Zcituhr. f. muih. Logik uld Grundlaqnr d . Math. Bd. 25, S. 4 4 7 - 4 6 4 (1979)

ON FUZZY LOGIC I11

Semrrntical completeness of some many-valued propositional calculi

by JAN PAVELKA in Praha (CSSR)

Introduction

The preceding two papers of this series

a) introduced the general concepts of complete-lattice-valued syntax and semantics on

b) exemplified these concepts by a class of complete-residueted-lattice-valued proposi-

Now we come to speak of the first non-trivial question which, in a way, puts to the test all the definitions presented so far, viz. the question of balance of syntax against semantics.

Every choice of an enriched complete residuated lattice & = ((L, 8, +), S ) of a certain type ( A T : A + N,, Ex: A + N*), plus a set P of propositional variables, poses the problem of axiomatizability of the 8-valued propositional calculus over P. According to [3], M. 25, the question actually goes like this:

(Q) Do there exist an L-fuzzy set of logical axioms A : F(P, L, A ) --f L and a set W of L-valued rules of inference in F(P, L, A ) such that for any L-fuzzy theory X: F(P, L, A ) + L over P and any formula e) E F(P, L, A ) the degree ( g y ( p , 8 ) X ) e), to which q~ follows from X in the L-semantical system ( F ( P , L , A ) , Y ( P , a)), equals exactly the degree ( q A , @ X ) q ~ , to which q~ is provable from X in the L-syntactical system ( F ( P , L, A ) , A, W ) ? l )

In this paper we investigate the case when the underlying lattice of B is a chain.

an abstract set of formulas ([3]);

tional calculi ([4]).

We answer (Q) for

(1) (2) I n comparison with (1)) where the answer is affirmative all over, the answer in the case (2) is mostly negative, but quite interesting, too: of all the 2"o isomorphism types of residuated lattices carried by I only the one represented by the EUKASIEWICZ inter- val L = (I, @, +) can serve as a basis for a semantically complete propositional calculus.

As expected, the answer to (Q) does not depend on the cardinality of P. It is also not surprising that - a t least as far as the results of this paper are concerned - it does not depend on the additional connectives (i.e. on the enrichment 8 of L) either.

all finite chains Cm+l = (0 = a, c a, c . . . c a , = l}; m 2 1 ; the unit interval I of real numbers.

1) Observe that this oxiomatizability is an attribute of the whole L-semantical system and does not SO far involve any notion of effectiveness.

448 JAN PAVELKA

After all, that was what we intended when we specified the restrictions placed on the interpretations of additional connectives, namely : to have more connectives a t our disposal in case a concrete field of application called for them, but to manage with the basic ones anyway.

About the technique employed: the negative part derives from the results of [3] - one considers the family of proofs necessary for attaining the syntactically indicated truth value of a formula and finds a discrepancy between its behavior and some prop- erties of the semantically induced consequence operation.

The positive part learns heavily on the work of H. RASIOWA and R. SIKORSKI. The main idea in the proofs of the conipleteness theorems is taken over from [ 5 ] : g' iven a fuzzy set X : F(P , L, A ) + L of non-logical axioms one first uses the syntax in ques- tion to obtain a factoralgebra 8 ( X ) = F(P, &)/z of the algebra of formulas F(P, &), which closely resembles the truth-value algebra ft7. Then the upper estimates of the semantically indicated values (Vy(,, 6+C) cp; cp E F(P, C, A ) , are obtained by means of an ultrafilter trick (like in 1151, our proofs are non-constructive), which reduces the LINDENBAUM-TARSKI algebra E ( X ) to ,8.

We assume that the reader is acquainted with [31 and [4]; we shall refer to them all the time.

1. An incompleteness theorem As we have already suggested, the axiomatizability of a (P , 8)-propositional calculus

depends above all on the base structure of 8, i.e. on the complete residuated lattice L = <L, 0, ->. Moreover, it may even happen that the nature of the underlying lattice L in itself rules out any possibility of making the ( P , 8)-calculus semantically complete, to wit

1.1. Propos i t ion . Let L be a complete lattice that satisfies the ascending chain con- dition but not the descending chain condition. Thew none of the (P , &)-propo.Gtional calculi with & carried by L is axionaatizable.

Proof . Suppose there exists an L-syntax { A , &) on E'(P, L, A ) such that the L-con- sequence operations and %?9(p,b) coincide. Now recall the L-rule of inference ro of [3], 7.1. By [3], 22, adding ro to 3 does not change the L-consequence operation and we have %7A,Lau{rol = %9(,,,6). Owing to [3], 18, there exists, for any X : F(P. L, A ) + L and any formula v, an (92 u (r,})-proof in F(P, L, A ) which targeth in q j and whose value w.r.t. X equals (V9(p,b)X) 9. Consequently, V9(p,G) is compact which contradicts [4], Proposition 3.7.

1.2. Example . Denote by N, the chain N of all natural numhers completed by the adjunction of a universal upper bound 00. Let N, denote the dual of N, so that 0 is the unit and co is the zero of N,. Examples of residuated lattices carried by N,: Put

*

- I

a + D if a , b ~ N (1.1) a & b =

if a = c O or b = c o ;

if a 2 b in N, (i.e. a 6 in N,)

if a < b = co in N,: (1.2) a+,6 = - a if a < b < co in N,

ON FUZZY LOGIC I11 449

if a , b E N if a = ocf or b = co; (1.3) a: @ * b =

if a 2 b in N,

if (1.4)

Then i t is easily verified that Ng) = (N,, residuated lattices in the sense of [4], Df. 1.6.

a -+2b = o,(b - a) / (a + 1)' if a < b < co in N, a < b = co in N,. I.. -

31)1) and N:) = {N,, Q 2 , -+2) are

Since Nm is an infinite descending-well-ordered chain, it hollows from 1.1. that neither fl$ nor fl:) nor any other residuated lattice carried by N, in fact, there are infinitely inany MR-couples on N,) can serve as a basis of a semantically complete N,-valued propositional calculus.

Now consider the general situation when the underlying lattice of & is a complete chain C. If C is not descending-well-ordered Propositional.1 does not give us a clue for deciding once for all MR-couples on C and we have to refine on our argument. To this end we first return to the general setting of [3].

1.3. Convent ion. Let C be a complete chain. Denote by char C the cardinal number defined as follows.

(1) If C is descending-well-ordered we put char C = 1,

(2) otherwise char C is the least cardinal number (x such that for any a E C \ (0 ) that does not have a predecessor in c' there exists an ascending-well-ordered chain B 5 C, = {x E C 1 x < a> with the properties

(i) B is oofinal with the segment C,; (ii) the ordinal type of B is a cardinal number ,9 6 (x.

1.4. Proposi t ion. Let C be a complete chain and let (A,&?) by a C-syntax on a set F . for any C-subset X : F + C and any 5 E F there exists a family 83 = ( w ( J ) ; j E J) of 9-proofs in F with the following properties:

*

- I

(1) card J 5 char C ;

( 2 ) each w ( J ) targets in x;

(3) (9-4 z = v (&X I j E J ) .

Proof , Owing to [3], Th. 16, we have

(1.5) whenever X E CF, x E F. Now fix X and x and consider the value a = (%?zi,L4X) x E C .

Assume a > 0 (if a = 0 then certainly X(x) = 0 and we can put 83 = ({{x))}). a) If a has a predecessor in C we can replace the join on the right side of (1.5) by

max hence there exists an 9-proof w ( O ) in F that targets in x and whose value G(O)X w.r.t. X is equal to a. We can set 83 = (w(O)>.

(U l,sX) x = V { & X I w is an d-proof in F that targets in x)

l) Notice that for any real number z > 1 the map n k z-", n E N; co k 0, is a complete (i.e. all --* I where I denotes the the Cowen interval meets and joins preserving) homomorphism

< I , a, -+'> (0 is the usual multiplioation of reals) of [4], Example 1.11. 29 Ztsehr. f . math. Logik

450 JAN PAVELKA

b) If a = V C , in C there exists a cardinal number j3 I char C and an embedding 6 H c,; E < B of b in C, such that a = V{cf I 5' < 8). For every 5' < @ choose an 9-proof w(€) in F which targets in x and whose value w.r.t. X satisfies the inequality &OX > = c e . Take 2B = {w@); 5' </I}. We have

which completes the demonstration.

I-syntactical system ( F , A , W ) the values 9-proofs in P.

Then the C-consequence operation V = VA,w has the property

(C7) for any C-subset X : P --+ C and any x E F , ( U X ) z = V{(U(X I a)) x I a E F , a finite}.

a 6 V{c, I 5' < B ) 5 V { @ X I E < B } 6 a ,

1.5. Example . For the unit interval of reds we have char I = q,. Thus for any x can be attained by sequences of

1.6. Corollary. Let C be a complete chain and let ( A , & ) be a C-syntax on a set F .

Proof. Given X and x, take the family 2B = {w(J ) ; j E J } guaranteed by 1.4. For j E J let w(j) = ( W ~ J ) , . . . , wiij ,) . Then

(%A,&) x = V{&(J)X I i E J> 5 v{(VA,rP(x I {r'@, . * * , Y;:,)})) x I i E J >

I V{(V.~,dX I G ) ) x I E; F , G finite} 5 (V-~,~PX) z.

1.7. Theorem. Let L = (0, 6, -+) be a complete residuated chain. Denote by z the order topology on C, given by the open basis a = {{x E C' 1 a < x < b} I a , b E C } .

If the residuation -+ in L is nol continuous as a map ( C , t) x ( C , t) + ( C , z) then no (P, &)-propositional calculus with d based on L is aziomatizable.

Proof. Recall from [4] that in any residuated lattice the residuation

[R 1)

(R2) By a straightforward though rather tedious discussion of local characters in ( C , z) one obtains l) the following

Lemma. The residuation in L = ( C , 6, +) is a continuous map ( C , t) x (C, z} -+

-+ ( C , t} iff it satisfies, beside ( R l ) and (R2) also the conditions

(R 1 !)

(RZ!)

then U9(p,8) fails to satisfy ((2'7) for any P and any d based on L.

preserves all meets in the second variable ;

sends all joins to meets in the first variable.

for any a E C the function b c* (a + b ) : C + C preserves all nonempty joins ; for any a E C the function b c, (b -+ a ) : C + C sends all nonenapty meets to joins.

Owing to 1.6 it now suffices to show that if + does not satisfy both ( R l ! ) and (R2!)

1. Suppose + does not satisfy (R1 !). There exist a E C and 0 + B E C with

c = V(a--+ b l b E B } < a + (VB) .

l) Hint: From (RI) and (R2) it follows that

(RO) One can uee this to show that -t is continuous on the square iff it is continuous in each variable separately.

+ is entitone in the fist and isotone in the second variable.

ON FUZZY LOGIC III 451

Given P and some 6 based on L we choose po E P and put

1 if e ) = b = e - p o , b E B 1 0 otherwise. X(cp) =

Then (‘Z?y(p,8)X) (a =- po) = a 3 (VB), yet (‘i?g(p,I)(X 1 G ) ) (a =S p,,) 5 c for any finite G g P(P, L, A ) . Conclusion: ‘ifg(p,g) does not satisfy ((27).

2. If (R2!) does not hold in L then there exist a E C and 0 + B g C with c = V { b + a I b E B ) < (AB) -+ a. Given P and b, again choose po E P and put

1 if c p = p , , + b , b ~ B I 0 otherwise. Y(q4 =

We have (ey(p , i )Y) (PO * a) = ( A B ) + a7 yet (@Y(P,&)( Y I a)) (PO => a) S c for any finite a! E F(P, L, A ) . Again, W9(p,8) does not satisfy (C7).

1.8. Corollary. As J. MENU and the author proved in [2], any residuated lattice carried by the unit interval I such that the residuation is continuous on I x I is nec- essarily isomorphic with the LUKASIEWICZ interval L = ( I , 0, +). Hence L and its isomorphs remain as the only plausible buses for axiomatizable I-valued popsi t ioual calculi.

2. Some universal sound rules of inference

Within the constraints that follow from the negative results of Section 1 let us now turn our attention in the opposite direction. Given a set P, a complete residuated lattice L = ( L , 0, -+) which satisfies ( R l ! ) and (R2!), and some € = ( L , 0) based on L we shall look for an L-syntax ( A , 92) on F ( P , L, A ) which is sound w.r.t. Y ( P , 8) on the one hand, but on the other hand is so strong that the L-consequence operations

In devising the L-fuzzy set A of logical axioms we shall use the estimates of tauto- logicaldegrees given in 141. Thus it remains to introduce the rules of inference. In ac- cordance with [3] an n-ary L-rule of inference on F(P, L, d) will consist of a partial n-ary operation r’ on F(P , L, A ) together with a total n-ary operation r” on L satisfy- ing the condition

(SC) In all cases we shall also check that the rule r = (r’, r”) in question is sound w.r.t. Y ( P , b), i.e. that

( S l ) Tr‘(V1,. . ., e),J 2 r”(Tvl, . . ., Tq,J holds whenever (v1 . . . p,,) ED^' and

where n is the arity of r.

L = (L, @, -+), tLe L-detachment rule

and @g(p,8) on F(P, L, A ) coincide.

in each variable, r” preserves all nonempty joins in L.

€ y“(, b),

2.1. Propos i t ion . For any P and’any enriched complete residuatrd Zattise d = ( L , 0),

29;

452 JAN PAVELKA

(i.e. rl = (ri, r 3 , D 4 = {(q, q => y ) I q, y EW, L, A ) ) , rl(q, q => y ) = y and r;‘(a, b ) = a 0 b for any a , b E L ) is a binary L-rule of inference on F(P , L, A ) sound w.r.t. 9 ( P , 8).

Proof. The distributive laws ( M l ) and (M2) ([4]) say that ry fulfils (SC). More- over, for any T E Y ( P , &) the adjointness inequality (A”) together with the commu- tativity of 8 yield

Tri(pl, q * y ) = T y 2 Tq 0 (Tq + T y ) = Tq 0 T(pl * y )

= r m q 3 T(q * y ) ) SO that rl is sound w.r.t. Y ( P , 8).

2.2. Remark . Observe that the inequality (M9) of [4] may be restated in the form

(2.1) Using (2.1) we shall show that unless L = (L, @, +) is a boolean algebra the L-detach- mnt rule in itself does not suffice for making an L-syntax on F(P , L, A ) complete w.r.t. Y ( P , 8). First we shall prove the following general proposition.

2.2.1. Propos i t ion . Let 1; be a complete lattice and let ( A , W ) bp an, L-syntax on a set F such that every r €9, Ar(r) = n , satisfies the condition

(2.2) Then for any X E L’ and any x E F the inequality

ry(a, b ) s a A b for all a, b E L .

r”(a,, . . . , a,,) 5 a, A . . . A a,, for all a,, . . ., a,, E L.

(4.3) (V>,,aX) x b v BX holds in L where I3 = VA.,B, b = V { X y I y E F } .

Proof. By (1.5) i t suffices to show that

(2.4) holds for any W-proof w = ( w l . . . w,,,) in F ( w l denotes the target formula of w).

We shall proceed by induction on the length of 20. Since ( ( 2 ) ) x’ = X ( z ) b and

((x, 0)) X = A ( x ) s B ( x ) hold for any X € L F and any x E F we can concentrate on the only non-trivial induction step when the last member w , of w has the form

GX 5 b v B ( w 1 )

A

n

W n , = ( x , r, (il, . * * > i , , ) ) . Assuming that (2.4) holds for all proofs with length less than m we obtain (cf. [3])

GX = r”(fi(,l)X, . . ., r”(b v B(rwl l ) , . . . , b v B(Fw,n))

= r‘ f (B(rw,l) , . . . , B(Twln)) v v{r’’(c:f. . . . . ctf) I 0 + M E 2’11, where

b if k E M . B ( r ~ i k ) if k $ l l f ’

k = 1 , . . . > n . c f =

From (2 .2 ) i t follows that r”(ci‘, . . ., ct‘) 5 b for any 0 .f; M E (1, . . ., n>. Next, since B is closed w.r.t. r we also have

r”(B(rwi,), . . . , B(rw;9b)) 5 Br’(rwil, . . . , r w,,,) = B(w1).

Hence GX 5 b v B(w7) and the demonstration is complete.

ON FUZZY LOQIC III 453

Now let us return to our previous situation. Given € which is not based on a boolean algebra we have to show that no L-syntax ( A , &) on F ( P , L, A ) such that all r E 9 fulfil (2.2) is complete w.r.t. Y ( P , 8). We shall need the following facts.

(1) If (A, ,%?,) and { A 2 , & ) are L-syntaxes on a set F and if A , s A , and

( 2 ) If a residuated lattice L = (L, 0, -+) satisfies the equality gl t g2 then ~ A ~ , B ~ 5 % A ~ , W , .

(2 .5) then L is boolean and @ = A.

obtained by the following argument: if L satisfies (2.5) then

1 = a v (a -+ 0)

((1) is an immediate consequence of the definition of %-4,W. The statement (2) is

x = z 0 1 = z @ [ x v (z -+ O ) ] = ( .@z)v[z@((Z-+O)] = ( z @ . X ) v O = x @ z

holds for any z E L hence @ is idempotent. Therefore x A y = (z A y)2 s x @ y for all z, y E L , and (M9) yields @ = A.)

Now suppose ( A , 2) is sound w.r.t. .Y(P, a). Since L is not a boolean algebra there is some a E L with a v (a -+ 0) < 1. Owing to (1) we can assume A = B = %9(p,8)0. Choose p , E P and put

a if q~ = Po { 0 otherwise. 9 q I =

By 2.2.1. it follows that (%.L,dX) (a =+- p,) 5 a v B(a po) = a v (a -+ 0) < 1 ; on the other hand, ( g : S p ( P , B ) X ) (a + p,) = 1 because T ( a po) = a -+ Tp, = 1 holds for any T E Y ( P , 8) with Tp,, 2 a. Indeed, ( A , 3) is not complete w.r.t. Y ( P , a).

The counterexainple used in the preceding argument may serve as a hint for the introduction of the following L-rules of inference on F(P, L, A ) .

2.3. Propos i t ion . For m y P, any t" = ( L , 0) such that L = (L, @, -+) fulii1.s (Rl!), and any a E L the (L , a)-lifting rule

r2a: L(-) b a = p a - + b

( i . e . D(r;a) = F(P , L, A ) , r;a(qI) = a L-rule of inference on F(P , L, A ) which is sound u1.r.t. Y ( P , 8).

p E F(P , L, A ) , a E L. Then Tria(p) = T(a =+ p) = a -+ T p = rya(Tpl).

exists a binary operation ( a , b) t+ a t 0 on L such that

(A) a v b z c iff a z b t c

holds for any a, b, c E L . From [4], Proposition 1.2, we know that the couple ( v , t) satisfies &) iff the inequalities

(i') a 2 b + ( a v b ) , (A") (a c b) v a >= b

hold in (L, V. -).

p, ri'a(b) = a + b for all b E L ) is a unary

Proof . The assumption ( R l ! ) says that r i a fulfils (SC). As to (S l ) , let T E Y ( P , 8),

2.4. Let L be a bounded lattice and let 2 denote its dual. 2 is brouwerian iff there

454 JAN I’AVELKA

Example . If L is a bounded chain then E is brouwerian and we have

0 if a z b i n L b otherwise.

a c b =

Proposi t ion. Let L be a complete lattice whose dual 1 is brouwerian. Then for any P and any d carried by L, and any a E L the (L, a)-elimination rule

r,a: --- - ( a P b )

(i.e. D(r;a).= {p v a I rp E F(P , L, A ) } , ria(rp v a) = q, rya(b) = a c b for all b E L ) i s a unary L-rule of inference on F(P, L, A ) which is sound w.r.t. 9 ( P , b).

Proof. The condition ( R l ) , fulfilled by the residuated lattice (l, V , t) says that for any a E L the function t t-+ a t t preserves all meets in E , that is, all joins in L. Hence rya satisfies (SC) for all u E L . If T E 9 ( P , 8) then by (A) we have for any p E F(P, L, A ) and any a E L ,

Tr;a(rpva) = T p ~ a a ( T r p v a ) = a c T ( r p v a ) = r y a ( T ( r p v a ) ) . 2.5. Propos i t ion . For any P , any 8 carried by L, and any a E L the (L, a)-con-

sistency-testing rule r4a = (ria, rya) where D(r:a) = {a} , r;a(a) = 0, and

0 if b 5 u.

1 if b S a rya(b) =

i.3 a unary L-rule of inference on F(P, L, A ) which is sound w.r.t. Y ( P , 8). Proof. Let a E L. For any K either b g a for some b E K so that V K g a and

L, K + 0

r y a ( V K ) = 1 = V{rya(b ) I b E K } or b a holds for all 6 E K so that V K 5 a and

r y a ( V K ) = 0 = V{rya(b ) I b E K } . Thus rya satisfies (SC) for every a E L. If T E 9 ( P , &), and a E L then Tr;a(a) = TO = = 0 = rya(a) = r:a(Ta) so that r4a is sound w.r.t. Y ( P , 8).

3. The completeness theorems

Also in this section we shall stick to the notation introduced in [3] and [4]. Accord- ing to [3] we shall often write “ X t., rp” instead of ‘‘(%A,sX) rp 2 a”. The symbols ai(rp, y, x) , &(rp, y ) , x i ( q ) , and t i (p, a, b , a‘, b’, n) will denote the schemata described in [4], 3.3 through 3.6. When the lattice L of truth values and the set A of additional connectives are clear from the context we shall abbreviate “F(P , L, A ) ” to “ F ” . 3.1. Theorem. Let P be a set; let 8 = ( ( L , @, +), 0) be an enriched complete re-

sidu.ated lattice of the type ( A T : A + N,, Ex: A -+ N*). Let (A,&!) be an L-syntax on the set P(P, L, A ) such that

(i) 92 contains rI and all r,a; a E L ; (ii) for any a , 6 E L , 0 t a u , 0 t n B h a & b, 0 kcr- .ba =- b, 0 l-,a;(a, b ) , i = 1, 2 ;

ON FUZZY LOQIC In 455

(iii) if d E A , Ar(cl) = n, and a,, . . ., a, E L then

0 kl&al,. . .,a,,);

(iv) for any rp, ly, x E F ( P , L, A ) ,

(v) if d ~ d , Ar(d) = n, and rpl) . . ., v,~, y l , . . .) y,, E F(P, L, A ) then

f l t - t ~ t ( v , y , ~ ) , i = 4 , . - - , 1 6 ;

0 t-l&(rpD - . * f vn, y1,. * * ) lyn) .

Then the following statements hold for any X : F ( P , L, A ) + L. (1 ) The relation d defined on F = F ( P , L, A ) by

(3.1) a s y iff Xt-le,*W i s a preorder on F with the property

(3.2) a $ q~ iff X tap,; a E L , rp E F . (2 ) The equivalence x on F defined by

(3.3) cp x y iff both 9 $ y and y s rp

i s a congruence on the algebra F(P, 8).

(3) For rp E F denote by fj the clam of e, modulo x. Put F ( X ) = F/x. Let

F ( x ) = ( F ( X ) ; {z; u EL}, A, V, 0, +, (6; d Ed}) be the factoralgebra of P (P , 8) by x . Then the algebra

q x ) = ( F ( x ) ; 6, i, A, V, 0, -+, (5; a EA})

is an enriched residuated lattice of the same type as 8.

phism 8 + 8 ( X ) , which furthermore preserves all joins.

implies X kU rp.

and y

(4) Denote by j the map L + F ( X ) that sends a to the class z. Then j is a homomor-

P r o o f . First observe that for any X EL^ and any rp E F , a E L the statement 0 t-, rp

(a) Thus for each rp E F the inequality rp d q~ is guaranteed by X t-, a,(rp). If y x we apply r1 to

t1 y * x , k1 a6(V, w) x) and obtain

(3.4) t-PT(l,l)=l ';(y =S xt U6(V> w) x ) ) = (V * y) * (p' => x). The application of rl to X k1 rp * y and (3.4) then yields X k1 e, => x hence rp s x . Conclusion: is reflective and transitive, i.e. a preorder on F . Clearly, (3.3) defines an equivalence on F. If u rp then we can apply rI to X I-, a and X k a 3 q, and obtain

X t - r l ~ , ~ u , l ~ = a rXa, 0 => q ~ ) = rp. Conversely, if X I-, rp then the application of r,a yields X t-r,,,,a(a)=l r;a(rp) = a => rp, which establishes (3.2).

1, q A y x g.1.b. (9, y), and rp v y x w 1.u.b. ( y , y ) hold in the preordered set ( F , d) whenever q ~ , y E F .

(b) Now we are going to show that 0 s rp

456 JAN PAVELKA

Since X t-, p holds for any p E F , the inequality 0 s p follows by (3.2). X F1 uJp) 1. Let us next verify e.g. that p A y is a greatest lower bound of {p, y>

and A y s y. says that p in ( F , s). From X t-, u,(p, y ) and X t1 a,(p, y ) we obtain p A y d Now assume x s p, x 2 y. By means of rl one easily derives from

X t-1 x * p, X x * W , X t i a,o(~, W, X) the conclusion X t-, x y A y. The proof of 91 v y x 1.u.b. (p, y) is analogous.

y. Then 5 is a partial ordering of F(X). Owing to (b) the connectives A and v preserve the preorder s on F , therefore they respect z . Moreover, if we put

(tp A y), that is, x

(c) For q, p E F ( X ) define $5 5 p iff p

- - @ A 6 = pAp, P V q = p V Y

for p, y E F it follows from (b) that

(3.5) L ( X ) = (F(x); 5, i, A, V)

is a bounded lattice.

(d) We have shown in (a) that for any p, y , x E F one can derive from the assump- x. Thus

on F . Next, applying the detach- tion y s x the conclusion (3.4), which by definition amounts to y * y s p * preserves in its second variable the preorder ment rule rl to

x t1 a,(% y , x) 3

X k1 (p => Y ) * ((Y A X ) * (p * % ) I -

x t-1 C,(Y * x , P * Y , 91 * x) we obtain

Consequently, if p s y then for any x E F we have y * x d p * x so that => in its first variable reverses the preorder s on F. Conclusion: * fulfils (RO) as a binary operation on ( F , s).

(p & y). Moreover, if rp s y * x holds for some x E F then one can apply rl to

(e) Let p, y, x E F . Due to X t, u,,(p, y , x) we have p s y

x F1p => (Y =. x) > x tl a,,(% y> x) and obtain X t, (p & y ) * x, i.e. p & y s x. Consequently, for any p, y E F the for- mula y & y is (up to the equivalence w ) the least element of ( F , d) with the property p s y * x. Hence it already follows that & satisfies (MO) on (F, 2) and (&, a) is an adjoint couple on (F, 2). The properties (MO), (RO) of & and *, resp. imply that both & and * respect z. Moreover, the adjointness condition (A) for & and * trans- fers to the poset (F(X), 5 ) so that setting

- @ @ q = p & y , @ - - + p = p = y

for any p, y E F we obtain an adjoint couple (0 , -+) on ( F ( X ) , 5 ) . (e) Now X t, o, says that for any @, 6, 2 E F ( X ) the inequality

(R3) 6 -+ 2 6 (7 + 6) --t (@ -P

(9 6 6) 0 I 5 q 0 (q 6 X) holds in ( F ( X ) , 0, +). By [4], Proposition 1.5, @ satisfies (M3), i.e.

ON FUZZY LOGIC JII 457

holds for all @, I& 2 E F ( X ) . From S I-, u, i t now follows that

(R4) for any $5, y, 2 E P ( X ) , cjj 5 y -+ f iff 7 5 $5 + 2. Again, we use [4], 1.5, and conclude that the operation 0 on F ( X ) is commutative. Consequently, p @ (y 0 2) = ( j 0 y ~ ) 0 p 5 2 @ (7 @ p) = ($5 @ p) @ 2 holds for all 4, y, f E F ( X ) hence Q is associative. Finally one can use r, to obtain from

tl u16(p)> tl ‘8(? w, y * p), ‘1 ‘dp? * w, w * 9)

the conclusion

(M4) Hence ( F ( S ) , @, 1) is a commutative monoid and finally

(3.6)

is a residuated lattice.

q 0 i = @ for all $5 E F ( X ) .

L(W = (L(X), 0, -+>

(f) By induction on n we can show that for any p,, . . . , cp,, E F and any a,, . . . , all E L the assumptions X taj vi; ,i = 1, . . . , n, imply X Fa,@. , ,@a,L p, & . . . & pI1. Indeed, for n = 1 the statement is trivial. Induction step: from X I-,, p l i ; i = 1, . . . , n + 1, it follows by the induction hypothesis that X pl & . . . 8: pIl , X Fa,+, pI,+,. Now recall X t, a,,(pl & . . . & p,&, P),,,~) and apply r, twice.

As a matter of fact we have just obtained, for every n 2 1, the derived n-ary L-rule of inference

,

1 a,, * * .,a,, a, @ . . . Q a,L i 91,. * * , pn

pl 8:. . . 8: pa

in our L-syntactical system ( F , A , 9).

1s ( p = = - y ) ~ ( y * p ) = p * y i f f X t , p , - y .

Then X i-, pi CJ y ; ; i = 1, . . . , n, and we can use (f) to obtain

(g) From (3.2) it follows that for any p, y E F , p x p iff 1 2 p =s y, y p iff

Let d E A , Ar(d) = n; p,, . . . , pI1, y , , . . . , y,, E F . Assume pi x y i ; i = 1, . . . , 11.

x El (ql 0 ylpl & . . . & (pl, 0 y, ,p where (k,, . . . , k,,> = Ez(d). Finally r1 applied to the last statement and the assump- tion

(3.7) x F14,(p,,. . . ? plL3 y11 * . * ) YIJ yields

X t i d(pi 9 . . ., ~p,,) 0 d(y1, . . * 7 ~ n ) 9

i.e. d(q,, . . . , p?,J x d(y , , . . . , y,). Thus all the additional connectives d ; d E A , respect x. Since the biresiduation in L ( X ) is defined as

p+b$3= ( @ + F ) A ( F + @ ) = p G y ,

(3.7) now says that the operation - (3.8) d(@i, . * 9 $5,J = 4vi 3 - ., pu); pi 7 . - . a $ 5 l h E F ( x )

458 JAN PAVELKA

on P(X) fits L ( X ) with the exponents k,, . . . , k,, . Therefore

(3.9) is an enriched residuated lattice of the type ( A r : A -+ N,, Ex: A -+ N*>.

(h) If a 2 b in L the assumption X I-(, b yields X t, b and by (3.2) we obtain a 2 b, e.e. ja = a s = jb in L ( X ) . The map j is isotone. This, together with the assump- tion (ii) of our theorem, implies j(a A b ) = ja A jb, j(a -+ b ) = ja -+ j b , and j(a @ b ) 5 ja @ jb for all a, b EL. To satisfy ourselves that j is a homomorphism L + L(X) we still have to show that ja @ jb 5 j(a @ b ) . Anyway, this is equivalent to ja 5 I - jb 3 j(a @ b) , which-follows from a 6 b 4 (a @ b) by the isotony of j .

For any d E A with Ar(d) = n, the assumption X t, &a,, . . ., an), a,, . . . , a, E L, implies that we always have j(od(ul, . . . , a,)) = d(ju,, . . . , ja,,) and j is therefore a homomorphism d -+ & ( X ) . .

L and any Q) E F the assumption

(3.10) a v ; a E K implies VK

& ( X ) = (L(X), {Z; d E A ) )

-

The proof of our theorem will be complete if we show that for any K

9. This, however, follows easily from (3.2) and the fact that (cf. [3])

(%A,.& 9 = v{a E L I x Fa q}* 3.2. Propos i t ion . Let, in addition to the assumptions of 3.1, one of the following

statements hold ~ O T L and ( A , 9). (vi) W contains { r p I a E L) ; (vii) each a E L \ {0,1> i s nilpotent in the semigroup (L , a).

Then #or any X : F ( P , L, d) -+ L either j : 6 --t b ( X ) is injective or 8 ( X ) i s degenerate. Proof. Let X EL" and suppose j is not injective. There exist distinct elements

a, b E L such that ja = jb, that is, a cf b < 1 yet j (a c-) b) = ja ++ jb = jl. (a) If W contains all the consistency-testing rules we can apply r4(a c+ b ) to

X a o b and obtain X t, 0. Hence j l 5 j0 in L ( X ) ; 8 ( X ) is degenerate. (b) If all the elements of L \ (0, l} are nilpotent take n 2 1 such that (a c-) b ) n = 0

in L. Then j l = ( j l )" = [j(a c-) b)]" = j ( (a c-) b)") = j0.

Again, B ( X ) is degenerate.

that b ( X ) is non-degenerate. Then 3.3. Propos i t ion . Assumptions as in 3.2. Assume that X : F ( P , L, A ) -+ L i s such

(1) A n y filter,) ,F in L ( X ) with the property

(3.11) .Fn j(L) = {jl}

can be extended to a filter 9 in L(X) that is maximal u1.r.t. inclusion and (3.11).

such 9 a j-ultrafilter) then for any 1: E F ( X ) , x $ 59 iff (3.12)

(2) If 59 i s a filter in L [ X ) that i s maximal w.r.t. (3.11) (in the sequel we shull call

there exist u E 9, a E L , and n E N, such that a < 1 and 9 @ 16 ja.

I ) see [4], Df. 2.7.

ON FUZZY LOGIC 111 459

Proof . (1) We only have to check the assumptions of ZORN'S Lemma. Let 8 be a chain of filters in L(X) and let each .F E 3 satisfy (3.11). Obviously, us is a filter in L ( X ) and we have

(U3) n j(L) = U { F n j(L) I 9 E $1 = {jl}. (2) Let 3' be a j-ultrafilter in L ( X ) . The condition (3.11) clearly rules out, for any

x E 9, the existence of any u, a , and n with the properties (3.12) because then we should have ja E 9. Conversely, if x E F ( X ) and there exist no u, a , and n with (3.12) then

3" = ( y E F ( X ) I y 5 x" @ TI for some u E 9, n E N4}

is a filter in L ( X ) that satisfies (3.11) and includes 9 u (z} hence 9' = 9 and z E 9.

3.4. Propos i t ion . Assumptions as in 3.2. Let, moreover, L be a chain and let u s have

(3.13)

lattice L ( X ) satisfies the identities

(3.14) (z + y) v ( y + x) = jl,

(2) If X : F(P, L, A ) in & ( X ) i s prime, that is. (3.15) x v y E 9 iff x E <! or y E 9

holds for any x, y E F ( X ) . Proof . (1) follows immediately from 3.1 and (3.13). (2) Let X and 9 satisfy our

assumptions and suppose neither x E 59 nor y E 9. Due to 4.3, there exist u, v E 3, a, b E L \ (l} , and rn, n E N+ such that

0 t-, &(cp, p) for any cp, p E F(P, L, A ) , H E N. (1) For any X : F ( P , L , A ) + L, any x, y E F ( X ) , and any n EN, the residuated

(x v y)" = xR v y". L i s such that & ( X ) is non-degenerate then any j-ultrafilter 9

2" @ u 5 j a , y n @ v 5 jb in L(X) . If we put k = max(m, n) it follows from (3.14) that

(Z v @ (U @ V ) = (x'$ v y',) @ (U @ V ) = (& @ u @ V ) v (yk Q u Q V )

I - (zm @ u) v (y" @ v) s ja v j b = j(a v b ) .

Since 9 is a filter we have u @ v E 9. Since L is a chain, a v b < 1. By (3.12), x v y 4 9. Now we are ready to prove the axiomatizability of all propositional calculi based on

finite residuated chains or the EUKASIEWICZ interval L. We shall first lay out the plan of the proof, common to Theorems 3.5 and 3.7, leaving open only those places which will require individual consideration.

We shall be given an enriched complete residuated chain 8 = ((C, @, +), 0) and a C-syntax ( A , 9i') on F(P , C, A ) which will satisfy the assumptions of 3.4 (therefore also those of 3.2 and 3.1) and we shall want to prove ( A , 9) complete w.r.t. 9 ( P , 8).

(I) We verify that ( A , 9 ) i s sound u3.r.t. 9 ( P , 8).

From [3], Proposition 24 we then obtain the inequality

(:3.16) (g ,,id) 5 ( g S ( l ' , S ) x )

for any X : F + L and any cp E F . Our aim is to show that all the inequalities (3.16) may be replaced by equalities. We fix X : F -+ L and cpo E F . If (U,,X) cpo = 1

460 JAN PAVELKA

there is nothing to be proved therefore assume ( g A , a X ) pa = a, < 1. Since C is a chain and (gY tp ,$ )X) yo = /\{Trp, I T E 9 ( P , &), T 2 X } i t suffices to show for

(3.17)

that

(3.18)

Assume given b according to (3.17). Because a, < 1, b ( X ) is non-degenerate hence by 3.2 the map j : & -+ F ( X ) is injective. By 3.1, a, is the greatest element) of C with ja,

b = a if a has a successor in C and b > a in the opposite case

there exists T E Y ( P , E ) with T 2 X , Tp, 5 b .

@, in L ( X ) . Owing to 3.4, L(X) satisfies (3.14).

(11) We verify that the filter S = {y I y 2 (@, -+ jb)" for some TL E N,)

in L ( X ) satisfies (3.11).

According to 3.3, 9 can be extended to a j-ultrafilter 9 in L ( X ) . By 3.4 the filter 9 is prime.

(111) We prove that for any x E F ( X ) there exists c E C srcch that 5 ct jc E 9. Recall from [4] that 9 defines a congruence on the enriched residuated lattice &(S)

arid denote by f the canonical homomorphism b ( X ) +> 8 ( X ) / 9 . Since 9 A j(C) = ( j l ) the composed homomorphism

f d A b ( X ) -If & ( X ) / 9

s one-to-one. By (111) it is also onto hence it is an isomorphism and the map

T = (f 0 j)-10 f 0 g: F(P , C, d) -+ C,

where g is the canonical homomorphism F(P, &) +> E ( X ) , preserves all connectives with the possible exception of the constants c; c E C, c + 0, 1. Nevertheless, because T(c) = [(fj)-l f g ] (c) = [(fj)-l fj] (c) = c for any c E C, T is indeed a homomorphism F(P, 6') -+ fb and hence belongs to Y ( P , a).

For any y E P we have X FAY(,+,) y hence j[X(y)] 5 p in L ( X ) and finally

T ( y ) = [(fj)-l fsl ( Y ) = [ W l I [ f ( ~ ) l 1 [ ( f j ) - l fJl [X(w)I = X ( Y ) . Conclusion: T 2 X . Moreover, from 4, -+ jb E 9 E 9 it follows by [4], 2.10, that

5 fj(b) in € ( X ) / 9 hence

T(po) = [(W fsl (9,) = [ ( f j ) - l fI 5 L(fj1-l f j l ( b ) = b . The proof is complete.

Convent ion. For given P and b = ( (L , 0, -+), 0) we put

Z =

ZA =

{ai(a, b) I i = 1,2; a, b E t} v {ai(y, y, X ) I i = 4, * . ., 16; q, W,

{&a,, . . .,ah@,) I d Ed, a,, . . ' >

E P(P, L, A ) ] >

EL}

v (a$(qi . . . ~)Ar(ct.) 3 y1 * * . Y.Ar(d) ) I d ~ 1 , * * * j ~ ) . 4 r ( d ) , y1, . . . ) Y n r ( d ) E F(P , L , d ) } ,

= { L ( y , W) I E N ; 9, y E F(P7 L, I}.

ON FUZZY LOGIC III 461

If L = C,,I+l, m >= 1, we put

= ( x , ( y ) 1 0 5 i S m - 1 , y E F ( P , CIII+lt A ) ) If L = I we put

4, = {~ , (p , a, b. a', b', n) I 1 > b' > b 2 0, 1 2 a > a' 2 0,

9, = ( ~ ~ ( y , a , b , a', b', n) I 1 > b' > b 2 0, 1 2 a' > a 2 0,

n E N,, n . a + b 5 n . a' + b' ; p E P(P, I , A ) } ,

n E N,, n * a - b 2 n * a ' - b'; cp E F ( P , I, A ) } . Observe that the sets of formulas defined above are always pairwise disjoint.

3.5. Theorem. Let L = (Cn,+l, @, -i) be any residuated (m + 1)-element chain, nL 2 1 . Given a set P of propositional variables and an enriched residicated lattice d bused on L set

if y = a . a E Clnfl a 0 b if cp = a & b ; a , b E C,,,+l

(3.19) ~ ( c p ) = a -+ b i f q~ = a =. 6 ; a, b E C,,r+l if ~ E C L J L " V A U X otherwise :

l a I:

(3.20)

Then fhp CI,L+l-syntax ( A , W) on F ( P , C I I l f l , A ) is complete with respect to the C,l,+l- semantics 9 ( P , 8).

W = {rt} u {r,a I i = 2, 3 , 4 , a E C,,t+l).

Proof. We shall adhere to our plan. Let

0 = b, < b, < . . . < b,, = 1

be the elements of Cmfl. We shall ju- t fill in the blocks (I), (11), and (111).

3.3, 3.4, 3.5.

k 5 m - 1. Should there exist c E CllIfl with c < 1, jc E 9 we would have

(I) The soundness of ( - 4 , 9 ) w.r.t. Y ( P , a) follows from 2.1, 2.3, 2.4, 2.5, and [4],

(11) Given X : F -+ C,l,+l and y o E F we assume that (%A,sX) q ~ , = b,, for some

(@o -+ j bk ) < = J ' c

X I-, 4 q 1 o ) = (90 * b/c) v (hL+i * q ~ o )

for some n E N,. By the definition of A we have

hence (Po + jb,,) v (jbhfl -+ lo) = j l in L ( X ) .

Since L ( X ) satisfies (3.14) it follows that

jl = ( j l ) l L = (@, -+ jb,)" v (jbh+l + @o)n 5 jc v (jbh+l + lo); in other words, X t, (bkfl X krd'rcl,=l bkfl

q~,) v c. Now we apply the elimination rule rgc and obtain vo, which contradicts the assumption that bk is the last element of

CnL+l such that X kbk TO.

462 JAN PAVELRA

(111) Given 2 E F ( X ) put

(3.21)

Clearly, 0 E 0,; D, is an initial segment of Cn,+l, and 1 E H,; H , is a final segment of C n l f l . By (3.14) we have (jc + 2) v ( x + j c ) = jl for any c E Cnl+,. Because 59 is prime we thus obtain D, u H, = CIIl+l. Suppose 0, n H, = 0. Then for some p 5 nz - 1, b, is the last element of D, while bp+l is the first element of H,. We have x = y' for some y E P . Because X I-, x,(y) we can repeat the argument from (11) and arrive a t the absurd conclusion that either 6, E H , or bp+l E 0,. Consequently there exists c, E D, n H,. Then 2 t.) jc, = (x + jc,) A (jc, + x ) E 9, which we aimed to prove.

3.6. Corollary. With P , 8, and (A,&?) as above there exists, for any given X : F ( P , Cm+l, A ) 4 CInfl and ~1 E F(P , C,lr+l, A ) , an .%'-proof w in F ( P , Cln+l, A ) such that wl = ~1 and 6X = (%9(p,l)X) p.

3.7. Theorem. Let t = ( I , 0, 4) 6~ the tukasiewicz interval with the rnultiplica- tion a @ b = 0 v (a + b - 1) and the residuation a + b = 1 A (1 - a + 6 ) . Let P be a set of propositional variables and let & bP an enriched residuated lattice based on L. Pwt

0, = {c E Clnf l I jc -+ x E S}, H , = {c E C,,l+l I x 4 jc E g}.

a if g , = a ; a ~ I a @ b if y = a & b ; a , b ~ Z

1 if ~ E G W . X ~ ~ A ~ Y ~ U J , (3.22) A(cp) = a + b if cp = b ; a , b E I 1 0 otherwise ;

(3.23) Then the I-syntax (A , 9) on F(P, I , A ) i s complete with respect to the I-semantics Y ( P , 8).

3 = (PI} u (r2a; a E I } .

Proof . Again we shall fill in the blocks (I), (11), and (111). (I) The soundness of ( A , 9) w.r.t. Y ( P , 8) follows from 2.1, 2.3, and [4], 3.3,3.4,3.6.

(11) Given X : F ( P , I , A ) 4 I and p, E F ( P , I , A ) we assume that (%A,9X) ~1, = = a, < 1 and consider all the elements b E I, b > a,. Fix one such 6 and suppose there exists c E I with c < 1 and jc E 3. Again,

(@, + jb)Iz 6 jc

for some n E N+ and we have

jl = (jl). = (q, + jb)" v (jb + Po)" 5 j c v (jb + @O)n 6 jc v (jb + @,).

Since c is zero or nilpotent in ( I , 0 ) there exists rn E N+ with c"' = 0 in (I, @). By (3.14) we have

jl = (j l)"L = (jc)In v (jb + @o)m = j0 v (jb + +J" 5 jb 4 Po. Consequently, jb Po, which contradicts the assumption 6 3 p,.

(111) Given x E P ( X ) we again define

D , = { c ~ Z ~ j c + x ~ 5 9 } , H , = { c ~ I 1 x + j c ~ 5 9 } .

Again, 0 E D,, 1 E H,, and 0, v H, = I . Now se shall show that both D, and H , are closed in the usual topology on I .

ON FUZZY LOGIC III 463

a) First we prove that I \ 0, is open in I . Fix c E I , c # 0,. Because the assump- tions e E D,, e' 5 e always imply e' E 0, i t suffices to find some c' < c with c' $0,. By (3.12) there exist u E 3, e E I , and n E N, so that e < 1 and (jc -+ 2)" @) u 5 je. Hence u 5 (jc + 2)" + je E 8. Choose e' such that e < e' < 1 and put C' = = c - (e' - e)/n. Take some y E F with $? = x. We have

X k1 ((c =. y)" =. e) =. ((c' 3 y)" =- e'), that is,

( j c -+ 2)" + j e 5 (jc' -+ 2)"

in L(X), therefore (jc' -+ 5)" -+ je' E 8, which clearly rules out jc' -+ x E 8. Conclusion: c' # 0,.

je'

b) The proof that I \ H, is open goes by a dual argument using 9,. Because I is connected there is some co E 0, n H,, which then satisfies x w j c , E 9. 3.8. Corol lary (see 1.5). With P , &, and ( A , W ) as above there exists, for any given

X : F(P, I, A ) -+ I and rp E F(P , I , A ) , a sequence {w(I'); n E N} of 9-proofs i n F(P, I , A ) so that each w(") targets in rp and d")X /" (g9 (p ,8 )X) q ~ .

3.9. Corollary. Since all the properties of the Lukasiewicz MR-couple (8, +) on I that were employed in the proof of 3.7 are preserved by isomorphisms, any (P, &)-props i - tional calculus based on an isomorph of L is axiomatizable. (In [ 2 ] it is proved that e.g. for any order-reversing involution f : I x I there exists a residuated lattice L = ( I , @, +') isomorphic to L so that a +' 0 = fa for all a E I . ) 3.10. Remark . In the particular case of the (m + 1)-element EUKASIEWICZ chain

L,jr+l we can also use the fact that all the elements b, , . . . , bin-, are nilpotent in (C,n+l, @) and achieve completeness with the detachment rule rl and the lifting rules r,bo, . . . , rzbln only.

3.11. Remark (cf. [l I). 1. Originally, the infinite-valued LUKASIEWICZ logic dealt only with H~ values. Nevertheless, since any substructure of ( I , @, +) is either finite or dense in I and since we need a complete lattice of truth values we have to deal with the whole of I.

2. In the original LUKASIEWICZ propositional calculi there are just two primitive connectives, namely the i m ~ ~ i c a € i o n and the negation. Anyway, if L = L,,,+l, m 5 1, or L = L then L satisfies the identities

-

a A b = ((a + 6 ) -+ (a -, 0)) -P 0,

a v b = (a -+ b ) + b,

a @ b = ( a + ( b - + O ) ) + O .

(3.24)

Thus, if we derive the logical connectives A, v, and & in accordance with (3.24), we obtain a modified (P , <L, 0))-propositional calculus which has, apart from the con- stants a, just one primitive * and which again is axiomatizable.

Added i n proof : I n 1977 the results of this paper were extended by the author to first order predicate calculi. As in [5], the proofs of the completeness theorems introduced a topology on the set of all j-ultrafilters in L ( X ) and used a topological argument to make surc that the desired factorization of L(X) is well-behaved with respect to quantifiers.

464 JAN PAVELRA

References [I] LUKASIEWICZ, J., Selected works. Studies in Logic and the Foundations of Mathematics, North-

[2] MENU. J., and J. PAVELKA, A note on tensor products on the unit interval. Comment. Math.

[3] PAVELKA, J., On fuzzy logic I: Many-valued rules of inference. This Zeitschr. 25 (1979), 45 -62. [4] PAVELKA, J., On fuzzy logic 11: Enriched residuated lattices and semantics of propositional

151 RASIOWA, H., and R. SIKORSKI, The mathematics of metamathematics. PWN, Warszawa 1963.

Holland Publ. Comp., Amsterdam, and PM'N, Warszawa 1970.

Univ. Carolinae 17 (1976), 71 -83.

calculi. This Zeitschr. 26 (1979), 119- 134.

(Eingegangen am 14. Marz 1977)