S defines a classical consequence ope ra to r C a, ob ta ined f rom ~ (C)~ th rough the equivalence be tween classical consequence opera to r and classical closure systems. Thus,

C"(A) = (-~ {T,~ I C ( T ) = T and A ~_ T~}.

A more opera t ional expression to Ca(A) is ob ta ined in the following three proposi t ions.

Proposition 5. Ca(A) = C(S~)~, where

a i f x • A, S ~ ( x ) = 0 otherwise.

Proof . I f x • Ca(A) then T(x) >1 a for every T verifying (?(T) = T a n d A c_ T~. Taking T = C(S~), then A ~_ (S~) , _ C(S~t), = T, and C ( T ) = T, hence T(x) >1 ~ and x e C(S~)~. Conversely , if x e C(S~) , , then C(STa)(x) >>- a, therefore for any T such that S~ ~_ T and C ( T ) = T is T(x) >I a and x E T,. In equivalence, for every T such that C'(T) = T and A ~_ T~ is x e T~. []

Proposition 6. I f C is defined by means o f an L-semantic on S, that is, if there exists a family 5 e ~_ L s such that

then C'(A) = ['-'/{T E :flA~_ T},

C"(A) ={x • S I T(x)>~a i fT (a )>~a , Va cA} .

Proof . Straightforward. []

Pavelka [8] gives a suitable definition of syntax in L-va lued logic. Namely, an L-rule of inference is a pair r = (r ' , r") where r ' is a partial n -a ry opera t ion on S and r" is an n-ary opera t ion in L preserving unions in each variable. A n L-syntax on S is a par (~/, Q) where ~ / i s an L-subse t of S, called the L-subset of logical axioms and Q is a finite set o f L-rules of inference. For every A, we deno te with A* the free semigroup genera ted by A. Let

r = S U (S x {0}) U (S x Q x N+)

and set, for every w E F , F w = w i f w e S a n d F w = x i f w E ( S x { O } ) U ( S X Q x N +) a n d x i s t h e first coord ina te of w. A n Q-proof is an e lement w = wl • • •Wm • F + such that if w k = (x, r, il . . . . . i,,) then the L-rule r is n-ary , k > 1, 1 <~ il . . . . . i, < k and x = r'(rwi, , . . . , Fwi, ). The posit ive integer m is called the length of w; it will be deno ted by l(w). We deno te by wq and name as the p roved formula of w the e lement FWm. Obviously, if w = wl • • •wm is an Q-proof , the w(~) = wl • • • wk, with k ~< m, is an Q - p r o o f also. The set of all Q-p roof s in S will be de no t ed by P(S, Q).

Let A be an L-subset of S. The value v(w, sg, A ) of the Q - p r o o f w with respect to the L-sys tem of hypothes is A is defined by induct ion on l(w). If l ( w ) = 1, then w • S U S X { 0 } and we set v(w, sq, A ) = A ( w ) if w E S and v(w, ~/, A) = ~/(' w) if w • S x {0}. If l (w) > 1, then

A(wm) if e S, Wm

v(w, s~, A) = 1 s~(Fw) if Wm e S x {0},

I. r"(v(w(i,), ~ , A) . . . . . v(w(i.), ~ , A )) if Wm= (X, r, il . . . . . i,).

3 2 8 J.L. Castro / Fuzzy logics as families of bivaluated logics

Intuitively v(w, ~l, A) is the degree of validity that the G-proof w assigns to the formula w q under the hypothesis A and the logical axioms ~/.

An L-subset T of S is closed with respect to ~ if, for every r E ~, T is closed with respect to r, i.e.,

T ( r ' ( x l , • . . , Xn) ) >! r " ( Z ( x l ) . . . . , T(xn) ) ,

where (xl, • • •, xn) is any element of the domain of r'. In this case, T is called an L-theory. The operator determinated by the syntax (~, R), C:~..~ defined by

C j(Z) = A {TI T is an L-subset of S closed with respect to ~ and T ~>A v ~'}

is a consequence operator and

C:~..~(A)(x) = sup{v(w, ~I, A) I w ~ P(S, ~) , w 1 = x}. [8].

Proposition 7. I f C is defined by means of an L-syntax (sg, ~) (see Pavelka), i.e., C = C~..~, then x ~ Ca(A) if and only if for every ~ <a , there exist a w E P(S, ~ ) such that wq=x, and v(w, ~, STO > 13.

Roughly speaking, x e Ca(A) if, and only if, for every fi < a there exist a G-proof of x with validity greater than fl under the hypothesis A at level a and the axioms ~/.

From these general results, we have a pseudo-syntax and a pesudo-semantic for the bivaluated logics associated to the different L-valued logic present in the literature.

Example 3. Let us consider the Lukasiewicz's logic on [0, 1], i.e., with valuation functions:

v(-~p) = 1 - v(p), v(p A q) = Min(v(p) , v(q)),

v ( p v q ) = M a x ( v ( p ) , v ( q ) ) , v ( p ~ q ) = M i n ( 1 , 1 - v ( p ) + v ( q ) ) .

The L-consequence operator obtained from that semantic is:

C(A)(x) = inf{v(x) l v is a valuation with v ( a ) ~ A(a), Va e P},

P being the set of propositions of the logic. Thus,

C~(p ~ q, p v q) = {x l v(x) ~ o~ if v(p ~ q) >~ a and v(p v q) >~ a}.

From 1 - v(p) + v(q) >t ~, v(p) >1 a or v(q) >! a follows v(q) >~ o~ - 1 + v(p) >~ a, and q • C"(p ~ q, p v q ) if and only if c~ = 1.

Example 4. Let us consider the propositional logic on [0, 1] with the following axioms [6]: (A1) p ~ p v q ] l , (A2) p v q ~ q v q ] l , (A3) p A q ~ p ] l , (A4) p A q ~ q ^ q ] l , (A5) q v p ~ q ] 1,

and the only rule of inference

P l a , P ~ O l b Q , a * b

being * a t-norm. In this case, when O<~a<½, q • C ~ ( q v p ) , but when ½<~<~1, q~ t C ~(qvp ) . This one is an

example of a non-trivial fuzzy logic in which the hypothesis

A classical set implies C(A) classical set,

J.L. Castro / Fuzzy logics as families of bivaluated logics

fails, since f rom {p [ 1, q [0} follows q with validity ½ by using (A5).

329

A quest ion is now considered: Are the L-consequence operators characterized by its associated family of classical consequence oeperators , i.e.,

C 1 = 6 ' 2 if and only if C 7 = C ~ for e v e r y a E L ?

The answer is negative in general, since C7 = C~ for every a • L if and only if c~(~]),~ = T~(~'2)~ for every a ~ L, but it is possible al though ~(C~) ~ ~(C2).

Nevertheless, if the index theories maps

are added then the families {(C a, index , ) ] a e L} characterize and determine the L-consequence opera tor C.

Equivalence Theorem. Let C be an L-consequence operator. Then the fami ly {(C'% index, ) ] a • L - {0}} verifies:

(i) For every T ~ @(C), i f a <~ [3 then T, ~_ Tt3. (ii) I f ~ ~_ ~ ( C ) and T = ~ ~ then T~ : (-'1 {T'~ I T ' • ~}.

(iii) For every T ~ @(C) and every x E S, there exist max{a ~ L Ix • T,}. Conversely, i f we have a fami ly {C(a) , Index(a ) I a • L - {0}} where C ( a ) is a consequence operator,

V a • L, and index(a ) is an indexation over the closure sys tems associated

I index(°~) ) ( ~ ( a ) , i~-->T?

such that (i) I f a <~ [3 then T f c_ T~/, and

(ii) l f J ~ I and O j E j T[ = Tz,, then f'~j~j Tt/ = Ti, ,, (iii) For every i • 1 and every x • S, there exists max{a • L Ix E TT}, are verified, then we can define

an L-consequence operator C and a mapp ing

I--~ @(C), i~--~ T~

such that C" = C ( a ) and T7 = Ti,. is defined as C(C~) where

= {T, l i ~ I}, T,(x) = max{a E L Ix E TT}.

Proof. The properties of {(C% index~) I a e L - {0}} are evident. Conversely, if {C(a), Index(a ) I a E L - {0}} satisfies the condit ions (i), (ii) and (iii), then set C = C ( ~ ) where

= {7? l i ~ I}, T,(x) = max{a ~ L Ix E T~'}.

is an L-closure system since if ~ ~_ % there exists a J ~_ 1 such that

IJ •J}, and for every a • L,

~x j e , I jeJ

by the definition of Tj and (ii). Hence ( O ~ ) = T/,, • ~. Finally, it is obvious that T~,. = T~' and thus C ( a ) = C ~. []

Roughly speaking, an L-consequence opera tor is equivalent to an L-indexed family of classical


Table 2

C t C1/2 C O

0 {a} {a} {a, b, c} {a} {a} {a} {a, b, c} {b} {a, b} {a, b} {a, b, c} {c} {a, b, c} {a, c} {a, b, c} {a, b} {a, b} {a, b} {a, b, c} {a, c} {a, b, c} {a, c} {a, b, c} {b, c} {a, b, c} {a, b, c} {a, b, c} {a, b, c} {a, b, c} {a, b, c} {a, b, c}

consequence operator together with a common indexation over its theories such that the indexation assigns the same index to intersections for every a E L. Observe that if L is finite, then condition (iii) always holds.

Example 5. Let S = {a, b, c}, L = {0, 0.5, 1} and let us consider the bivaluated logic for every a e L, and the indexation as given in Tables 2 and 3.

The L-consequence operator associated will be (see Table 4):

C({a

C({a

C({a

C({a

C({a

(~({a

O,b

O,b

O,b

O,b

O,b

a, b

0, c [ 0}={a [1, b [0, c [0},

0, c [ 1}={a [ 1, b [ 1, c I1},

0.5, c [ 0.5} = {a [1, b [ 1 , c 11},

1, c [ 0} ={a I 1, b [ 1, c [0},

1, c [ 1})={a [1, b[1, c [1},

/3, c I 3'}) = C({a 10, b I/3, c

C({a [0, b [0, c [0.5}) ={a [ 1, b [0, c [0.5},

(?({a [0, b [0.5, c 10}) = {a [ 1, b I 1, c 10},

C({a 10, b [0.5, c [ 1})= {a [1, b[1, c [1},

C({a [O,b [ 1, c [ 0.5})= {a [1, b[1, c [1},

[ y} for every a,/3, y E L.

Table 3

,~ T~, 7~ ' T~ T '~3

1 {a} {a} {a, b} {a, b, c} -~ {a} {a, c} {a, b} {a, b, c} 0 {a, b, c} {a, b, c} {a, b, e} {a, b, c}

4. Checking properties of L-valued logics

The equivalence theorem is useful for checking properties of L-valued logics by checking the properties of its associated bivaluated logics.

The first property we study is the compactness.

Table 4

To ~ 7"2 T;

all all q[1 all b[0 hi0 b[1 b[1 cl0 cl0.5 cl0 cll


T h e o r e m 10. Let C be an L-consequence operator. I f C is compact, then C ~ is compact for every a E L.

Proof. Let us suppose that C is compact and show that C ~ is compact. Let A ~_ S. Then

C~(A) - = c(s )o. (1)

From the compactness of C' follows that for each x E S there exist a finite subset G ~_S such that C(A)(x ) = C(A I G))(x). Let A ~_ S, and let G a finite subset of S such that

I G ) ( x ) = (2)

Let us check that

S7,1 c = S7~oo. (3)

If x E A N G then (S~ I G)(a) = a = S~nc(a) , and if x ~ A fq G then (S~t ] G)(a) = 0 = S~nc,(a). From (1), (2) and (3) follows C"(A) = C(S~) , and x e C'~(A) if and only if

C(SAv.IG)(X) ~ Ol iff x ~ C~(A n G),

hence C" is compact . []

Other interesting proper ty is the decidability. The notion of decidability of an L-subset was introduced by Biacino and Ger la [1].

Let us assume L is a compelete , infinitely distributive lattice with an antitone involution -~:L--* L, and S is codified by natural numbers, S ~ ~. Then we can identify any A E L s with a partial mapping f rom ~ to L. Let L ' be a substructure of L dense in L, i.e., every e lement of L is the least upper bound of a family of e lements of L ' . We assume also that L ' is codified by elemens of ~ and the operat ions v, ^ and -n are effectively computable in L' . Thus, we can talk about recursive maps f rom ~ " ~ L ' in the classical sense.

Definition 6 [1]. A L-subset A ~ L ~ is said to be recursive enumerable if there exists a recursive map h : [~ × ~ --* L ' such that

A(x) = sup{h(x, n) In ~ N}.

An L-subset A is said to be decidable if A and -~A, where -hA(x)= ~(A(x) ) for every x E ~, are recursive enumerable .

Definition 7. We will say that an L-consequence opera tor C is decidable if every theory of the logic is decidable, that is,

C(A) is decidable for every A E L s.

T h e o r e m 11. Let C be an L-consequence operator. I f C is decidable, then C a is' decidable for every a ~ L, that is, C"(A) is decidable for any A ~_ S and any a E L.

Proof. If C is decidable then, for every A ~ L s, the mappings

C ( A ) : N - o L and ~(~(A):N---~L

are recursive enumerable , hence there exist two recursive maps hi(A), h2(A):~J x ~J---> L ' such that

C(A)(x ) = sup{hl(A)(x, n) In ~ ~} and -~4(A)(x) = sup{ha(A)(x, n) In E ~}

hence, for every A ~_ S,

Ca(A) = {x ] O(S~)(x) >i ~} = {x ] 3n E ~ h,(S~)(x, n) >t a}


is recursively enumerable and

C"(A) c = {x Ix ~ C~(A)} = {x ] C(S~)(x) < a}

= {x I >>- = {x r 3n • h2(S )(z, n) >1

is recursively enumerable , therefore C"(A) is decidable. []

Remark 5. These two results can be used in order to prove that a multivalued logic C has not such a property. For example, in order to prove that C is not compact is enough to prove that C a is not compact for some a • L.

A c k n o w l e d g e m e n t

The author thanks the referees for the valuable comments that have improved this paper.

References

[1] L. Biacino and G. Gerla, Fuzzy subsets: a constructive appraoch, Fuzzy Sets and Systems 45 (1992) 161-168. [2] D.J. Brown and R. Suszko, Abstracts logics, Dissertationes Mathematicae 102 (1973) 9-42. [3] J.L. Castro and E. Trillas, Sobre pre6rdenes y operadoes de consequencias de Tarski, Theoria 11 (1989) 419-425. [4] J.L. Castro and E. Trillas, On the semantic of implication, Proceedings of First International Conference on Fuzzy Logic &

Neural Networks, Iizuka (Japan) (1990) 719-723. [5] J.L. Castro, Contribuci6n al estudio de modelos 16gicos para la Inteligencia Artificial, Ph.D. Thesis, Universidad de Granda

(1991). [6] J.A. Goguen, The logic of inexact concepts, Synthese 19 (1969) 325-373. [7] V. Murali, Lattice of fuzzy subalgebras and closure systems in I x, Fuzzy Set and Systems 41 (1991) 101-111. [8] J. Pavelka, On Fuzzy Logic I, Zeitschr. f Math. Logik und Grundlagen d. Math. Bd.25 (1979) 45-52. [9] J. Pavelka, On Fuzzy Logic II, Zeitschr. f Math. Logik und Grundlagen d. Math. Bd.25 (1979) 119-134.

[10] J. Pavelka, On Fuzzy Logic III, Zeitschr. f Math. Logik und Grundlagen d. Math. Bd.25 (1979) 447-464.

fuzzy logics as families of bivaluated logics

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