pleteness theorem for propositional fuzzy logic was ... tension of Pavelka's proof to rst-order fuzzy logic ... by P. H jek for fuzzy propositional logic in 4] and ...... book. Kluwer, Dordrecht 1994. 13] Nov k, V.: Paradigm, Formal Properties and.
Proc. Int. Conference FUZZ-IEEE/IFES'95, Yokohama 1995, 1461{1468
A New Proof of Completeness of Fuzzy Logic and Some Conclusions for Approximate Reasoning Vilém Novák University of Ostrava Faculty of Sciences Dept. of Mathematics Bráfova 7 701 00 Ostrava, Czech Republic and Institute of Information and Automation Theory Academy of Sciences of the Czech Republic Pod vodárenskou vì¾í 4, 186 02 Praha 8, Czech Republic
1 Introduction | a brief survey
ately uncountable, which disquali es it for study of recursive properties. However, as both papers demonstrate, it is possible to introduce only countably many names for rational numbers being dense subset of [0; 1] so that we may study recursive properties of fuzzy logic using classical tools. The latter was demonstrated by P. Hájek in [4]. In [14], we have demonstrated that the language can be further simpli ed when introducing a special unary connective p (square root). This idea was proposed by U. Höhle. We will not repeat all the arguments in favour of this system of fuzzy logic. They have been presented many times in the cited works, most concisely in [13]. Let us state only that the theory below and in cited works is a strong formal tool for stydying fuzzy logic in general, i.e., many-valued logic, theory of approximate reasoning and fuzzy set theory. Moreover, it also makes us possible to get di erent light on classical logic since fuzzy logic is its direct generalization.
In this paper, we turn again to the problem of the completeness of rst-order fuzzy logic. The completeness theorem for propositional fuzzy logic was proved by J. Pavelka in [15]. Later, some other variations appeared [18] proving, however, completeness only for formulas in the degree 1. Extension of Pavelka's proof to rst-order fuzzy logic was done by the author in [5, 6]. The proof, however, is quite complicated based on ultra lter trick and on several tricky axioms. An open question remained, whether this proof can be done in a different way following classical Henkin construction of a model from the constats. This has been done by P. Hájek for fuzzy propositional logic in [4] and independently on him for rst-order one by the author. This result, besides others, is presented in this paper. Let us remark, that motivation of both proofs was di erent. The main goal of Hájek was to prepare material for study of recursive properties of fuzzy logic. Some of the results, also inde- 2 Language and axioms pendently, have been presented in [12]. The main problem consits in introducing names for all the As usual, we will consider the set of truth values to truth values being special atomic formulas in the form a nite or uncountable residuated chain language. If we work with the interval [0; 1] of L = hL; _; ^; ; !; 1; 0i truth values then the language becomes immedi1
The common abbreviations of formulas :A, where L is either a nite chain or L = [0; 1], ! is a Lukasiewicz implication and is a Lukasiewicz A_B, A^B, A & B, A,B, (9x)A, Ak are introduced (see [5, 6, 12, 13]). As explained in these product. We denote works, syntax of fuzzy logic is evaluated by syntacan = a| {z a} : tic truth values. An evaluated formula is a couple n?times aA This basis for the de nition of the operation p : [0;is1]a?! where A 2 FJ and a 2 [0; 1]. The (syntactic) truth [0; 1]. We put value a is an evaluation of the formula A in the pa = a + 1 syntax of fuzzy logic. 2 We introduce the following rules of inference: for every a 2 [0; 1]. The language of rst-order fuzzy logic consists (a) Modus ponens of: aA; bA)B rMP : a
b B (i) Variables x; y; : : :. (ii) Constants c, d, r,: : :. (b) b-lifting rule (iii) n-ary functional symbols f; g; : : :. aA rRb : b ! a b)A : (iv) Symbol 0 for the truth value false . (v) n-ary predicate symbols p; q; : : :. (c) Generalization (vi) A binary connective ), a unary connective aA y ) . r G : a (8x)A (vii) A symbol for a general quanti er 8. In fuzzy logic, we deal with fuzzy sets of ax(viii) Auxiliary symbols. ioms. Concerning a fuzzy set of logical axioms AL , various systems have been proposed. J. Pavelka in However, it is useful to introduce names a for [15] deals with more than thirty of them adding a all the truth values a 2 L since many formal re ec- new axiom whenever he needed it. The author in tions can be simpli ed. Due to the above remarks, [6] has reduced them signi cantly. However, some we can work with them keeping in mind that nally, redundancies still remained. As pointed out by Háwe can get rid of them. jek [4] and Gottwald [3], it is possible to reduce the fuzzy set of propositional logical axioms to that of Terms are de ned as usual. Rose-Rosser [17]
p
Formulas
(R1) j= A)(B )A) (a) If t1 ; : : :; tn are terms and p an n-ary pred- (R2) j= (A)B))((B )C))(A)C)) icate symbol then p(t1; : : :; tn) and 0 are (atomic) formulas. (R3) j= (:B ):A))(A)B) (b) If A; B are formulas then A)B, A and (R4) j= ((A)B))B))((B )A))A) (8x)A are formulas. Note that all these formulas are tautologies in the A set of all the terms of a language J is denoted degree 1. The following theorem is an explicit forby MJ and a set of all the well-formed formulas by mulation of possibility to replace propositional axFJ . The set of all terms without variables is de- ioms from [6] aby the above ones. Its proof is noted by MV . straightforward but tedious.
p
y) It is possible to consider also a set need it for the purpose of this paper.
fcj ; j 2 Jopg of additional nj -ary connectives as is done elsewhere.
2
We do not
Theorem 1 All the schemes of formulas (T2){ if Ai has been obtained using a rule of inference r. (T8), (D1){(D15), (D21){(D24) from [6] can be The an is is a value of the proof w. The value of proved in the degree 1 using modus ponens and the the proof w will usually be written as b-lifting inference rules. Val(w):
The axioms (TK) and (TI1, TI2) from [6] are Recall that a theory T in the language J of rstthe mentioned tricky axioms which are not needed order fuzzy logic (a fuzzy theory ) is a triple in the completeness proof below and, they are omitted from further consideration. T = hAL ; AS ; Ri Let us present the following additional tautolo- where AL FJ , and AS FJ are fuzzy sets of gies. logical and special axioms, respectively and R is a set of inference rules containing at least the rules (T1) j= (a)b),(a b) RMP , rG and rRb, b 2 L. By J(T) we denote the where a b denotes the symbol (atomic for- language of fuzzy theory. A fuzzy predicate calculus mula) for the truth value a ! b when a and is the fuzzy theory with AS = ;. If w is a proof of a formula in the theory T then its value will be b are given. denoted by ValT (w). (T2) j= (8x)A)Ax [t] for any term t. Theorem 2 (Closure theorem) Let A 2 FJ (T ) and A0 be its closure. Then (T3) j= (8x)(A)B),(A)(8x)B) T `a A i T `a A0 : provided that x is not free in A. (T4) j= (9y)(Ax [y])(8x)A)n A fuzzy theory is consistent if ValT (wA ) ValT (w:A ) = 0 The fuzzy set of logical axioms is then speci ed as follows: holds for every formula A 2 FJ (T ) and all proofs 8 wA of A and w:A :A, respectively. a 2 L if A := a > > > > > < 1 if A is any of formulas of the Theorem 3 A theory T is contradictory i T ` A AL (A) = > form (R1){(R4), (T1){(T4) holds for every formula A 2 F . J (T ) > > > > : 0 otherwise A fuzzy theory is complete if it is consistent and
!
!
T `a A implies T ` A)a holds for every closed formula A and every a, a 2 L.
An evaluated proof (or shortly, a proof) of a formula A from a fuzzy set AS of special axioms is a sequence of evaluated formulas w := a0 A0 ; a1 A1 ; : : :; an An such that An := A and every evaluated formula in the proof is ai =AL (Ai ) Ai if Ai is a logical axiom, ai = AS (Ai ) Ai if Ai is a special axiom, or ai Ai := rsem (ai1 ; : : :; ain )rsyn(Ai1 ; : : :; Ain ); i1 ; : : :; in < i where r is an n-ary sound rule of inference. To stress how the given formula has been obtained within the proof, we will often write its members in the form ai Ai P where P is LA or SA if Ai is a logical or special axiom respectively (or it is their name), or P is r
Lemma 1 Let T be consistent and T `a A and T `:a :A. Then T is complete. proof: Let wA be a proof of A, Val(wA ) = b a and w:A be a proof of :A, Val(w:A ) = b0 :a. Then w := b A wA ; a ! b a)A rRa ; W W i.e., fVal(w); wA g = a ! ba b = 1, i.e., T ` a)A. Similarly, we obtain T ` :a):A. Furthermore, h i w := :a ! b0 :a):A ; w A 1 (:a):A))(A)a) R3 ; h i :a ! b0 A)a ; :
rMP
3
i.e., fVal(w); w:A g = :a ! b :a b0 = 1, i.e., Theorem 5 Let T be a complete theory. Then a T ` A)a. 2 canonical structure D0 for J(T) is a model of T W
W
0
such that
Since the opposite implication has been proved in [6], we obtain that a fuzzy theory T is complete i T `a A and T `:a :A. Hence, we may de ne a complete theory in a classical way using only negation. A language J 0 is an extension of J if J J 0 . Obviously, FJ FJ . Let T = hAL ; AS ; Ri, T 0 = hA0L ; A0S ; Ri be theories in the respective languages J and J 0. Put AS (A) = AS (A) if A 2 FJ and AS (A) = 0 otherwise. If AS A0S then T 0 is an extension of T. To simplify the notation, we will write AS instead of AS and understand that AS (A) = 0 for all A 2 FJ ? FJ . An extension T 0 is a conservative extension of T if T 0 `b A and T `a A implies a = b for every formula A 2 FJ (T ) . The extension T 0 is a simple extension of T if J(T 0 ) = J(T). A fuzzy theory T is Henkin if Henkin axioms AS (Ax [r])(8x)A(x)) = 1 (1) are added to the fuzzy set of special axioms where r is a special constant for the formula (8x)A(x). The proof of the following two theorems can be found in [6].
T `a A i D0(A) = a holds for every formula A 2 FJ (T ) .
3 Completeness theorems
0
In this section, we present the completeness theorem together with several other lemmas. We begin with the validity theorem proved in [6].
Theorem 6 (Validity theorem) If T `a A and
T j=b A then a b.
0
Lemma 2 If a fuzzy theory T 0 = T [ f aAg is
contradictory then to every formula B there is m such that
T ` Am )B:
Let T 0 be contradictory. By Theo` B for every formula B and then for every b < 1 there is a proof wB0 with ValT (wb) b. By deduction theorem, there are m and a proof wAm )B in T such that proof:
rem 3, T 0
0
ValT (wAm )B ) ValT (wB ) b: Theorem 4 Let T be a consistent theory, K a set of special constants for all the closed formulas (8x)A and let AH be a fuzzy set of Henkin axioms From it follows that T ` Am )B. 0
(1) where AH (C) = 0 if C is not a Henkin axiom. Then the theory
2
Lemma 3 Let T `a A and b > a. Then
TH = T [ A H
T 0 = T [ f 1 A)bg
is a conservative extension of the theory T .
Let T : FJ ?! L be a homomorphism preserving all meets and joins, where FJ is an algebra of formulas (see [6]). Then a structure D0 = hD0 ; pD0 ; : : :; fD0 ; : : :; u; : : :i de ned by D0 = M V ; D0 (t) = t; t 2 MV fD0 (t1 ; : : :; tn) = f(t1 ; : : :; tn) pD0 (t1 ; : : :; tn) = T(p(t1 ; : : :; tn)) is a canonical structure for the language J (for details see [6]).
is a consistent extension of T . proof: In this proof we will refer to some of the formal theorems (T1){(T8), (D1){(D24) from [6]. Let T 0 be contradictory and c < 1. By Lemma 2 there is m such that
T ` (A)b)m )c:
(2)
At the same time, using (T7) and (T8) we have T ` (A)b)m _(b)A)m ; 4
T ` A)b for every b > a. Let T `c A)a and write down the proof T ` (A)b)m _(b)A) w := 1 A)b SA ; : : :; b ! a A)a rMP ; using (D15), (D6) and (D7). Using (2), (D1) and From it follows that c b ! a for every b > a. W W (D6) we obtain (b ! (b ! a) c from which But then a a 0 1 and put T = T [ f A)bg. From Lemma 3 we know that T0 is consistent and by maximality of T we conclude that f 1 A)bg AS . Hence, proof:
In this section, we show some conclusions of the completeness property for the theory of approximate reasoning. Its formalization via the above fuzzy logic theory has been presented in several papers. This is possible due to two important features of it, namely that it deals with fuzzy sets of axioms and that all the truth values are equally important, 5
Analogously, if A := c(B) then we may put g(A) 2 c?1(f(B)) since c is surjective. Since the formulas from F0 are independent, no subformula of B is already assigned a value. Hence, we may proceed in the above way for all the formulas from F0 to obtain the function g for all closed the atomic formulas contained in the formulas from F0 . For the rest of formulas from FA we can de ne the function g arbitrarily. Since the algebra of all the formulas from FJ is free on FA , we may uniquely extend g into D : FJ ?! L and due to the de nition, we have D(A) = f(A) for every A 2 F0. 2
i.e., there are no designated truth values. The main goal of the logical system is to achieve maximal truth value (provability degree), no matter what it is. Recall from [9, 10, 13, 11] that approximate reasoning leads to a special rst-order fuzzy theory AR T, which is given by a fuzzy set of special axioms AR AS = R1 [ [ Rm : (3) e
e
The Ri , i = 1; : : :; m are fuzzy sets of formulas e
Ri = Ai )Bi = f ctsAi;x [t])Bi;y [s]; t; s 2 MV g e g (4) representing imprecise statements about the relation between two phenomena. This theory is obtained using translation rules from a part of natural language to the formal language of fuzzy (manyvalued) logic. So far, most elaborated are natural language syntagms leading to, so called, simple formulas. A simple formula is de ned as follows:
A natural question arises whether the computation formulas used in the applications fuzzy logic give correct and the highest possible truth degree of the conclusion. What is to say, we require to nd a fuzzy set of conclusions which are theorems in the given theory of approximate reasoning. The following theorems partly give an answer to this question. (i) An atomic formula which is di erent from a, Their proofs are based on the completeness propa 2 L is simple. erty above. (ii) Let c be a surjective connective (i.e., the function c : L ?! L assigned to it is surjective) Theorem 10 Let T = f aA; A 2 F0 ; a 2 Lg. and A be a simple formula. Then c(A) and Then A are simple formulas. T `a A Simple formulas A(x1; : : :; xn) and holds for every A 2 F0. B(y1 ; : : :; yn ) are independent if the formulas proof: Due to Lemma 4, there is a truth valA(x1 ; : : :; xn) and Bx1 ;:::;xn [y1; : : :; yn] have no common subformulas. We will denote the set of uation D such that all the simple formulas by F0, F0 FJ . By F0 , D(A) = a F0 F0 we denote a set of closed independent simple formulas. for every A 2 F0 . Then D j= T and the theorem follows from the completeness theorem. 2 Lemma 4 Let f : F0 ?! L be an arbitrary func-
:
tion. Then there is a structure D such that
Lemma 5 Let T = f a A; c (A)B)g where A; B 2 F0. Then T `a c B:
D(A) = f(A) holds for every A 2 F0. proof: Let FA FJ be a set of all closed atomic formulas. We de ne a function g : FA ?! L as follows. Let A 2 F0 be atomic. Then A 2 FA and we put g(A) = f(A): Let A := :B where B 2 FA then g(A) = :f(B):
proof: Due to Lemma 4, there is a truth valuation D such that D(A) = a and D(B) = a c. But then D(A)B) = a ! a c c and thus, D j= T. The lemma follows from the completeness theorem. 2
6
Theorem 11 Let T = f at Ax [t]; cts (Ax [t])By [s]); t; s 2 MV g
set of simple independent formulas. Find a truth valuation
D(Aj;x [t]) = ak;t t 2 M1 ; j = 1; : : :; m _ (ak;t cts); D(Bj;y [s]) =
where A(x) and B(y) are independent simple formulas. Then
t2M1
T `c By [s] W for every s 2 MV where c = t2MV (at cts ). proof:
s 2 M2; j = 1; : : :; m: Then
Realize that formulas from the set
bs ak;t !
fAx [t]; By[s]; t; s 2 MV g
=
are closed and independent. Choose s 2 MV and write a proof wts := at Ax [t] SA ; cts Ax [t])By [s] SA ; at ctsBy [s]rMP : Then c
_
t2MV
ValT (wts) =
_
t2MV
t2MV
j =1
t2M1
(ak;t cts ) =
D(Aj;x [t])Bj;y [s]) =
m ^
= D( (Aj;x [t])Bj;y [s])); j =1
i.e., D j= T. But since wts := ak;t Ak;x [t] SA ; : : :; ak;t cts By [s] rMP
(at cts ):
we have
bs
Using Lemma 4 we nd a truth valuation
D(Ax [t]) = at t 2 MV _ D(By [s]) = (at cts)
m ^
_
_
t2M1
(ak;t cts);
which gives the theorem.
s 2 MV :
s 2 M2
2
These two theorems have impact on the approximate reasoning. In principle they state that the provability degree derived in this theory where the _ IF{THEN rules are considered as linguistic impli(at cts ) cts D(Ax [t])By [s]) = at ! cations is the highest possible one when no other t2MV information than a set of IF{THEN rules is given. holds for every t; s 2 MV , i.e., D j= T. The the- Similar theorems can be proved also for the fuzzy orem then follows from the completeness theorem. graph approach to approximate reasoning (cf. [13]. 2 But then
Theorem 12 Let
5 Discussion m
^ T = f ak;t Ak;x[t]; cts ( (Aj;x [t])Bj;y [s])); In this paper, we returned to the basic property of fuzzy logic which is its completeness, provided that j =1 L = [0; 1] or it is a nite chain. The goal was to simt 2 M1 ; s 2 M 2 g plify quite complicated proofs presented in [6, 15], and also to put a di erent light on fuzzy logic. Note for some k, 1 k m, where Aj (x), j = 1; : : :; m that similar motivation and results in slightly modare independent simple formulas and M1 ; M2 i ed proofs have recently been obtained (indepenMV , M1 \ M2 = ;. Then dently) also by P. Hájek [4]. s 2 M2 T `bs Bk;y [s]; Except for a theoretical interest, the above theW ory has also practical impact. Many arguments in where bs = t2M1 (ak;t cts). its favour have already been given. In [5, 13] we gave arguments for accepting this theory as a genproof: It can be veri ed that the set K = eral framework for fuzzy logic and fuzzy set theory. fAj;x[t]; Bj;y [s]; j = 1; : : :; m; t 2 M1 ; s 2 M2 g is a Also the fact that approximate reasoning can be
7
developed as a special case of this logic is quite inM. Èerný and J. Nekola, Eds.: Fuzzy Apteresting. We only remark that this formal theory proach to Reasoning and Decision Makshould be considered as the rst stage. It is possiing. Academia, Prague and Kluwer, Dordrecht ble that further development will probably depart 1992. to higher order logic and inspire itself also by nonmonotonic, default, and other branches of logic. [10] Novák, V.: Fuzzy Logic As a Basis of Approximate Reasoning. In: Zadeh, L. A., Kacprzyk, Last but not least, this theory gives good backJ. Fuzzy Logic for the Management of ground for research in fuzzy expert systems since Uncertainty . J. Wiley & Sons, New York logical part of their reasoning may be seen as prov1992. ing in a formal theory of approximate reasoning. [11] Novák, V.: A Formal Integrated Theory of Approximate Reasoning. Proc. of Vth IFSA Congress, Seoul, Korea 1993. References [12] Novák, V.: Ultraproduct Theorem and Recur[1] Chang, C. C. and H. J. Keisler: Model Thesive Properties of Fuzzy Logic. In: U. Höhle ory, North{Holland, Amsterdam 1973. and E. P. Klement, Eds. Non-Classical Log-
ics and Their Applications | A Handbook. Kluwer, Dordrecht 1994.
[2] Goguen, J. A.:The logic of inexact concepts, Synthese 19(1968-69), 325{373.
[13] Novák, V.: Paradigm, Formal Properties and Limits of Fuzzy Logic. Int. J. of General Systems (to appear). [14] Novák, V.: Fuzzy Logic revisited. EUFIT'94, Aachen 1994. [15] Pavelka, J.: On fuzzy logic I, II, III, Zeit. Math. Logic. Grundl. Math. 25(1979), 45{52; 119{134; 447{464. [16] Rasiowa, H. and R. Sikorski: The Mathematics of Metamathematics. PWN, Warszawa 1963. [17] Rose, A. and J. B. Roser: Fragments of many-valued statement calculi. Trans. A.M.S. 87(1958), 1{ 53. [18] Schwartz, D. G.: Axioms for a Theory of Semantic Equivalence. Fuzzy Sets and Systems 21(1987), 319{349. [19] Shoen eld, J. R.: Mathematical Logic. Addison{Wesley, New York 1967. [20] Zadeh, L.A.: The concept of a linguistic vari-
[3] Gottwald, S.: Mehrwertige Logik. Akademie-Verlag, Berlin 1989. [4] Hájek, P.: Fuzzy Logic and Arithmetical Hierarchy. Preprint. Institute of Informatics, Academy of Sciences of Czech Republic, Prague 1994. [5] Novák, V.: Fuzzy Sets and Their Applications, Adam{Hilger, Bristol, 1989. [6] Novák, V.: On the Syntactico-Semantical
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