E L S E V I E R Fuzzy Sets and Systems 85 (1997) 63 72

F U | | Y sets and systems

Rules of inference in fuzzy sentential logic

Esko Turunen Lappeenranta University of Technology, P.O. Box 20, FIN-53851 Lappeenranta, Finland

Received August 1994; revised August 1995

Abstract

Rules of inference in fuzzy sentential logic are studied, in particular, some instances of fuzzy rule of inference are established. An example of how to apply these rules is given.

Keywords: Fuzzy sentential logic; MV-algebra

1. Introduction follows:

The aim of this study is to introduce some fuzzy rules of inference. Each of them has a counterpart in classical logic. We consider injective MV-algebra valued fuzzy sentential logics as these fuzzy infer- ence systems are proved to be semantically com- plete (cf. [9]). It is of special importance to realize that in the unit interval the only semantically com- plete fuzzy sentential logics are the injective MV- algebra valued fuzzy logics. It is well-known that any nilpotent Archimedean T-norm generates an injective MV-algebra; moreover, injective MV- algebras in the unit interval are exactly the structures isomorphic with the Lukasiewicz MV- algebra.

By a rule of inference in classical logic, we mean (cf. [8]) an operation which with a finite sequence of formulae ~ . . . . . ~. (1 ~< n) in a formalized lan- guage associates another formula fl in this language in such a way that fl is a logical consequence of formulae al . . . . . ~,. This fact is usually denoted as

0~1, . . . ,0~n (,)

Formulae cq, ... ,7, are called the premises and /~ the conclusion of(*). By saying that a formula/~ is a logical consequence of a set S of formulae we mean that if every formula ~ belonging to S is acknowledged to be true, then/~ must also be ac- cepted as true. Thus, the most important property of the rule of inference is soundness, i.e., rule of inference preserves truth.

One of the main problems in defining fuzzy logic is how to define fuzzy rules of inference. Clearly, to be able to introduce a fuzzy rule of inference, we first have to define truth and validity in fuzzy logic. By starting with these semantical concepts we elim- inate the possible inconsistency of our logic, i.e. that not everything is provable - a fact which is unfortu- nately all too often neglected in various fuzzy infer- ence systems.

0165-0114/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved SSDI 0 1 6 5 - 0 1 1 4 ( 9 5 ) 0 0 3 2 4 - X

64 E. Turunen / Fuzzy Sets' and Systems 85 (1997) 63- 72

In this paper we follow Pavelka [7] and define a fuzzy rule of inference as consisting of two com- ponents. The first component operates on formulae and is, in fact, a rule of inference in the usual sense; the second component operates on truth values and says how the truth value of the conclusion is to be computed from the truth values of the premises such that the degree of truth is preserved.

After recollecting and proving some necessary algebraic definitions and results, we introduce several new fuzzy rules of inference and give a prac- tical example of how to use them. The advantage of our approach is the semantical completeness of our logic; the degree of validity of any formula co- incides with the degree of deduction of the formula.

2. Preliminaries

In this section we recollect some well-known results on algebra and prove some new ones. These results will be used in Section 3. Let (L, ~<, A, v ,0 , 1) be a complete, bounded lattice with the least element 0 and the greatest element 1, and endowed by a binary operation 63, called mul- tiplication, such that the following holds:

63 is commutative, associative and isotone on L x L, (1)

a63 0 = 0 , a631 = a , (2)

V (ai 63 b) = ( V ai) 63 b. (3) i~F \ i ~ F /

Define a binary operation --* on L, called residua- tion, by the formula

a ~ b = V{c la63 c <~ b}. (4)

Then the system L = (L, ~<, /x, v , 63, --% 0, 1) is called complete residuated lattice and (63, --* ) is an adjoint couple on L (cf. [7]). In a complete re- siduated lattice L, it holds that

a Q b ~ < c if and only if a~<b ~ c ,

aG(a ~b)<~ b,

(a ~ b) 63 (b ~ c) <~ (a --* c),

a63b<<.a^b<-..a,b<<.avb.

(5)

(6)

(7)

(8)

By biresiduation we shall understand the derived operation

(a ~ b) = (a ~ b) ^ (b ~ a). (9)

We define

a* = a ~ 0 (10)

and say that a* is the complement of an element a of L.

By (7), it holds that

(a ~ b ) Qb*~<a* . (11)

Recall (cf. [6]) that an MV-aloebra is an algebra L = (L, Q, @,*,0, 1) of type (2, 2, 1,0,0) such that the following holds:

a @ 0 = a, (12)

a @ b = b @ a , (13)

a • 1 = 1, (14)

a * * = a , (15)

0* = 1, (16)

a 63 b = (a* E) b*)*, (17)

(a* E) b)* 0) b = (b* ~) a)* @ a. (18)

The operations G and @ are commutative, isotone and associative. Given an MV-algebra L, we define a binary operation ~ on L by

a~<b if and only i f a * G b = l (19)

and obtain a partial ordering. Moreover, the opera- tions A and v , defined by

a ^ b = (a ~ b*) Q b, (20)

a v b = (a Q b * ) ~ b (21)

coincide, then with the lattice operations meet and join, respectively. The corresponding lattice (L, ~<, ^ , v ) is a distributive lattice. If this lattice is complete we say that the original MV-algebra is complete. Belluce [1] proved that complete MV- algebras are infinitely distributive, i.e., that

V (a i^b)=( i~raJ)Ab (22)

E. Turunen / Fuzzy Sets and Systems 85 (1997) 63-72 65

always holds. In [4] the following connection between (complete) MV-algebras and (complete) residuated lattices is proved. Any (complete) MV- algebra (L, Q ,O,* ,0 , 1) defines a (complete) re- siduated lattice (L, ~<, ^ , v , Q, -%0, 1), where the lattice operations ~<, ^ , v are defined via (19),(20) and (21), respectively, and the residuation

is defined by a --* b = a* ~) b. Conversely, a (complete) residuated lattice (L, ~<, ^ , v , Q, ~ , 0, 1) such that

a v b = ( a ~ b ) ~ b (23)

always holds can be viewed as a (complete) MV- algebra (L, Q, E), *,0, 1), where the unary opera- tion *, called complementation, is defined by (10) and the operation ~ , called addition, is defined by

a @ b = a * ~ b . (24)

It holds in any MY-algebra L (cf. [3]) that

a* ~) b* = (a @ b)*, (25)

a,b <~ a v b <~ a G b , (26)

a Q a* = 0, (27)

a 6) (b @ c) <<. b @ (a Q c). (28)

Proposition 2.1. I t holds in an MV-aloebra for all elements a, b, c and d that

(a @ b) fi) a* ~< b, (29)

(a ~) b) Q) (a* ~) c) @ (b* t~ d) ~< c @ d. (30)

Proof. Since (a ~ b) q) a* = a* q) (a ~ b) ~< b ~) (a*@ a ) = b ~ 0 = b, we have (29). To establish (30) we realize first that

(a @ b) G (b* ~ d) ~< a G [b (?) (b* G d)] (by (28))

~< a ~ [d (~ (b G b*)] (by (28))

= a ~) [d ~) 0] (by (27))

= a ~) d (by (12)).

Thus, (a E) b) @ (a* ~) c) Q (b* ~ d) ~< (a G d) Q (a* ~ c). Moreover,

(a @ d) O (a* @ e) ~< d @ [a fi) (a* q) c)] (by (28))

~< d ~ [c 0) (a G a*)] (by (28))

~< d @ [c • 0] (by (27))

~< d (~ c (by (12)).

Therefore (30) holds true. []

Let (L, Q, t~, *, 0, 1) be a complete MV-algebra. The meet operation ^ , defined by (20), is of course commutative, associative and isotone on L x L and a ^ 0 = 0, a ^ 1 = a holds. This together with (22) implies that by defining another residuation =~ on L x L by

a =:, b = V { c l a ^ c <~ b}, (31)

we obtain another complete residuated lattice (L, ~<, ^ , v , =~,0, 1) and another adjoint couple ( ^ , =~) on L. By [8], this residuated lattice is a pseudo-Boolean algebra. We set

a + = a ~ 0 (32)

and say that a + is the pseudo-complement of an element a e L. Then (cf. [8]) it holds that

a ~ a ++ , (33)

a ÷ A b ÷ = (a v b) +, (34)

a + vb ÷ ~< (aAb) +, (35)

a ^ a ÷ = 0, (36)

a v 0 = a. (37)

Proposition 2.2. It holds in a pseudo-Boolean alge- bra for all elements a,b,c and d that

(a v b) A a ÷ ~< b, (38)

(a v b) ^ (a =~ c) ^ (b =:- d) ~< c v d. (39)

Proof. By distributivity, (36), (37) and (8), we have ( a v b ) ^ a + = ( a ^ a + ) v ( b ^ a +) = Ov(b Aa ÷ ) = b ^ a ÷ ~< b. Thus, (38) holds. To prove (39) we rea- son first that a ^ (a =:, c) ~< c, b ^ (b => d) ~< d, therefore

[ a ^ ( a =:, c ) ^ (b =~ d)]

v [ b ^ ( a =~ c )^ (b =,. d)] ~< c v d .

By distributivity, (a v b) ^ (a ~ c) ^ (b ~ d) ~< c yd. Hence (39) holds. []

66 E. Turunen / Fuzzy Sets' and Systems 85 (1997) 63-72

Comple te and divisible MV-algebras are called injective MV-algebras (for the definition of divis- ibility, cf. [5]). In the unit interval injective MV- algebra coincide with complete MV-algebra . Here we give examples of complete MV-algebras in the unit interval as the unit interval is the most impor- tant set of values of truth in the appl icat ion of fuzzy logic.

Example 2.1. The most wel l -known complete MV- algebra defined on [0, 1] is the Lukasiewicz struc- ture. The MV-opera t ions are defined by a (5) b = max{0, a + b - l} ,a ~ b = min{1 ,a + b}, a* = 1 - a. Notice that the opera t ion A coincides with the min-opera t ion and the opera t ion v co- incides with the max-opera t ion . The residuation

is defined by a ~ b = m i n { 1 , 1 - a + b } and the residuation ~ is defined by a ~ b = 1 if a ~< b and b otherwise. It is wel l-known that any complete MV-algebra defined on [0, 1] is i somorphic with the Lukasiewicz structure.

Example 2.2. Let f : [0, 1] ~, [0,k], 0 < k < oc, be a strictly increasing cont inuous function with f (0 ) = 0 , f (1) = k. Define the pseudo-inverse o f f by Jp(x) = f ~(x) if x e [0, k] and 1 otherwise. Define now the MV-opera t ions by a ~ b = J p ( f ( a ) + f(b)), a* = Jp(f(1) - f ( a ) ) , a L') b = (a* • b*)* for a,b e [0, 1]. We obtain a complete MV-algebra . The function f is called the generator of this MV- algebra.

F r o m now on we let L be a fixed injective MV- algebra. We are ready to introduce the semantic concepts of L-valued fuzzy sentential logic. The basic symbols of the language under considerat ion consist of

(i) an infinite set P = {p~]i ~ ~} of propositional variables (as 'it is raining', ' John is tall', ' this apple is yellow', etc.),

(ii) a family {a la e L} of nullary opera t ions called inner truth values and generalizing the con- tradict ion symbol I of classical logic,

(iii) logical connectives ~ (implication), [] (dis- .junction), ® (conjunction), ~ (negation), i , (intu- itionistic implication), w (intuitionistic disjunction), c~ (intuitionistic conjunction), ..~ (intuitionistic nega- tion).

Proposi t ional variables and inner truth values are atomic formulae. The set of well-formed for- mulae U: is composed in the following way:

(a) a tomic formulae are in t:, (b) if ct and // are in ~z then also ct ~ fl, ~ [ ] fl,

2 ® f l , ~ c t , c~ ~ /J,~w/~,c~c~/~ and ~ ~ are in 0:. We also introduce the abbreviat ions c~ ¢~ fl for

(2 ~ [3)c~(fl ~ ~) (equivalence) and ~ ~ fl for (2 ~-~ f l)~(fl w-~ 2) (intuitionistic equivalence).

A valuation is a function v : ~: "~ L such that

v(pi) is defined for all proposi t ional variables pi,

(40)

v(a) = a for all inner t ruth values a, (41)

v(~ ~ fl) = v(~) ~ v([3), (42)

v(2 [ ] / 9 = v(~) • v(fl), (43)

v(ct ® fl) = v(~) @ v(//), (44)

v(-n ~) = v(2)*, (45)

v(~ ~ fl) = v(ct) ~ v(fl), (46)

v(e wfl) = v(e) v v(fl), (47)

v(2 c~fl) = v(~) A V(fl), (48)

V( ~ C~) = V(~) +, (49)

It is easy to see that for any valuat ion v holds

v(~ ~ / 9 = [v(c0 --, v(/~)] ^ [v ( /b --, v (~) ] , (50)

v(~ ¢=~/~) = [v(~) ~ v(/~)] ^ [v(/~) ~ v(~)], (51)

If Y is a set of formulae, called a set of non-logical axioms, v is a valuat ion and #(~) ~< v(~) holds for any formula ~, we say that the valuat ion v satisfies T, or equally, that 1- is satisfiable. We associate with each formula 2 a value csem]]-(@ in L, defined by

csem~]-(~) = A {v((x)[v satisfies 7} .

If cs°r~#(C0 = a, we write 11 ~ac~ and say that a for- mula ~ is valid with the degree a (with respect to 1-).

The axiomatizabi l i ty of this logic has been shown in [9]. We call the connectives ~----~, w, r~ and ~ in- tuitionistic connectives as they have m a n y proper- ties known to hold in intuitionistic logic. It is easy to see, for example, that ~1 ~ ~ c~ , ~ does not generally hold, thus corresponding to the si tuat ion

E. Turunen / Fuzzy Sets and Systems 85 (1997) 63-72 67

in intuitionistic logic. The intuitionistic connectives have been in t roduced mainly to have a wider var- iety of connectives available in applicat ion. Not ice that the implicat ion ~ is theoretically the mos t impor t an t connective; in fact, all the other connect- ives can be defined by means of this connect ive and the inner t ruth value 0.

3. Fuzzy rules of inference

We are now ready to in t roduce the main results of this study. Fol lowing Pave lka [7] we set the following.

Definition 3.1. An n-ary fuzzy rule of inference is a system

0 q , . . . , c t n a l , . . . ,an r =

rsYn(~l . . . . ,0tn)' rsem(al, ... ,an)'

where formulae al . . . . ,0~n are the premises and for- mula rSyn(O~l, . . . , C t n ) is the conclusion. Values al . . . . . an and r S e m ( a l , . . . , a n ) E Q_ are the corres- ponding values of truth. The mapp ing rSem: ~_n ,,, D_ is semi-continuous on each variable, that is,

r sem(a 1 . . . . , V a k j . . . . . an) j~F

= V rs~m(al . . . . . akj . . . . . a,) j eF

holds. Not ice that the m a p p i n g r s~m is isotone on each variable as it is semi-continuous. Moreover , we assume that for each valuat ion v holds (sound- ness)

rSem(v(~l) . . . . . V(0~n)) ~ v(rSYn(0~l, . . . , 0~n) ).

Proposition 3.1. The following systems are fuzzy rules of inference:

(I) Generalized modus ponens:

~, ~ fl a,b

fl ' a 6) b

(II) Generalized intuitionistic modus ponens:

~, ~ ~ fl a , b

fl ' a A b

(III) Generalized modus tollendo tollens:

~ f l , ~ fl a,b --ao~ ' a @ b

(IV) Generalized intuitionistic modus tollendo tollens:

~ fl, o~ l , fl a, b ,.~a a A b

(V) Generalized hypothetical syllogism:

ct ~ fl, fl ~ Z, a,b ~ Z a @ b

(VI) Generalized intuitionistic hypothetical syllo- gism:

ct ~---~ fi, fl D 'Z , a,b ct l ~ Z a ^ b

(VII) Generalized adjunction law 1:

~,fl a,b

c t®f l ' a Q b

(VIII) Generalized intuitionistic adjunction law l:

o~,fl a,b

ctnfl ' a A b

(IX) Generalized commutation law 1:

~® fl a

f l®cx' a

(X) Generalized intuitionistic commutation law 1:

c~nfl a

fl ("~ O~ ~ a

(XI) Generalized equivalence law 1:

o~ <:¢. fl a

~ fl' a

(XII) Generalized intuitionistic equivalence law 1:

O~ I-'--~ fl ' a

(XIII) Generalized equivalence law 2:

f ~-~ O~ ' a

68 E. Turunen / Fuzzy Sets and Systems 85 (1997) 63 72

(XIV) Generalized intuitionistic equivalence law 2:

O~¢:::::~fl a

l I-----> ,~ ~ a

(XV) Generalized equivalence law 3:

O~ <:*" fl a

(fl ~ ct)n(c( ~ fl)' a

(XVI) Generalized intuitionistic equivalence law 3:

O~ < :" fl a

(fl ~ ~)n(~, , fl)' a

(XVII) Generalized equivalence law 4:

(fl ~ ~)n(~ ~ fl) a

(XVIII) Generalized intuitionistic equivalence law 4:

(fl, ) ct)n(ct, , fl) a

~¢= *f l ' a

(XIX) Generalized simplification:

C( a

(XX) Generalized intuitionistic simplification:

OtNfl a

c( a

P r o o f . The semi-continuity of the rules (IX)-(XX) is obvious. The semi-continuity of the rules (I), (III), (V) and (VII) follows from (3). The semi- continuity of the rules (II),(IV),(VI) and (VIII) fol- lows from (22).

We prove the soundness of the odd numbered rules by using the properties of the adjoint couple (O,--*}. The soundness of the even numbered rules can be proved in a similar manner by using the properties of the adjoint couple (A, ~ >. Let v be a valuation. Then we have

(1) rSem(v(~),V(~F-~fl))

= v(cO 0 [v(~) ~ v(fl)] (by (42))

~<v(fl) (by (6))

= v ( r ' " ( ~ , ~ ~ D )

(Ill) r S e m ( v ( - - l f l ) , V ( ~ F-~fl))

=v(f l )*O[v(~) ~v( f l ) ]

4v(~)*

= v ( ~ ~)

= v(rSy"(-q fl, ~ ~ fl)).

(V) rSem(v(~ ~ fl), V( f l ~ Z))

= [v(~) -~v(f l ) ]oEv(f l ) --,v(z)] < Ev(~) --,v(z)]

= v ( ~ x )

= v ( r ' " ( ~ f l , f l ~ x ) ) .

(VII) rSem(v(~),V(fl))=V(~)QV(fl)

= [ v ( ~ ® f l ) ]

= v(r'"(~,fl)).

(by (45),(42))

(by (ii))

(by (45))

(by (42))

(by (7))

(by (42))

(by (44))

P r o p o s i t i o n 3.2. The following systems are fuzzy rules of inference.

= v ( ~ ) @ v ( f l ) (by (44))

<v(~) (by (8)) =v(rS'"(~,D). []

(IX) By a similar argument and the commutativ- ity of O.

(XI) rSem(v(~ ~:~fl))

= v(~ ~ f l )

= [v(c 0 -- v(fl)] ^ [v(fl) ~ v(cO] (by (50))

~< [v(c 0 --* v(fl)] (by (8))

= v(~ ~ fl) (by (42))

= v(rSyn(o(~:>fl)).

(XIII), (XV) and (XVII) are trivial.

(XIX) rSem(v(o~ @ fl)) ---- V(Gt ~ fl)

E. Turunen / Fuzzy Sets and Systems 85 (1997) 63-72 69

(XXI) Generalized rule of introduction of double negation:

o~ a

--n--ha' a

(XXII) Generalized rule o f elimination of double negation:

- - 1 - - I O~ a

o~ a

(XXIII) Generalized De Morgan law 1:

-n ~ ® - q fl a

-a ( ~ [] fl ) ' a

(XXIV) Generalized De Morgan law 2:

(~ [] fl) a I

" -as®- -a f t ' a

(XXV) Generalized De Morgan law 3:

(--7 ~ ) [] (-7 fl ) a -1 (~ @ fl) ' a

(XXVI) Generalized De Morgan law 4:

--1 (~ @ fl) a

-n ~ []--n fl' a

(XXVlI) Generalized commutation law 2:

o~ [] fl a

f l [ ] ~ ' a

(XXVIII) Generalized addition law 2:

o~ a I

~ [] fl ' a

(XXlX) Generalized modus tollendo ponens:

--n~,o~[]fl a,b

fl ' a @ b

(XXX) Generalized disjunctive syllogism:

a[] f l , ot ~ Z, fl ~ 6 a ,b ,c

Z [ ] 6 ' a @ b @ c

Proof. The semi-continuity of the rules (XXI)- (XXVIII) is trivial. The semi-continuity of the rules (XXIX) and (XXX) follows from 3.

We prove the soundness of each rule. Let v be a valuation. Then it holds that

(XXI) rSem(v(00) ----- V(~)

= v(~)** (by (15))

= v(--a---1 ~) (by (45))

= v(rSy"(~)).

(XXII) By a similar argument.

(XXIII) rSem(v(--a ~ ® -a fl))

= v(-n ~ @ - 7 fl)

= v(~)* @ v(fl)* (by (44),(45))

= [v(~) @ v(fl)]* (by (25),(15))

= v(--a (~ [] fl)) (by (43), (45))

= v(r~y"(-n ~ @ - 7 fl)).

(XXIV) By a similar argument.

(XXV) rSem(v(-q ~ [] --7 fl))

= v ( ~ ~ [ ] ~ fl)

= v(~)* G v(fl)* (by (43),(45))

= [v(~) @ v(fl)]* (by (17),(15))

= v(-q (~ ® fl)) (by (44), (45))

= v ( r S ' " ( ~ ~ [ ] - I fl)).

(XXVI) By a similar argument.

(XXVII) r~m(v(~ [] fl))

= v ( ~ • ~ )

= v(a)<~ v(fl) (by (43))

= v(fl) G v(~) ( 0 comm.)

= v(fl [ ] ~) (by (43))

= v ( r ' " ( ~ [ ] ~)) .

(XXVIII) r~¢m(v(~)) = v(~)

v(~) @ v(fl)

= v(z [ ] fl)

= v(r~'"(~)).

(XXlX) r~°"(v(~ [] fl), v(-~ ~))

= v(~ [ ] fl) Q v(-~ ~)

(by (8))

(by (43))

70 E. Turunen / Fuz zy Se ts and Sys tems 85 (1997) 63 - 72

= [v(a)@ v(fl)] @ v(a)* (by (43),(45))

<. v(fl) (by (29))

= v(rSy"(~[]fl,~ ~)).

(XXX) rs°m(v(~f l ) ,v(~ ~ Z), v(fl ~ ~))

= v(~ [ ] fl) (~ v(~ ~ Z) (9 v(f l ~ ~)

= [v(~) ® v ( f l ) ] <.~, [v (~) - , v ( z ) ]

@ [v(fl) ~ v(6)] (by (42),(43))

~< v(zm 6) (by (30))

= v ( r S Y " ( ~ f l , ~ Z, f l ~ ) ) . []

Proposi t ion 3.3. The following systems are fuzzy rules o f inference.

(XXXI) Generalized rule o f introduction o f intu- itionistic double negation:

a

,~ ~ o¢ ' a

(XXXII) Generalized intuitionistic addition law 2:

O( a

O~ kJ f l ' a

(XXXIII) Generalized intuitionistic disjunctive simplification:

O( t.-J C( a

$( a

(XXXIV) Generalized intuitionistic De Morgan law 1:

~ c ~ f l a ~ ( ~ u f l ) ' a

(XXXV) Generalized intuitionistic De Morgan law 2:

~(ctwfl) a

~ n ~,, fl ' a

(XXXVI) Generalized intuitionistic De Morgan law 3:

~ ( ~ n f l ) ' a

(XXXVII) Generalized intuitionistic commutation law 2:

~fi a

f l k.2,~ ' a

(XXXVIII) Generalized intuitionistic modus tol- lendo ponens:

~ , , ~ w f l a,b

fi ' a A b

(XXXIX) Generalized intuitionistic disjunctive syllogism:

~wfl, ~ l ~ Z, fl l , iJ a ,b ,c

Zw6 ' a A b A c

Proof . The semi-continuity of the rules (XXXI)- (XXXVII) is obvious and the semi-continuity of the rules (XXXVIII) and (XXXIX) follows from (22). We demonstrate the soundness of each rule. Let v be a valuation. Then the following holds.

(XXXI) r~¢m(v(~)) = v(g)

~< v(~) ++ (by (33))

= v ( ~ 7 ) (by (49))

= v (W"(~) ) .

(XXXII) rSCZ(v(a)) = v(~)

< v(~)vv( f l ) (by (8))

= v(~ wfl) (by (47))

= v ( r S y n ( o 0 ) .

(XXXIII) By easy verification.

(XXXlV) r~¢~(v( ~ ~ c~ ~ fi))

= v( ~ a n ~f l )

= v(c¢) +/x v(fl) + (by (48),(49))

= Iv(co) v v(fl)] + (by (34))

= v( ~ (c~ wfl)) (by (47), (49))

= v(r~Y"( ~ m ~fl)) .

E. Turunen / Fuzzy Sets and Systems 85 (1997) 63-72 71

(XXXV) By a similar argument.

( X X X V I ) rSem(v( ~ ~ U ~ 9))

=v(~~u~~)

= v(a) + v v(fl) + (by (47),(49))

~< [v(a) A v(fl)] + (by (35))

= v( ~ (c¢ c~fl)) (by (48), (49))

---- v(rSyn ( ~O{U ~ f l ) ) .

(XXXVII) Follows from the commutativity of v .

( X X X V I I I ) r S e m ( v ( ~ U f l ) , V ( ~ ~ ) )

= ~ ( ~ u / ~ ) ^ v ( ~ ~ )

= [v(~) v v ( / b ] ^ v(~) +

4v(~) = v(rSyn(~ufl, " ~ ) ) .

(XXXIX) rSem(v(~Ufl),V(~ ~ Z),V(fl I • 6))

= [ ~ ( ~ ) v v ( ~ ) ] ^ [v(~) ~ v (z ) ]

,x [v(fl) ~ v(6)] (by (46),(47))

<~ v(z ) v v(6) (by (39))

= v(z u6) (by (47))

= v(rsyn(°cwfl, ~ I ) Z, fl ~ 6)).

(by (47), (49))

(by (38))

[]

We have introduced some sound fuzzy rules of inference each having a counterpart in classical logic. It may happen that a formula :~ is a con- clusion from various premises; therefore, various values of truth may be associated with :¢. This leads us to the concepts of zero-order fuzzy theory, •eta- proof degree of deduction, etc., in fuzzy sentential logic. These and related topics have been studied in detail in [9]. Instead of giving exact definitions here we conclude this study by an example which dem- onstrates how these rules of inference can be used to implement an efficient theorem prover.

Problem. Consider the following premises: (A) If wages rise or prices rise then there is

inflation.

(B) If there is inflation then either the Govern- ment holds it down or people suffer.

(C) If people suffer then the Government loses its popularity.

(D) The Government does not hold down the inflation and people do not suffer.

Assume, moreover, that the premises (A) and (D) are absolutely true while (B) and (C) are only par- tially true, say 0.9-true and 0.8-true, respectively, in the scale [0, 1].

Using Lukasiewicz-valued logic, we have to find a metaproof of the highest degree for a conclusion:

(E) Wages do not rise.

Solution. We start by introducing the following abbreviations: P = wages rise, Q = prices rise, R = there is inflation, S--- the Government holds down the inflation, T = people suffer, W = the Government loses its popularity, ~ = implies, [] = or, @ = and, --7 = not.

Then we define a zero-order fuzzy theory ~ with the followng non-logical axioms corresponding to the premises (A)-(D):

T((P[] Q) ~ R) = 1, 1-(R ~ ( S • T)) = 0.9,

q]-(T ~ W ) ~ 0 . 8 , 7 ( ~ S ® ~ W ) = 1,

q]-(~) = 0 elsewhere.

Let v be a valuation such that v(P)= 0.3,v(Q) = O,v(R) = 0.3, v(S) = O,v(T) = 0.2, v(W) = 0.

Then Y(a)~< v(:¢) for any formula ~. This means that 7 is satisfiable and therefore consistent. We also realize that v(~ P) = 0.7, thus the conclusion (E) is valid with the degree less or equal to 0.7.

Next we write the following metaproof¢o for the formula ~ P:

(a) (P[ ] Q) ~ R 1 (assumption), (b) R ~ (S [] T) 0.9 (assumption), (c) T ~ W 0.8 (assumption), (d) ~ S ® --3 W 1 (assumption), (e) -7 W 1 (Rule XIX & (d)), (f) -7 S 1 (Rule XIX & (d)), (g) --7 T 0.8 (Rule III & (c),(e)), (h) --7 S ®--7 T 0.8 (Rule VII & (f), (g)), (i) ~(SRq T) 0.8 (Rule XXIII & (h)), (j) ~ R 0.7 (Rule III & (i),(b)), (k) --q(P[]Q) 0.7 (Rule III & (j),(a)),

72 E. Turunen / Fuzzy Sets and Systems 85 (1997) 63-72

(1) --1 P ®--7 Q 0.7 (Rule XXIV & (k)), (m) -7 P 0.7 (Rule XIX & (1)).

The degree of this m e t a p r o o f o , denoted by Valv(o 0, is 0.7.

Could there exist ano ther m e t a p r o o f for --7 P, say o9', such that Val-r(~o)< Val~(~o'); in part icular, what is the degree of deduction o f - 1 P, denoted by q/- I--b ~ P, where the value b e L is defined by

b = V [Val r (o ) I o is a m e t a p r o o f for ~ P}?

Since the fuzzy theory q]- is consistent it is complete,

i.e.

ql- t-a ~ if and only if q]- I=a ct for any formula cc

We conclude that 1]- k-o.7-7 P as well as q]- ~ 0 . 7 ~ P.

Remark . To prove the completeness of any consis- tent zero-order fuzzy theory, besides Rules I, VII and two others not ment ioned in this study, certain forms of logical axioms are required (cf. [9]). Be- cause of soundness, int roducing new rules of infer- ence does not change the degree of deduct ion of any formula but facilitates the finding of metaproofs .

References

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