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A triangular-norm-based propositional fuzzy logic
Fuzzy Sets and Systems 136 (2003) 55 – 70 www.elsevier.com/locate/fss
A triangular-norm-based propositional fuzzy logic San-Min Wanga; ∗ , Bao-Shu Wanga , Guo-Jun Wangb a
School of Computer Science and Engineering, XiDian University, No. 2 South Taibai Road, Xi’an 710071, People’s Republic of China b Institute of Mathematics, Shaanxi Normal University, Xi’an 710062, People’s Republic of China Received 5 October 2000; received in revised form 18 March 2002; accepted 27 March 2002
Abstract In 1997, Wang introduced the formal system L∗ for a type of fuzzy propositional calculus given by a particular t-norm di3erent from the three famous ones (Lukasiewicz, G6odel, product). Recently, Pei proved the completeness theorem for L∗ with respect to W8 -semantic by developing a theory of algebraic systems. In this paper, we aim to prove it by essentially metamathematical method. Furthermore, we discussed originally the c 2002 Elsevier Science completeness for the schematic extension L∗n of L∗ with respect to Wn -semantic. B.V. All rights reserved. Keywords: Non-classical logics; Formal system L∗ ; Schematic extension L∗n ; Completeness; Triangular norm
1. Introduction Spurred by the success in applications, especially in fuzzy control, fuzzy logic aroused the interest of many famous scholars and a series of important results have been created . In 1992, a t-norm was introduced in Ref. [7] as follows: min{x; y} if (x) + (y) ¿ 1; T (x; y) = 0 otherwise; where is a strictly increasing continuous function from the interval onto itself such that (0) = 0 and (1) = 1. In Ref. [1], it was called nilpotent minimum and its R-implication was given as ∗
Corresponding author. E-mail addresses: [email protected], [email protected] (S.-M. Wang).
c 2002 Elsevier Science B.V. All rights reserved. 0165-0114/03/$ - see front matter PII: S 0 1 6 5 - 0 1 1 4 ( 0 2 ) 0 0 1 4 3 - 4
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follows: Imin0; (x; y) =
1
if x 6 y; −1
(max(1 − (x); (y))) otherwise:
It was estimated by Ref. [1] that the nilpotent minimum can be admitted into investigations of many theoretical and practical problems soon by its propitious characteristics. Since 1997, many creative works have been done starting with them as follows: Grstly, a formal system L∗ for a type of fuzzy propositional calculus, based on an instance of Imin0; which was called R0 -implicator by the author, was constructed in Ref. [10]; secondly, the generalized tautological theory of its corresponding semantical system, which was called the revised Kleene system by the author, was established in Ref. [11]; after that, the full implicational triple I method of fuzzy reasoning, which aims to lay a theoretical foundation for fuzzy controllers developing, was presented in Ref. [12], etc. Recently, Pei has proved the completeness theorem for L∗ with respect to W8 -semantic [6] by developing a theory of algebraic systems R0 corresponding to L∗ and Wang constructed the formal system KL∗ for the fuzzy predicate calculus corresponding to L∗ and proved the completeness theorem for KL∗ with respect to W8 -interpretation [15], and this laid a Grm foundation for the results given by Wang [10,11,12]. In this paper, we aim to prove it by metamathematical method and by this the completeness for KL∗ with respect to W8 -semantic can also be proved. Essentially, our proof is a type of generalization of Henkin’s technique by which the classical two-valued completeness had been proved and its basic idea is from Section 5.3 of [2]. However, it is more subtle and delicate by complexity of the system L∗ itself. Furthermore, we discussed originally the completeness for the schematic extension Ln∗ of L∗ with respect to Wn -semantic.
2. Preliminaries Now we introduce the fuzzy propositional calculus PC(T0 ). Proposition 2.1. Let the binary operation T0 on the unit interval [0; 1] be de8ned as follows: ∀x; y ∈ [0; 1];
T0 (x; y) =
0
x + y 6 1;
min(x; y) x + y ¿ 1:
Then (i) T0 is a triangular norm [7; 1]. (ii) The residuum → of T0 is the R0 -implicator [10; 1], i.e. for any x; y ∈ [0; 1];
x → y = R0 (x; y) =
1
x 6 y;
max(1 − x; y) otherwise:
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(iii) If a unary operation + is de8ned as follows: ∀x ∈ [0; 1]; + x = x → 0, then + is the standard fuzzy negation, i.e., + x = 1 − x for any x ∈ [0; 1]. (iv) If a binary operation ∨ is de8ned as follows: ∀x; y ∈ [0; 1]; x ∨ y = + ((((x → (x → y)) → x) → x) → + ((x → y) → y)), then ∨ is the standard fuzzy disjunction, i.e. x ∨ y = max(x; y) for any x; y ∈ [0; 1]; (v) ∀x; y ∈ [0; 1]; T0 (x; y) = + (x → + y). The proof is trivial and omitted. These show propitious characteristics of the nilpotent minimum further. Incidentally, we obtain (iv) through a computer running for half an hour. The identity (iv) is looking unfamiliar although it is the simplest in deGning ∨ by → and +. Alternatively, one can use the formula + ((((x → y) → y) → ((y → x) → x)) → + ((x → y) → y)), and this had been mentioned in [4] by rich Klement and Navara. Easy calculation shows that they are equivalent. By this proposition, we can introduce an algebra W8 of type (+; →) on [0; 1], where + is the standard fuzzy negation and → is R0 -implicator. Its subalgebras ([0; 1] ∩ Q; +; →) and ({0; 1=n − 1; : : : ; n − 2=n − 1; 1}; +; →) are denoted by W and Wn , respectively, where n is an integer and n¿4. The algebras W8 ; W and Wn are called the revised Kleene Systems [11]. Denition 2.2. The language of the propositional calculus PC(T0 ) given by T0 has the set S of propositional variables p1 ; p2 ; : : :, and connectives +; →. The set F(S) of well-formed formulas (w:f:’s for short) in PC(T0 ) is deGned inductively as follows: (i) pn is a w:f: for each n¿1, (ii) If ’ and are w:f:’s, then +’; ’ → are w:f:’s; (iii) F(S) is generated only by (i) and (ii). In PC(T0 ), the associated truth-value functions of + and → are the standard fuzzy negation and the R0 -implicator, respectively. Using basic logical connectives + and →, we can deGne additional logical connectives as follows: (i) p ∨ q for + ((((p → (p → q)) → p) → p) → + ((p → q) → q)), (ii) p ∧ q for + (+ p ∨ + q), (iii) p ↔ q for (p → q) ∧ (q → p). Then each of these has associated truth-value function as follows: (i) x ∨ y = max(x; y), (ii) x ∧ y = min(x; y), max{1 − y; x} if x¡y; if x = y; (iii) x ↔ y = 1 max{1 − x; y} otherwise: In addition to these, 18 stands for p → p and 08 stands for + (p → p). Denition 2.3. A W8 -evaluation in PC(T0 ) is a mapping e : F(S) → W8 such that e(+ ’) = + e(’) and e(’ → ) = e(’) → e( ) for any ’; ∈ F(S). The set of all W8 -evaluations is denoted by #(W8 ) and called W8 -semantic of PC(T0 ).
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Denition 2.4. (i) Let ’ be a w:f: ’ is a W8 -tautology of PC(T0 ) (denoted by |= ’) if e(’) = 1 for each e ∈ #(W8 ); (ii) let T ⊆ F(S) and e ∈ #(W8 ); e is a W8 -model of T if e( ) = 1 for any ∈ T ; (iii) let T ⊆ F(S) and ’ a w:f: if e(’) = 1 for each W8 -model e of T , then we say T yields semantically ’ and write T |= ’. Denition 2.5. The formal deductive system L∗ of PC(T0 ) is deGned as follows [10]: (i) The set of w:f: F(S) is given in DeGnition 2.2. (ii) Let p; q; r be distinct propositional variables. The following are axiom schemes of L∗ : (A1 ) p → (q → p), (A2 ) (+ p → + q) → (q → p), (A3 ) (p → (q → r)) → (q → (p → r)), (A4 ) (p → q) → ((r → p) → (r → q)), (A5 ) p → + + p, (A6 ) p → p ∨ q, (A7 ) p ∨ q → q ∨ p, (A8 ) (p → r) ∧ (q → r) → (p ∨ q → r), (A9 ) (p ∧ q → r) → (p → r) ∨ (q → r), (A10 ) (p → q) ∨ ((p → q) → + p ∨ q), (A11 ) p → p ∧ p, (A12 ) (p → q) → (p ∧ r → q ∧ r), (iii) The rule of inference in L∗ is modus ponens (MP for short): from ’ and ’ →
infer
.
Denition 2.6. (i) A theory is a set of w:f:’s, called special axioms of the theory. A proof in a theory T is a sequence ’1 ; : : : ; ’n of formulas such that ’i either is an axiom of L∗ or is a special axiom of T or follows from some preceding ’j ; ’k by MP. (ii) A formula ’ is provable in T (notation: T ’) if it is the last member of a proof in T . (iii) For any w:f:’s ’ and , if T ’ → and T → ’, we say ’ is T -provably equivalent to and write ’ ≈T . Especially, if T = ∅ and T ’, we say ’ is a theorem of L∗ and write ’; if T = ∅ and ’ ≈T , we say ’ is provably equivalent to and write ’ ≈ . Proposition 2.7. following are theorems or rules ofL∗ [10]: (D0 ) (D1 ) (D2 ) (D3 ) (D4 ) (D5 ) (D6 ) (D7 )
’ → ’, + + ’ → ’, {’ → ; → &} ’ → &, (’ → ) → (+ → + ’), { → ’; & → '} (’ → &) → ( → '), & ∧ ’ → ’ ∧ &, {’ → ; ’ → &} ’ → ∧ &, {’; } ’ ∧ ,
S.-M. Wang et al. / Fuzzy Sets and Systems 136 (2003) 55 – 70
(D8 ) (D9 ) (D10 ) (D11 ) (D12 ) (D13 ) (D14 )
{’ → ; & → (} ’ ∨ & → ∨ (, ’ ∧ → ’; ’ ∧ → , (’ → ) ∨ ( → ’), + ’ ∨ → (’ → ), ’ ∨ ’ ≈ ’, (’ → ) ≈ (+ → + ’), (’ → ( → &)) ≈ ( → (’ → &)).
Proof. (D0 ): (1◦ ) ’ → ((’ → + ’) → ’) by (A1 ), (2◦ ) (’ → ((’ → + + ’) → ’)) → ((’ → + + ’) → (’ → ’)) (3◦ ) (’ → + + ’) → (’ → ’) by (1◦ ); (2◦ ) and (MP), (4◦ ) ’ → + + ’ by (A5 ), (5◦ ) ’ → ’ by (3◦ ); (4◦ ) and (MP).
by (A3 ),
(D1 ): (1◦ ) + ’ → + + + ’ by (A5 ), (2◦ ) (+ ’ → + + + ’) → (+ + ’ → ’) by (A2 ), (3◦ ) + + ’ → ’ by (1◦ ); (2◦ ) and (MP). (D2 ): (1◦ ) ’ → Assumption, (2◦ ) → & Assumption, (3◦ ) ( → &) → ((’ → ) → (’ → &)) by (A4 ), (4◦ ) (’ → ) → (’ → &) by (2◦ ); (3◦ ) and (MP), (5◦ ) ’ → & by (1◦ ); (4◦ ) and (MP). (D3 ): (1◦ ) (’ → ) → ((+ + ’ → ’) → (+ + ’ → )) by (A4 ), (2◦ ) ((’ → ) → ((+ + ’ → ’) → (+ + ’ → ))) → ((+ + ’ → ’) → ((’ → ) → (+ + ’ → ))) by (A3 ), (3◦ ) (+ + ’ → ’) → ((’ → ) → (+ + ’ → )) by (1◦ ); (2◦ ) and (MP), (4◦ ) (’ → ) → (+ + ’ → ) by (D1 ); (3◦ ) and (MP), (5◦ ) → + + by (A5 ), ◦ (6 ) ( → + + ) → ((+ + ’ → ) → (+ + ’ → + + )) by (A4 ), (7◦ ) (+ + ’ → ) → (+ + ’ → + + ) by (5◦ ); (6◦ ) and (MP), (8◦ ) (’ → ) → (+ + ’ → + + ) by (4◦ ); (7◦ ) and (D2 ), (9◦ ) (+ + ’ → + + ) → (+ → + ’) by (A2 ), (10◦ ) (’ → ) → (+ → + ’) by (8◦ ); (9◦ ) and (D2 ). (D4 ): (1◦ ) → ’ Assumption, (2◦ ) & → ' Assumption,
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(3◦ ) (4◦ ) (5◦ ) (6◦ ) (7◦ ) (8◦ )
(& → ') → ((’ → &) → (’ → ')) by (A4 ), (’ → &) → (’ → ') by (2◦ ); (3◦ ) and (MP), ((’ → &) → (’ → ')) → (’ → ((’ → &) → ')) by (A3 ), ’ → ((’ → &) → ') by (4◦ ); (5◦ ) and (MP), → ((’ → &) → ') by (1◦ ); (6◦ ) and (D2 ), (’ → &) → ( → ') by (7◦ ); (A3 ) and (MP).
(D5 ): (1◦ ) + ’ ∨ + & → + & ∨ + ’ by (A7 ), (2◦ ) (+ ’ ∨ + & → + & ∨ + ’) → (+ (+ & ∨ + ’) → + (+ ’ ∨ + &)) (3◦ ) + (+ & ∨ + ’) → + (+ ’ ∨ + &) by (1◦ ); (2◦ ) and (MP), (4◦ ) & ∧ ’ → ’ ∧ & by (3◦ ) and (Def : 2:2). (D6 ): (1◦ ) ’ → & Assumption, (2◦ ) ’ → Assumption, (3◦ ) (’ → &) → (’ ∧ ’ → & ∧ ’) by (A12 ), (4◦ ) ’ ∧ ’ → & ∧ ’ by (1◦ ); (3◦ ) and (MP), (5◦ ) (’ → ) → (’ ∧ & → ∧ &) by (A12 ), (6◦ ) ’ ∧ & → ∧ & by (2◦ ); (5◦ ) and (MP), (7◦ ) ’ → ’ ∧ ’ by (A11 ), (8◦ ) ’ → & ∧ ’ by (4◦ ); (7◦ ) and (D2 ), (9◦ ) & ∧ ’ → ’ ∧ & by (D5 ), (10◦ ) ’ → ’ ∧ & by (8◦ ); (9◦ ) and (D2 ), (11◦ ) ’ → ∧ & by (6◦ ); (10◦ ) and (D2 ). (D7 ):
(1◦ )
(2◦ ) (3◦ ) (4◦ ) (5◦ ) (6◦ ) (7◦ )
Assumption, Assumption, → (’ → ) by (A1 ), ’→ by (1◦ ); (3◦ ) and (MP), ’ → ’ by (D0 ), ’→’∧ by (4◦ ); (5◦ ) and (D6 ), ’∧ by (2◦ ); (6◦ ) and (MP). ’
(D8 ): (1◦ ) ’ → Assumption, ◦ (2 ) & → ' Assumption, (3◦ ) → ∨ ' by (A6 ), (4◦ ) ' → ' ∨ by (A6 ), (5◦ ) ’ → ∨ ' by (1◦ ); (3◦ ) and (D2 ), (6◦ ) & → ' ∨ by (2◦ ); (4◦ ) and (D2 ), ◦ (7 ) ' ∨ → ∨ ' by (A7 ), (8◦ ) & → ∨ ' by (6◦ ); (7◦ ) and (D2 ),
by (D3 ),
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(9◦ ) (’ → ∨ ') ∧ (& → ∨ () by (5◦ ); (8◦ ) and (D7 ), (10◦ ) (’ → ∨ ') ∧ (& → ∨ () → (’ ∨ & → ∨ ') by (A8 ), (11◦ ) ’ ∨ & → ∨ ' by (9◦ ); (10◦ ) and (MP). (D9 ): +’→+’∨+ by (A6 ), (2◦ ) + → + ’ ∨ + by (A6 ); (A7 ) and (D2 ), (3◦ ) (+ ’ → + ’ ∨ + ) → (+ (+ ’ ∨ + ) → + + ’) by (D3 ), (4◦ ) (+ → + ’ ∨ + ) → (+ (+ ’ ∨ + ) → + + ) by (D3 ), (5◦ ) + (+ ’ ∨ + ) → + + ’ by (1◦ ); (3◦ ) and (MP), (6◦ ) + (+ ’ ∨ + ) → + + by (2◦ ); (4◦ ) and (MP), ◦ ◦ (7 ) ’ ∧ → ’ by (5 ); (D1 ) and (D2 ), (8◦ ) ’ ∧ → by (6◦ ); (D1 ) and (D2 ).
(1◦ )
(D10 ): (1◦ ) ’ ∧ → ’ ∧ by (D0 ), (2◦ ) (’ ∧ → ’ ∧ ) → (’ → ’ ∧ ) ∨ ( → ’ ∧ ) by (A9 ), (3◦ ) (’ → ’ ∧ ) ∨ ( → ’ ∧ ) by (1◦ ); (2◦ ) and (MP), (4◦ ) ’ ∧ → ; ’ ∧ → ’ by (D9 ), (5◦ ) (’ ∧ → ) → ((’ → ’ ∧ ) → (’ → )) by (A4 ), (6◦ ) (’ ∧ → ’) → (( → ’ ∧ ) → ( → ’)) by (A4 ), (7◦ ) (’ → ’ ∧ ) → (’ → ) by (4◦ ); (5◦ ) and (MP), (8◦ ) ( → ’ ∧ ) → ( → ’) by (4◦ ); (6◦ ) and (MP), (9◦ ) (’ → ’ ∧ ) ∨ ( → ’ ∧ ) → (’ → ) ∨ ( → ’) by (7◦ ); (8◦ ) and (D8 ), (10◦ ) (’ → ) ∨ ( → ’) by (3◦ ); (9◦ ) and (MP). (D12 ): (1◦ ) + ’ → + ’; + ’ → + ’ by (D0 ), (2◦ ) + ’ → + ’ ∧ + ’ by (D6 ), (3◦ ) ’ ∨ ’ → ’ by (2◦ ); (D3 ); (D1 ) and (D2 ). (D11 ): (1◦ ) + ’ → (+ → + ’) by (A1 ), (2◦ ) + ’ → (’ → ) by (1◦ ); (A2 ) and (D2 ), (3◦ ) → (’ → ) by (A1 ), (4◦ ) + ’ ∨ → (’ → ) by (2◦ ); (3◦ ); (D8 ); (D12 ) and (D2 ). Easily,we can get (D13 ) by (A2 ) and (D3 ), and (D14 ) by (A3 ). Theorem 2.8 (the substitution theorem). Let w:f: ’ be built up from w:f:’s ’1 ; : : : ; ’t by means of + and → denoted by ’(’1 ; : : : ; ’t ). If ’i ≈T i for integer i from 1 to t, then ’ ≈T ’ ( 1 ; : : : ; t ), where ’( 1 ; : : : ; t ) is given by replacing each occurrence of ’i by i .
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Proof. The proof is by induction on the number n of connectives in building up ’ from ’1 ; ’2 ; : : : ; ’t . For the base step suppose that n = 0; then t =1; ’ = ’(’1 ) = ’1 and ’( 1 ) = 1 . Thus ’ ≈ ’( 1 ) from ’1 ≈T 1 . Now suppose that the theorem holds for any n6k. For n = k + 1, there are two cases to be considered: Case 1: ’ is + ’ (’1 ; : : : ; ’t ). Applying the hypothesis of induction we have ’ (’1 ; : : : ; ’t ) ≈T ’ ( 1 ; : : : ; t ). By D3 and A2 , we can get + ’ (’1 ; : : : ; ’t ) ≈T + ’ ( 1 ; : : : ; t ) i.e., ’ ≈T ’( 1 ; : : : ; t ). Case 2: ’ is ’ (’1 ; ’2 ; : : : ; ’t ) → ’ (’1 ; ’2 ; : : : ; ’t ). Applying the hypothesis of induction we have ’ (’1 ; : : : ; ’t ) ≈T ’ ( 1 ; : : : ; t ) and ’ (’1 ; : : : ; ’t ) ≈T ’ ( 1 ; : : : ; t ). By D4 we can get ’ (’1 ; : : : ; ’t ) → ’ (’1 ; : : : ; ’t ) ≈T ’ ( 1 ; : : : ; t ) → ’ ( 1 ; : : : ; t ), i.e., ’ ≈T ’( 1 ; : : : ; t ). Hence by the principle of mathematical induction the theorem holds whatever the number of connectives in building up ’ from ’1 ; ’2 ; : : : ; ’t . The above proof is analogous to that of Theorem 3.2.23 on p. 79 of [10]. Proposition 2.9. The following is a theorem of L∗ : (D15 ) (’ → (’ → (’ → ))) ≈ (’ → (’ → )) Proof. (1◦ ) (’ → + ’) → ((’ → (’ → + ’)) → (’ → + ’)) by (A1 ), (2◦ ) ((’ → + ’) → + ’) → ((’ → (’ → + ’)) → (’ → + ’)) by (A4 ), (3◦ ) (’ → + ’) ∨ ((’ → + ’) → + ’) by (A10 ); (D12 ) and (Th: 2:8), (4◦ ) (’ → + ’) ∨ ((’ → + ’) → + ’) → ((’ → (’ → + ’)) → (’ → + ’)) by (1◦ ); (2◦ ); (D8 ); (D12 ) and (Th: 2:8), (5◦ ) (’ → (’ → + ’)) → (’ → + ’) by (3◦ ); (4◦ ) and (MP), (6◦ ) (’ → (’ → + ’)) ≈ (’ → + ’) by (5◦ ) and (A1 ), (7◦ ) (’ → (’ → (’ → ))) ≈ (’ → (’ → (+ → + ’))) by (D0 ); (D13 ) and (Th: 2:8), (8◦ ) (’ → (’ → (’ → ))) ≈ (+ → (’ → (’ → + ’))) by (7◦ ); (D14 ) and (Th: 2:8), (9◦ ) (’ → (’ → (’ → ))) ≈ (+ → (’ → + ’)) by (8◦ ); (6◦ ) and (Th: 2:8), (10◦ ) (’ → (’ → (’ → ))) ≈ (’ → (’ → )) by (9◦ ); (D14 ); (D13 ) and (Th: 2:8). Theorem 2.10 (the generalized deduction theorem). Let T be a theory of L∗ and ’ and and T ∪ {’} . Then T ’ → (’ → ).
w:f:’s
Proof. The proof is by induction on the number n of w:f:’s in the sequence forming the deduction of from T ∪ {’}. For the base step suppose that this sequence has one member. That member must be itself, and so either is an axiom of L∗ or is a member of T ∪ {’}. Case 1: is an axiom of L∗ or ∈ T . The following is then a deduction of ’ → (’ → ) from T : (1◦ ) axiom of L∗ or a member of T , (2◦ ) → (’ → ); (’ → ) → (’ → (’ → )) by (A1 ), (3◦ ) ’ → (’ → ) by (1◦ ); (2◦ ) and (MP).
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Case 2: is ’. By D0 we can easily get its proof. This ends the base step. Now suppose that the theorem holds if n¡k, where k is an integer and k¿1. Then, similarly, we can get the theorem hold if is an axiom of L∗ , or ∈ T , or = ’. Now suppose that is obtained from two previous w:f:’s in the deduction by an application of MP. These two w:f:’s must have the forms & and & → . By the hypothesis of induction, we get T ’ → (’ → &) and T ’ → (’ → (& → )). The required deduction of ’ → (’ → ) from T may now be given as follows: (1◦ ) (2◦ ) (3◦ ) (4◦ ) (5◦ ) (6◦ ) (7◦ )
’ → (’ → &) Assumption, ’ → (’ → (& → )) Assumption, & → (’ → (’ → )) by (2◦ ) and (D14 ), (& → (’ → (’ → ))) → ((’ → &) → (’ → (’ → (’ → )))) by (A4 ), (’ → &) → (’ → (’ → )) by (3◦ ); (4◦ ); (MP); (D15 ) and (Th: 2:8), ’ → (’ → (’ → )) by (1◦ ); (5◦ ) and (D2 ), ’ → (’ → ) by (6◦ ); (D15 ) and (Th: 2:8).
Thus, the theorem holds if n = k. Hence by the principle of mathematical induction the theorem holds whatever the number of w:f:’s in the deduction of from T ∪ {’}. Proposition 2.11 (see Theorem 2.3 [4]). Let {’ ∨ } &.
T1 ∪ {’} &
and
T2 ∪ { } &,
then
T1 ∪ T2 ∪
Proof. By Theorem 2.10 and T1 ∪ {’} & we can get T1 ’ → (’ → &) then T1 ∪ T2 ∪ {’ ∨ } ’ → (’ → &). Similarly, T1 ∪ T2 ∪ {’ ∨ } → ( → &). Now we prove {’ → (’ → &); → ( → &); ’ ∨ } &. (1◦ ) (2◦ ) (3◦ ) (4◦ ) (5◦ ) (6◦ ) (7◦ ) (8◦ ) (9◦ ) (10◦ ) (11◦ ) (12◦ ) (13◦ ) (14◦ ) (15◦ ) (16◦ ) (17◦ ) (18◦ ) (19◦ )
’ → (’ → &) Assumption, → ( → &) Assumption, ’∨ Assumption, → (’ → (’ → &)) by (1◦ ); (A1 ) and (MP), ’ → (’ → ( → &)) by (4◦ ); (D14 ) and (Th: 2:8), ’ → ( → ( → &)) by (2◦ ); (A1 ) and (MP), → (’ → ( → &)) by (6◦ ); (D14 ) and (Th: 2:8), ’ ∨ → (’ → ( → &)) by (5◦ ); (7◦ ); (D8 ); (D12 ) and (Th: 2:8), ’ → ( → &) by (3◦ ); (8◦ ) and (MP), ’ ∨ → ( → &) by (2◦ ); (9◦ ); (D8 ); (D12 ) and (Th: 2:8), → & by (3◦ ); (10◦ ) and (MP), → ( → (’ → &)) by (7◦ ); (D14 ) and (Th: 2:8), ’ → ( → (’ → &)) by (4◦ ); (D14 ) and (Th: 2:8), ∨ ’ → ( → (’ → &)) by (12◦ ); (13◦ ); (D8 ); (D12 ) and (Th: 2:8), → (’ → &) (3◦ ); (14◦ ) and (MP), ’ ∨ → (’ → &) (1◦ ); (15◦ ); (D8 ); (D12 ) and (Th: 2:8), ’ → & by (3◦ ); (16◦ ) and (MP), ’ ∨ → & by (11◦ ); (17◦ ); (D8 ); (D12 ) and (Th: 2:8), & by (3◦ ); (18◦ ) and (MP).
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Theorem 2.12 (the soundness theorem of L∗ ). Let T be a theory and ’ be a w.f. If T ’, then T |= ’. Proof (Wang [10]): The proof is by induction on the number of w:f:’s in a sequence of w:f:’s which constitutes a proof of ’ in T . We can easily get that all axioms are tautologies and MP keeps tautologies, i.e., if ’ and ’ → are tautologies then is a tautology. By these we can get the proof of this theorem. 3. The completeness theorem for L∗ 8
8
Now we introduce the truth constant 12 into the language of L∗ such that e( 12 ) = 12 for any W8 -evaluation e of L+∗ and denote the extended set of w:f:’s by F(S + ). Accordingly, we introduce 8 8 8 8 the axiom ( 12 → + 21 ) ∧ (+ 21 → 12 ) into the set of axioms of L∗ and denote the new formal system by L+∗ . And if ’ is provable in T over L+∗ we denote it by T + ’. Denition 3.1 (Menu and Pavelka [5], Wang [10]). Let M be an algebra of type (+ →): M is said to be a linearly ordered R0 -algebra if there is a linearly ordering relation 6 on M such that (i) M is bounded with respect to 6 and 0M and 1M stand for the least element in M and the greatest one, respectively; (ii) + is an order-reversing involution on M with respect to 6 and there is an element denoted by 1=2M ; s:t: + 1=2M = 1=2M ; (iii) ∀x; y ∈ M; x → y = 1 if x6y; x → y = max(+ x; y) otherwise. Let M be a linearly ordered R0 -algebra. We can deGne the following operations on M: ∀x; y ∈ M; x ⊗ y = +(x → + y); x ∨ y = max(x; y); x ∧ y = min(x; y). Easily, we can get a linearly ordered R0 -algebra is a residuated lattice in which the adjoint couple is ⊗ and →. But it is not a BL-algebra since the following does not hold: x ∧ y = x ⊗ (x → y) which was given in [9]. Denition 3.2. Let T be a theory over L∗ . (i) T is consistent if there is no a w:f: ’ such that T ’ and T + ’, (ii) T is complete if for any ’; ∈ F(S); T ’ → or T → ’. Proposition 3.3. Let T be a theory. Then T is consistent if and only if there is a w.f. ’ unprovable in T. Proof. We only prove the suOciency. Assume T is inconsistent. Then there is a w:f: such that T and + . By (A1 ), we get → (+ ’ → ), then T + ’ → , then + → ’ by (D3 ); (D1 ) and (D2 ), then ’. A contradiction to ’ being unprovable in T ! Hence T is consistent. Proposition 3.4. Let T be a theory over L∗ and ’ unprovable in T. Then there is a complete consistent theory T8 over L+∗ such that T ⊆ T8 and ’ is unprovable in T8 .
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65
Proof. Let {((n ; n )} be the sequence of all w:f: pairs in F(S + ). We construct inductively the sequence {Tn } of theories as follows: Let T0 = T . Then ’ is unprovable in T0 over L+∗ . Assume Tn−1 has been constructed such that T0 ⊆ Tn−1 and ’ is unprovable in Tn−1 over L+∗ , then Tn is decided by Tn−1 and ((n ; n ). If ’ is unprovable in Tn−1 ∪ {(n → n } over L+∗ , put Tn = Tn−1 ∪ {(n → n }; otherwise Tn = Tn−1 ∪ { n → (n }, in which case, assume Tn + ’. Then Tn−1 + ’ by (D10 ) and Theorem 2.11. A contradiction! Thus, ’ is unprovable in Tn over L+∗ . Now let T8 be the union of all Tn . Then clearly T8 is complete and Tn ⊆ Tm if 06n6m for any integers n and m. Assume T8 + ’, then there is an integer n such that Tn + ’. A contradiction! Thus, ’ is unprovable in T8 over L+∗ : T8 is also consistent by Proposition 3.3. Proposition 3.5 (Wang [10]). Let T8 be a complete consistent theory over L+∗ and ≈+ be T8 + + + + + 8 T -provably equivalent relation on F(S ). Let [’]T8 = { ∈ F(S ) | ≈T8 ’} for any ’ ∈ F(S ) and [F]+ = {[’]+ | ’ ∈ F(S + )}. Then: T8 T8 + (i) ≈T8 is a congruent relation on F(S + ). (ii) The quotient algebra [F]+ given by the congruence ≈+ on F(S + ) is a linearly ordered set in T8 T8 which the order relation 6 is de8ned as follows: ∀’; ∈ F(S + ); [’]+ 6[ ]+ if and only if T8 T8 8 T + ’ → . 8 +8 ; 1[F]+ = [1] 8 +8 ; (iii) + is an order-reversing involution on [F]+ with respect to 6 and 0[F]+8 = [0] T8 T T T T8 + + + [’]+ ∨ [ ] = max([’] ; [ ] ). T8 T8 T8 T8 + + + (iv) [’]+ → [ ] = 1 if [’] 6[ ] ; [’]+ → [ ]+ = max(+ [’]+ ; [ ]+ ) otherwise. 8 8 8 T T T T8 T8 T8 T8 T8 + + 8 8 if and only if T8 + ’. (v) [’]T8 = [1] T 8 with respect to 6. (vi) 12 [F]+ = [ 12 ]+ T8 T8
is an equivalent relation on F(S + ). By (D4 ) and (D13 ) Proof. (i) By (D0 ) and (D2 ) we can get ≈+ T8 we can get it is congruent. (ii) We deGne 6 on [F]+ as follows: ∀’; ∈ F(S + ); [’]+ 6[ ]+ if and only if T8 + ’ → . T8 T8 T8 Easily, we can get 6 is partially ordered by (D0 ) and (D2 ). And it is linearly ordered since T8 is complete. 8 +8 , i.e. 1[F]+ = [1] 8 +8 . Similarly, we (iii) ∀’ ∈ F(S + ); T8 + ’ → 18 by (D0 ) and (A1 ). Then [’]+ 6[1] T8 T T T8 8 +8 by (D0 ); (D13 ) and (A1 ); + is an involution on [F]+8 by (D1 ) and (A5 ) and can get 08 [F]+ = [0] T8
T
T
it is order-reversing with respect to 6 by (D3 ); [’]+ ∨ [ ]+ = max([’]+ ; [ ]+ ) holds by (A6 ) and T8 T8 T8 T8 (D8 ). (iv) ∀’; ∈ F(S + ); [’]+ 18 if and 6[ ]+ if and only if T8 + ’ → if and only if ’ → ≈+ T8 T8 T8 + + + + + only if [’ → ]T8 = 1[F]+8 if and only if [’]T8 → [ ]T8 = 1[F]+8 . If [’]T8 [ ]T8 , then [’ → ]+ ¡1[F]+8 T8 T T T + + + and [’ → ]T8 ∨ [(’ → ) → + ’ ∨ ]T8 = 1[F]+8 by (A10 ). Hence [(’ → ) → + ’ ∨ ]T8 = 1[F]+8 . Then T T [’ → ]+ 6[+ ’ ∨ ]+ and [+ ’ ∨ ]+ 6[’ → ]+ by (D11 ). Thus [’]+ → [ ]+ = [’ → ]+ = [+ ’ T8 T8 T8 T8 T8 T8 T8 ∨ ]+ = max(+ [’]+ ; [ ]+ ). T8 T8 T8 (v) and (vi) are trivial and their proof are omitted. These show [F]+ is a linearly ordered R0 T8 algebra.
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Proposition 3.6. Let M be a countable linearly ordered R0 -algebra. Then there is a countable densely linearly ordered R0 -algebra M such that there is an isomorphic embedding i : M → M of type (+; →). Proof. (i) ∀u ∈ M , put Cu = {(u; 0)} if u has no successor in M; Cu = {(u; r) | 06r¡1; r ∈ Q} otherwise. Let M = u∈M Cu and for (u; r); (v; s) ∈ M put (u; r)6(v; s) if and only if u¡M v or (u = v and r6s) (lexicographic product). Clearly (M ; 6) is a bounded and densely ordered chain in which the least and the greatest elements are (0; 0) and (1; 0), respectively. (ii) ∀(u; r) ∈ M , put + (u; 0) = (+ u; 0) if r = 0; + (u; r) = (+ u ; 1 − r) otherwise, where u is the successor of u in M . Trivially, we get + is an order-reversing involution on M and +(1=2M ; 0) = (+ 1=2M ; 0) = (1=2M ; 0). (iii) ∀(u; r); (v; s) ∈ M , put (u; r) → (v; s) = (1; 0) if (u; r)6(v; s); (u; r) → (v; s) = max(+ (u; r); (v; s)) otherwise. Then M is a densely linearly ordered R0 -algebra by (i)–(iii). (vi) Trivially, we can get the mapping f(u) = (u; 0) embeds M isomorphically to M . Proposition 3.7. Let M be a countable densely linearly ordered R0 -algebra. Then there is an isomorphic embedding f of M into W8 . Proof. Let M1 = {x ∈ M | x¡1=2M }; M2 = {x ∈ M | x¿1=2M }; W1 = {x ∈ W | x¡ 12 } and W2 = {x ∈ W | x¿ 12 }. Then M1 ; M2 ; W1 and W2 are countable densely linearly ordered sets. It is well known that any two countable densely linearly ordered sets with the greatest and least elements are orderisomorphic. Let f be an order isomorphism of M1 onto W1 . Now we construct f8 : M → W , put 8 = f(x) if x ∈ M1 ; f(x) 8 = + f(+ x) if x ∈ M2 and f( 1 ) = 1 . Trivially, we get f8 is an algebraic f(x) 2M 2 isomorphism of M onto W and, the identical mapping i : W → W8 is an isomorphic embedding. Thus i ◦ f8 is an isomorphic embedding of M into W8 . Theorem 3.8 (the strong completeness theorem for L∗ ). Let T be a consistent theory over L∗ and ’ ∈ F(S) and T |= ’, then T ’. Proof. Assume ’ be unprovable in T over L∗ . By Proposition 3.4, there is a complete consistent theory T8 over L+∗ such that T ⊆ T8 and ’ is unprovable in T8 over L+∗ . By Proposition 3.5, [F]+ T8 is a linearly ordered R0 -algebra. Let e be the natural homomorphism from F(S + ) onto [F]+ , i.e. T8 + + 8 8 + if and only if T for any ∈ F(S ). Then e( ) = [ 1] = 1 . Thus e(’)¡1. By e( ) = [ ]+ + [F]T8 T8 T8 + 8 Propositions 3.6 and 3.7, there is an isomorphically embedding f from [F]T8 into W . Thus, f ◦ e is 8 ) = f ◦ e( ) for any ∈ F(S). Trivially, e8 is a W8 -evaluation of L∗ a W8 -evaluation of L+∗ . Put e( and a model of T . And e(’) 8 = f ◦ e(’) = f(e(’))¡1. A contradiction to T |= ’. Thus T ’. In the preceding theorem, put T = ∅, we get Theorem 3.9 (the completeness theorem for L∗ ). Let w:f: ’ be a W8 -tautology of L∗ . Then ’ is a theorem of L∗ , i.e. |= ’ implies ’.
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4. The completeness theorem for Ln∗ In this section, we will consider the axiomatization of the propositional calculi PC(Wn ) given by Wn . Denition 4.1. Let n be an integer and n¿4. The formal system Ln∗ of PC(Wn ) is the schematic extension (see [2, 2.3.20]) of L∗ by adding the following to its axiom schemes: (A13 ) /(p1 ; : : : ; pn ) where w:f: /(p1 ; : : : ; pn )(/ for short) is deGned as follows: (1◦ ) n = 2m. Let /0 = + ((pn → + pn ) → + (+ pn → pn )) → pn ∧ + pn and let /1 = p1 ∨ + p1 and let k be an integer. If 26k6m, then /k = (pk → /k −1 ) ∨ pk . If m + 16k62m − 1, then /k = /k −1 ∧ (pk → pk ). Then /(p1 ; : : : ; pn ) = /2m−1 ∧ /0 ; (2◦ ) n = 2m + 1. Let /2 = p1 ∨ (p1 → + p2 ) ∨ (+ p2 → p2 ) and let k be an integer. If 36k6m, then /k = (pk → /k −1 ) ∨ pk . If m+16k62m+1; /k = /k −1 ∧ (pk → pk ). Then /n = /(p1 ; : : : ; pn ). Let T be a theory and ’ w:f: T ∪ {/} ’ is denoted by T n ’. For any w:f:’s ’ and , if n ’ → short).
and n
→ ’ we say ’ is T -provably equivalent to
Ln∗
and write ’ ≈ T
(’ ≈nT
for
Denition 4.2. (i) A Wn -evaluation of Ln∗ is a mapping e : F(S) → Wn such that e(+ ’) = + e(’) and e(’ → ) = e(’) → e( ) for any ’; ∈ F(S). The set of all Wn -evaluation is denoted by #(Wn ) and called Wn -semantic of Ln∗ ; (ii) a w:f: ’ is a Wn -tautology of Ln∗ (denoted by |=n ’) if e(’) = 1 for each e ∈ #(Wn ), and the set of all Wn -tautologies of Ln∗ is denoted by T (Wn ); (iii) let T be a theory and e ∈ #(Wn ): e is a Wn -model of T if e(’) = 1 for any ’ ∈ T ; (iv) if e(’) = 1 for each Wn -model e of the theory T , then we say T yields semantically ’ and write T |=n ’. Denition 4.3 (Wang [11]). (i) Let 0 ∈ Wn . If e(’)¿0 for each Wn -evaluation e, we say ’ is an 0-tautology of Ln∗ . The set of all 0-tautologies of Ln∗ is denoted by 0 − T (Wn ). (ii) Let ’ ∈ 0 − T (Wn ). If there is a Wn -evaluation e such that e(’) = 0, we say 0 is an exact 0-tautology. The set of all exact 0-tautologies is denoted by [0] − T (Wn ). Proposition 4.4. Let 12 60¡1 and let ’ = ’(p1 ; : : : ; pt ) be a w.f. If ’ ∈ [0]−T (Wn ) and p ∈ {p1 ; : : : ; pt }, then (p → ’) ∨ p ∈ [0 + 1=n − 1] − T (Wn ). We can use the proof given for Theorem 9 of [11]. Proposition 4.5. (i) Let ∅ = M ⊆ Wn . Then M is a subalgebra of Wn if and only if {0; 1} ⊆ M and x ∈ M implies +x ∈ M for any x ∈ Wn . (ii) Let n be an odd integer, n¿3. For any integer k; Wk is isomorphic to a subalgebra of Wn if and only if 26k6n.
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(iii) let n be even integer and n¿2, then for any integer k; Wk is isomorphic to a subalgebra of Wn if and only if 26k6n and k is an even one. (iv) If M is isomorphic to a subalgebra of Wn , then T (Wn ) ⊆ T (M ). Proof. (i) Necessity is proved by 0 = + (x → x) and 1 = x → x and suOciency by x → y ∈ {1; + x; y}. (ii) Necessity is trivial and omitted. SuOciency is proved as follows: Case 1. (k is an even integer). We construct an isomorphically embedding f from Wk to Wn by letting f(i=k − 1) = i=n − 1 if 06i6k=2 − 1 and f(i=k − 1) = 1 − f(1 − i=k − 1) otherwise. Case 2. (k is an odd integer). An isomorphically embedding f from Wk to Wn is constructed by letting f(i=k − 1) = i=n − 1 if 06i¡k − 1=2 and f(1=2) = 1=2 and f(i=k − 1) = 1 − f(1 − i=k − 1) otherwise. (iii) is similar to (ii) and omitted. (iv) Let i be an isomorphic embedding from M to Wn and ’ ∈ T (Wn ): ∀e ∈ #(M ); i ◦ e ∈ #(Wn ). Then i ◦ e(’) = i(e(’)) = 1. Thus e(’) = 1. Then ’ ∈ T (M ). Hence T (Wn ) ⊆ T (M ). Proposition 4.6. (i) If n is an odd integer and k is an integer, then / ∈ T (Wk ) if and only if 26k6n. (ii) If n is an even integer and k is an integer, then / ∈ T (Wk ) if and only if 26k6n and k is an even integer. Proof. (i) Let n = 2m + 1. By trivial calculation, we can get /2 ∈ [m + 2=2m] − T (W2m+1 ) ∩ [m + 2=2m+1]−T (W2m+2 ). By Proposition 4.4, we can get /3 ∈ [m+3=2m]−T (W2m+1 ) ∩ [m+3=2m+1]− T (W2m+2 ); : : :, and /m ∈ [2m=2m] − T (W2m+1 ) ∩ [2m=2m + 1] − T (W2m+2 ). Thus /m ∈ T (W2m+1 ) and /m ∈ T (W2m+2 ). Trivially, we can get / ∈ T (W2m+1 ) and / ∈ T (W2m+2 ). Since W2m+2 is isomorphic to a subalgebra of Wk for k ¿ 2m+2; T (Wk ) ⊆ T (W2m+2 ) by Proposition 4.5. Therefore, / ∈ T (Wk ) for k¿2m + 2. For 26k62m + 1; Wk is isomorphic to a subalgebra of W2m+1 . Then T (W2m+1 ) ⊆ T (Wk ). Therefore, / ∈ T (Wk ) for 26k6n. (ii) Let n = 2m. By trivial calculation, we can get /1 ∈ [m=2m − 1] − T (W2m ) ∩ [m + 1=2m + 1] − T (W2m+2 ); /0 ∈ T (W2k ) and /0 ∈ T (W2k+1 ) for any integer k¿1. By Proposition 4.4, we can get /2 ∈ [m + 1=2m − 1] − T (W2m ) ∩ [m + 2=2m + 1] − T (W2m+2 ); : : :, and /m ∈ [2m − 1=2m − 1] − T (W2m ) ∩ [2m=2m + 1] − T (W2m+2 ). Then /m ∈ T (W2m ) and /m ∈ T (W2m+2 ). Trivially, we can get / ∈ T (Wk ) if and only if k is an even integer and 26k6n. Proposition 4.7. Let k be an integer and k¿2. Then / ∈ T (Wk ) implies there is an isomorphically embedding i : Wk → Wn of type (+; →). We can prove it by Propositions 4.6 and 4.5. Theorem 4.8 (the soundness theorem for Ln∗ ). Let T be a theory and ’ ∈ F(S). Then T n ’ implies T |=n ’. We can use the proof given for Theorem 2.12 and / ∈ T (Wn ).
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Denition 4.9. Let M be an algebra of type (+; →). M is said to be a linearly ordered Rn -algebra if there is a linearly ordering relation 6 on M such that (i) M is bounded with respect to 6 and, 0M and 1M stand for the least element in M and the greatest, respectively; (ii) + is an order-reversing involution on M with respect to 6; (iii) ∀x; y ∈ M; x → y = 1 if x6y; x → y = max(+ x; y) otherwise; 8 1 ; : : : ; xn ) = 1, where 1(x 8 1 ; : : : ; xn ) is the truth function corresponding to the 13th axiom (iv) 1(x scheme /(p1 ; : : : ; pn ) of Ln∗ and x1 ; : : : ; xn are arbitrary elements in M . Proposition 4.10. Let M be a linearly ordered Rn -algebra. Then there is an isomorphic embedding i : M → Wn of type (+; →). Proof. (i) Let |M | = k + 1¡∞ and M = {x0 ; x1 ; : : : ; xk }, where 0 = x0 ¡x1 ¡ · · · ¡xk = 1. We can deGne a mapping f : M → Wk+1 by letting f(xi ) = i=k for any xi ∈ M . Clearly, f is an isomor8 1 =k; : : : ; in =k) = 1(f(x 8 8 phism of type (+; →). Since 1(i i1 ); : : : ; f(xin )) = f(1(xi1 ; : : : ; xin )) = 1 for any n (i1 =k; : : : ; in =k) ∈ Wk+1 , Wk+1 is a linearly ordered Rn -algebra and / ∈ T (Wk+1 ). By Proposition 4.7, there is an isomorphic embedding g : Wk+1 → Wn of type (+; →). Hence i = g ◦ f is an isomorphic embedding from M to Wn of type (+; →). (ii) Assume |M | = ∞. Let {x1 ; : : : ; xn } ⊆ M , where x1 ¡x2 ¡ · · · ¡xn ¡1. Trivially, we can get {x1 ; : : : ; xn } ∪ {+ x1 ; : : : ; + xn } ∪ {0; 1} is a linearly ordered Rn -subalgebra of M and its cardinal number is ¿n, a contradiction to (i). Hence |M |¡∞. Proposition 4.11. (i) Let T be a theory and ’ unprovable in T. Then there is a complete consistent theory T8 such that T ⊆ T8 and ’ is unprovable in T8 . (ii) Let T8 be a complete consistent theory over Ln∗ and ≈nT8 be T8 -provably equivalent relation on F(S). Let [’]nT8 = { ∈ F(S) | ≈nT8 ’} for any ’ ∈ F(S) and [F]nT8 = {[’]nT8 | ’ ∈ F(S)}. Then (1◦ ) ≈nT8 is a congruent relation on F(S). (2◦ ) The quotient algebra [F]nT8 given by ≈nT8 is a linearly ordered Rn -algebra in which the order relation 6 is de8ned as follows: ∀’; ∈ F(S); [’]nT8 6[ ]nT8 if and only if T8 n ’ → . 8 n8 if and only if T8 n ’. (3◦ ) [’]nT8 = [1] T The proofs of (i) and (ii) are similar to that of Propositions 3.4 and 3.5, respectively. Theorem 4.12 (the strong completeness theorem for Ln∗ ). Let T be a consistent theory over Ln∗ and ’ ∈ F(S). If T |=n ’, then T n ’. The proof is similar to that of Theorem 3.8. Theorem 4.13 (the completeness theorem for Ln∗ ). Let w:f: ’ be a Wn -tautology of Ln∗ . Then ’ is a theorem of Ln∗ , i.e. if |=n ’. Then n ’.
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5. Concluding remarks Since Wang introduced the formal deductive system L∗ for the fuzzy propositional calculus in 1997, more and more studies show it to have many good properties and important applications. Firstly, since the nilpotent minimum and the corresponding implication, as pointed out in Ref. [1], combine advantageous properties of Lukasiewicz-like t-norms (e.g. the law of contradiction holds, the corresponding R- and S-implication coincide) and those of minimum itself (e.g. easy usage of 0-cuts in practice), L∗ based on them combines advantageous properties of the G6odel R-fuzzy logic RG (e.g. the deduction theorem is analogous to the classical one) and the Lukasiewicz R-fuzzy logic RL (e.g. +; ∨, and ∧ are the standard fuzzy negation, disjunction and conjunction). Secondly, the important issues of compactness [13], deduction, soundness and completeness for L∗ have been studied and very nice and deep logical results have been proved, which lay a Grm foundation for the investigations of practical problems. Thirdly, the formal system KL∗ of the fuzzy predicate calculus corresponding to L∗ has been constructed and the completeness for KL∗ with respect to W8 -interpretation has been proved. It should be notable that Ref. [12] shows L∗ as well as KL∗ to be powerful tools in analyzing topics of Zadeh’s agenda, such as fuzzy modus ponens, composition al rule of inference, fuzzy functions and fuzzy control, etc. Finally, the completeness theorem for Ln∗ with respect to Wn -semantic has practical signiGcance since no computer could deal with inGnite data. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
J.C. Fodor, Contrapositive symmetry of fuzzy implications, Fuzzy Sets and Systems 69 (1995) 141–156. P. HRajek, Metamathematics of Fuzzy Logic, Kluwer Academic Press, Dordrecht, 1998. A.G. Hamilton, Logic for Mathematicians, Cambridge University Press, London, 1978. E.P. Klement, M. Navara, A survey on di3erent triangular norm-based fuzzy logics, Fuzzy Sets and Systems 101 (2) (1999) 241–251. J. Menu, J. Pavelka, A note on tensor products on the unit interval, Comment. Math. Univ. Carol. 17 (1976) 71–82. ∗ D.W. Pei, G.J. Wang, The completeness of the formal system L , Science in China (Series E) 32 (2002) 56 – 64 (in Chinese). P. Perny, B. Roy, The use of fuzzy outranking relations in preference modelling, Fuzzy Sets and Systems 49 (1992) 33–53. A. Rose, J.B. Rosser, Fragment of many-valued statement calculi, Trans. Amer. Math. Soc. 87 (1958) 1–53. E. Turunen, Mathematics Behind Fuzzy Logic, Physica-Verlag, Wurzburg, 1999. G.J. Wang, A formal deductive system for fuzzy propositional calculus, Chinese Sci. Bull. 42 (1997) 1521–1526. G.J. Wang, Theory of -(0-tautologies) in revised Kleene systems, Sci. China, Ser. E 41 (1998) 188–195. G.J. Wang, The full implicational triple I method for fuzzy reasoning, Sci. China, Ser. E 42 (1999) 43–53 (in Chinese). G.J. Wang, Several compactness theorems for fuzzy semantic, Chinese Sci. Bull. 44 (1999) 1275–1279. G.J. Wang, Non-Classical Mathematical Logic and Approximate Reasoning, Science Press, Moscow, 2000. (in Chinese) F. Esteva, L. Godo, Monoidal t-norm based logic: towards a logic for left-continuous t-norms, Fuzzy Sets and Systems 124 (2001) 271–288.
A triangular-norm-based propositional fuzzy logic San-Min Wanga; ∗ , Bao-Shu Wanga , Guo-Jun Wangb a
School of Computer Science and Engineering, XiDian University, No. 2 South Taibai Road, Xi’an 710071, People’s Republic of China b Institute of Mathematics, Shaanxi Normal University, Xi’an 710062, People’s Republic of China Received 5 October 2000; received in revised form 18 March 2002; accepted 27 March 2002
Abstract In 1997, Wang introduced the formal system L∗ for a type of fuzzy propositional calculus given by a particular t-norm di3erent from the three famous ones (Lukasiewicz, G6odel, product). Recently, Pei proved the completeness theorem for L∗ with respect to W8 -semantic by developing a theory of algebraic systems. In this paper, we aim to prove it by essentially metamathematical method. Furthermore, we discussed originally the c 2002 Elsevier Science completeness for the schematic extension L∗n of L∗ with respect to Wn -semantic. B.V. All rights reserved. Keywords: Non-classical logics; Formal system L∗ ; Schematic extension L∗n ; Completeness; Triangular norm
1. Introduction Spurred by the success in applications, especially in fuzzy control, fuzzy logic aroused the interest of many famous scholars and a series of important results have been created . In 1992, a t-norm was introduced in Ref. [7] as follows: min{x; y} if (x) + (y) ¿ 1; T (x; y) = 0 otherwise; where is a strictly increasing continuous function from the interval onto itself such that (0) = 0 and (1) = 1. In Ref. [1], it was called nilpotent minimum and its R-implication was given as ∗
Corresponding author. E-mail addresses: [email protected], [email protected] (S.-M. Wang).
c 2002 Elsevier Science B.V. All rights reserved. 0165-0114/03/$ - see front matter PII: S 0 1 6 5 - 0 1 1 4 ( 0 2 ) 0 0 1 4 3 - 4
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follows: Imin0; (x; y) =
1
if x 6 y; −1
(max(1 − (x); (y))) otherwise:
It was estimated by Ref. [1] that the nilpotent minimum can be admitted into investigations of many theoretical and practical problems soon by its propitious characteristics. Since 1997, many creative works have been done starting with them as follows: Grstly, a formal system L∗ for a type of fuzzy propositional calculus, based on an instance of Imin0; which was called R0 -implicator by the author, was constructed in Ref. [10]; secondly, the generalized tautological theory of its corresponding semantical system, which was called the revised Kleene system by the author, was established in Ref. [11]; after that, the full implicational triple I method of fuzzy reasoning, which aims to lay a theoretical foundation for fuzzy controllers developing, was presented in Ref. [12], etc. Recently, Pei has proved the completeness theorem for L∗ with respect to W8 -semantic [6] by developing a theory of algebraic systems R0 corresponding to L∗ and Wang constructed the formal system KL∗ for the fuzzy predicate calculus corresponding to L∗ and proved the completeness theorem for KL∗ with respect to W8 -interpretation [15], and this laid a Grm foundation for the results given by Wang [10,11,12]. In this paper, we aim to prove it by metamathematical method and by this the completeness for KL∗ with respect to W8 -semantic can also be proved. Essentially, our proof is a type of generalization of Henkin’s technique by which the classical two-valued completeness had been proved and its basic idea is from Section 5.3 of [2]. However, it is more subtle and delicate by complexity of the system L∗ itself. Furthermore, we discussed originally the completeness for the schematic extension Ln∗ of L∗ with respect to Wn -semantic.
2. Preliminaries Now we introduce the fuzzy propositional calculus PC(T0 ). Proposition 2.1. Let the binary operation T0 on the unit interval [0; 1] be de8ned as follows: ∀x; y ∈ [0; 1];
T0 (x; y) =
0
x + y 6 1;
min(x; y) x + y ¿ 1:
Then (i) T0 is a triangular norm [7; 1]. (ii) The residuum → of T0 is the R0 -implicator [10; 1], i.e. for any x; y ∈ [0; 1];
x → y = R0 (x; y) =
1
x 6 y;
max(1 − x; y) otherwise:
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(iii) If a unary operation + is de8ned as follows: ∀x ∈ [0; 1]; + x = x → 0, then + is the standard fuzzy negation, i.e., + x = 1 − x for any x ∈ [0; 1]. (iv) If a binary operation ∨ is de8ned as follows: ∀x; y ∈ [0; 1]; x ∨ y = + ((((x → (x → y)) → x) → x) → + ((x → y) → y)), then ∨ is the standard fuzzy disjunction, i.e. x ∨ y = max(x; y) for any x; y ∈ [0; 1]; (v) ∀x; y ∈ [0; 1]; T0 (x; y) = + (x → + y). The proof is trivial and omitted. These show propitious characteristics of the nilpotent minimum further. Incidentally, we obtain (iv) through a computer running for half an hour. The identity (iv) is looking unfamiliar although it is the simplest in deGning ∨ by → and +. Alternatively, one can use the formula + ((((x → y) → y) → ((y → x) → x)) → + ((x → y) → y)), and this had been mentioned in [4] by rich Klement and Navara. Easy calculation shows that they are equivalent. By this proposition, we can introduce an algebra W8 of type (+; →) on [0; 1], where + is the standard fuzzy negation and → is R0 -implicator. Its subalgebras ([0; 1] ∩ Q; +; →) and ({0; 1=n − 1; : : : ; n − 2=n − 1; 1}; +; →) are denoted by W and Wn , respectively, where n is an integer and n¿4. The algebras W8 ; W and Wn are called the revised Kleene Systems [11]. Denition 2.2. The language of the propositional calculus PC(T0 ) given by T0 has the set S of propositional variables p1 ; p2 ; : : :, and connectives +; →. The set F(S) of well-formed formulas (w:f:’s for short) in PC(T0 ) is deGned inductively as follows: (i) pn is a w:f: for each n¿1, (ii) If ’ and are w:f:’s, then +’; ’ → are w:f:’s; (iii) F(S) is generated only by (i) and (ii). In PC(T0 ), the associated truth-value functions of + and → are the standard fuzzy negation and the R0 -implicator, respectively. Using basic logical connectives + and →, we can deGne additional logical connectives as follows: (i) p ∨ q for + ((((p → (p → q)) → p) → p) → + ((p → q) → q)), (ii) p ∧ q for + (+ p ∨ + q), (iii) p ↔ q for (p → q) ∧ (q → p). Then each of these has associated truth-value function as follows: (i) x ∨ y = max(x; y), (ii) x ∧ y = min(x; y), max{1 − y; x} if x¡y; if x = y; (iii) x ↔ y = 1 max{1 − x; y} otherwise: In addition to these, 18 stands for p → p and 08 stands for + (p → p). Denition 2.3. A W8 -evaluation in PC(T0 ) is a mapping e : F(S) → W8 such that e(+ ’) = + e(’) and e(’ → ) = e(’) → e( ) for any ’; ∈ F(S). The set of all W8 -evaluations is denoted by #(W8 ) and called W8 -semantic of PC(T0 ).
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Denition 2.4. (i) Let ’ be a w:f: ’ is a W8 -tautology of PC(T0 ) (denoted by |= ’) if e(’) = 1 for each e ∈ #(W8 ); (ii) let T ⊆ F(S) and e ∈ #(W8 ); e is a W8 -model of T if e( ) = 1 for any ∈ T ; (iii) let T ⊆ F(S) and ’ a w:f: if e(’) = 1 for each W8 -model e of T , then we say T yields semantically ’ and write T |= ’. Denition 2.5. The formal deductive system L∗ of PC(T0 ) is deGned as follows [10]: (i) The set of w:f: F(S) is given in DeGnition 2.2. (ii) Let p; q; r be distinct propositional variables. The following are axiom schemes of L∗ : (A1 ) p → (q → p), (A2 ) (+ p → + q) → (q → p), (A3 ) (p → (q → r)) → (q → (p → r)), (A4 ) (p → q) → ((r → p) → (r → q)), (A5 ) p → + + p, (A6 ) p → p ∨ q, (A7 ) p ∨ q → q ∨ p, (A8 ) (p → r) ∧ (q → r) → (p ∨ q → r), (A9 ) (p ∧ q → r) → (p → r) ∨ (q → r), (A10 ) (p → q) ∨ ((p → q) → + p ∨ q), (A11 ) p → p ∧ p, (A12 ) (p → q) → (p ∧ r → q ∧ r), (iii) The rule of inference in L∗ is modus ponens (MP for short): from ’ and ’ →
infer
.
Denition 2.6. (i) A theory is a set of w:f:’s, called special axioms of the theory. A proof in a theory T is a sequence ’1 ; : : : ; ’n of formulas such that ’i either is an axiom of L∗ or is a special axiom of T or follows from some preceding ’j ; ’k by MP. (ii) A formula ’ is provable in T (notation: T ’) if it is the last member of a proof in T . (iii) For any w:f:’s ’ and , if T ’ → and T → ’, we say ’ is T -provably equivalent to and write ’ ≈T . Especially, if T = ∅ and T ’, we say ’ is a theorem of L∗ and write ’; if T = ∅ and ’ ≈T , we say ’ is provably equivalent to and write ’ ≈ . Proposition 2.7. following are theorems or rules ofL∗ [10]: (D0 ) (D1 ) (D2 ) (D3 ) (D4 ) (D5 ) (D6 ) (D7 )
’ → ’, + + ’ → ’, {’ → ; → &} ’ → &, (’ → ) → (+ → + ’), { → ’; & → '} (’ → &) → ( → '), & ∧ ’ → ’ ∧ &, {’ → ; ’ → &} ’ → ∧ &, {’; } ’ ∧ ,
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(D8 ) (D9 ) (D10 ) (D11 ) (D12 ) (D13 ) (D14 )
{’ → ; & → (} ’ ∨ & → ∨ (, ’ ∧ → ’; ’ ∧ → , (’ → ) ∨ ( → ’), + ’ ∨ → (’ → ), ’ ∨ ’ ≈ ’, (’ → ) ≈ (+ → + ’), (’ → ( → &)) ≈ ( → (’ → &)).
Proof. (D0 ): (1◦ ) ’ → ((’ → + ’) → ’) by (A1 ), (2◦ ) (’ → ((’ → + + ’) → ’)) → ((’ → + + ’) → (’ → ’)) (3◦ ) (’ → + + ’) → (’ → ’) by (1◦ ); (2◦ ) and (MP), (4◦ ) ’ → + + ’ by (A5 ), (5◦ ) ’ → ’ by (3◦ ); (4◦ ) and (MP).
by (A3 ),
(D1 ): (1◦ ) + ’ → + + + ’ by (A5 ), (2◦ ) (+ ’ → + + + ’) → (+ + ’ → ’) by (A2 ), (3◦ ) + + ’ → ’ by (1◦ ); (2◦ ) and (MP). (D2 ): (1◦ ) ’ → Assumption, (2◦ ) → & Assumption, (3◦ ) ( → &) → ((’ → ) → (’ → &)) by (A4 ), (4◦ ) (’ → ) → (’ → &) by (2◦ ); (3◦ ) and (MP), (5◦ ) ’ → & by (1◦ ); (4◦ ) and (MP). (D3 ): (1◦ ) (’ → ) → ((+ + ’ → ’) → (+ + ’ → )) by (A4 ), (2◦ ) ((’ → ) → ((+ + ’ → ’) → (+ + ’ → ))) → ((+ + ’ → ’) → ((’ → ) → (+ + ’ → ))) by (A3 ), (3◦ ) (+ + ’ → ’) → ((’ → ) → (+ + ’ → )) by (1◦ ); (2◦ ) and (MP), (4◦ ) (’ → ) → (+ + ’ → ) by (D1 ); (3◦ ) and (MP), (5◦ ) → + + by (A5 ), ◦ (6 ) ( → + + ) → ((+ + ’ → ) → (+ + ’ → + + )) by (A4 ), (7◦ ) (+ + ’ → ) → (+ + ’ → + + ) by (5◦ ); (6◦ ) and (MP), (8◦ ) (’ → ) → (+ + ’ → + + ) by (4◦ ); (7◦ ) and (D2 ), (9◦ ) (+ + ’ → + + ) → (+ → + ’) by (A2 ), (10◦ ) (’ → ) → (+ → + ’) by (8◦ ); (9◦ ) and (D2 ). (D4 ): (1◦ ) → ’ Assumption, (2◦ ) & → ' Assumption,
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(3◦ ) (4◦ ) (5◦ ) (6◦ ) (7◦ ) (8◦ )
(& → ') → ((’ → &) → (’ → ')) by (A4 ), (’ → &) → (’ → ') by (2◦ ); (3◦ ) and (MP), ((’ → &) → (’ → ')) → (’ → ((’ → &) → ')) by (A3 ), ’ → ((’ → &) → ') by (4◦ ); (5◦ ) and (MP), → ((’ → &) → ') by (1◦ ); (6◦ ) and (D2 ), (’ → &) → ( → ') by (7◦ ); (A3 ) and (MP).
(D5 ): (1◦ ) + ’ ∨ + & → + & ∨ + ’ by (A7 ), (2◦ ) (+ ’ ∨ + & → + & ∨ + ’) → (+ (+ & ∨ + ’) → + (+ ’ ∨ + &)) (3◦ ) + (+ & ∨ + ’) → + (+ ’ ∨ + &) by (1◦ ); (2◦ ) and (MP), (4◦ ) & ∧ ’ → ’ ∧ & by (3◦ ) and (Def : 2:2). (D6 ): (1◦ ) ’ → & Assumption, (2◦ ) ’ → Assumption, (3◦ ) (’ → &) → (’ ∧ ’ → & ∧ ’) by (A12 ), (4◦ ) ’ ∧ ’ → & ∧ ’ by (1◦ ); (3◦ ) and (MP), (5◦ ) (’ → ) → (’ ∧ & → ∧ &) by (A12 ), (6◦ ) ’ ∧ & → ∧ & by (2◦ ); (5◦ ) and (MP), (7◦ ) ’ → ’ ∧ ’ by (A11 ), (8◦ ) ’ → & ∧ ’ by (4◦ ); (7◦ ) and (D2 ), (9◦ ) & ∧ ’ → ’ ∧ & by (D5 ), (10◦ ) ’ → ’ ∧ & by (8◦ ); (9◦ ) and (D2 ), (11◦ ) ’ → ∧ & by (6◦ ); (10◦ ) and (D2 ). (D7 ):
(1◦ )
(2◦ ) (3◦ ) (4◦ ) (5◦ ) (6◦ ) (7◦ )
Assumption, Assumption, → (’ → ) by (A1 ), ’→ by (1◦ ); (3◦ ) and (MP), ’ → ’ by (D0 ), ’→’∧ by (4◦ ); (5◦ ) and (D6 ), ’∧ by (2◦ ); (6◦ ) and (MP). ’
(D8 ): (1◦ ) ’ → Assumption, ◦ (2 ) & → ' Assumption, (3◦ ) → ∨ ' by (A6 ), (4◦ ) ' → ' ∨ by (A6 ), (5◦ ) ’ → ∨ ' by (1◦ ); (3◦ ) and (D2 ), (6◦ ) & → ' ∨ by (2◦ ); (4◦ ) and (D2 ), ◦ (7 ) ' ∨ → ∨ ' by (A7 ), (8◦ ) & → ∨ ' by (6◦ ); (7◦ ) and (D2 ),
by (D3 ),
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(9◦ ) (’ → ∨ ') ∧ (& → ∨ () by (5◦ ); (8◦ ) and (D7 ), (10◦ ) (’ → ∨ ') ∧ (& → ∨ () → (’ ∨ & → ∨ ') by (A8 ), (11◦ ) ’ ∨ & → ∨ ' by (9◦ ); (10◦ ) and (MP). (D9 ): +’→+’∨+ by (A6 ), (2◦ ) + → + ’ ∨ + by (A6 ); (A7 ) and (D2 ), (3◦ ) (+ ’ → + ’ ∨ + ) → (+ (+ ’ ∨ + ) → + + ’) by (D3 ), (4◦ ) (+ → + ’ ∨ + ) → (+ (+ ’ ∨ + ) → + + ) by (D3 ), (5◦ ) + (+ ’ ∨ + ) → + + ’ by (1◦ ); (3◦ ) and (MP), (6◦ ) + (+ ’ ∨ + ) → + + by (2◦ ); (4◦ ) and (MP), ◦ ◦ (7 ) ’ ∧ → ’ by (5 ); (D1 ) and (D2 ), (8◦ ) ’ ∧ → by (6◦ ); (D1 ) and (D2 ).
(1◦ )
(D10 ): (1◦ ) ’ ∧ → ’ ∧ by (D0 ), (2◦ ) (’ ∧ → ’ ∧ ) → (’ → ’ ∧ ) ∨ ( → ’ ∧ ) by (A9 ), (3◦ ) (’ → ’ ∧ ) ∨ ( → ’ ∧ ) by (1◦ ); (2◦ ) and (MP), (4◦ ) ’ ∧ → ; ’ ∧ → ’ by (D9 ), (5◦ ) (’ ∧ → ) → ((’ → ’ ∧ ) → (’ → )) by (A4 ), (6◦ ) (’ ∧ → ’) → (( → ’ ∧ ) → ( → ’)) by (A4 ), (7◦ ) (’ → ’ ∧ ) → (’ → ) by (4◦ ); (5◦ ) and (MP), (8◦ ) ( → ’ ∧ ) → ( → ’) by (4◦ ); (6◦ ) and (MP), (9◦ ) (’ → ’ ∧ ) ∨ ( → ’ ∧ ) → (’ → ) ∨ ( → ’) by (7◦ ); (8◦ ) and (D8 ), (10◦ ) (’ → ) ∨ ( → ’) by (3◦ ); (9◦ ) and (MP). (D12 ): (1◦ ) + ’ → + ’; + ’ → + ’ by (D0 ), (2◦ ) + ’ → + ’ ∧ + ’ by (D6 ), (3◦ ) ’ ∨ ’ → ’ by (2◦ ); (D3 ); (D1 ) and (D2 ). (D11 ): (1◦ ) + ’ → (+ → + ’) by (A1 ), (2◦ ) + ’ → (’ → ) by (1◦ ); (A2 ) and (D2 ), (3◦ ) → (’ → ) by (A1 ), (4◦ ) + ’ ∨ → (’ → ) by (2◦ ); (3◦ ); (D8 ); (D12 ) and (D2 ). Easily,we can get (D13 ) by (A2 ) and (D3 ), and (D14 ) by (A3 ). Theorem 2.8 (the substitution theorem). Let w:f: ’ be built up from w:f:’s ’1 ; : : : ; ’t by means of + and → denoted by ’(’1 ; : : : ; ’t ). If ’i ≈T i for integer i from 1 to t, then ’ ≈T ’ ( 1 ; : : : ; t ), where ’( 1 ; : : : ; t ) is given by replacing each occurrence of ’i by i .
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Proof. The proof is by induction on the number n of connectives in building up ’ from ’1 ; ’2 ; : : : ; ’t . For the base step suppose that n = 0; then t =1; ’ = ’(’1 ) = ’1 and ’( 1 ) = 1 . Thus ’ ≈ ’( 1 ) from ’1 ≈T 1 . Now suppose that the theorem holds for any n6k. For n = k + 1, there are two cases to be considered: Case 1: ’ is + ’ (’1 ; : : : ; ’t ). Applying the hypothesis of induction we have ’ (’1 ; : : : ; ’t ) ≈T ’ ( 1 ; : : : ; t ). By D3 and A2 , we can get + ’ (’1 ; : : : ; ’t ) ≈T + ’ ( 1 ; : : : ; t ) i.e., ’ ≈T ’( 1 ; : : : ; t ). Case 2: ’ is ’ (’1 ; ’2 ; : : : ; ’t ) → ’ (’1 ; ’2 ; : : : ; ’t ). Applying the hypothesis of induction we have ’ (’1 ; : : : ; ’t ) ≈T ’ ( 1 ; : : : ; t ) and ’ (’1 ; : : : ; ’t ) ≈T ’ ( 1 ; : : : ; t ). By D4 we can get ’ (’1 ; : : : ; ’t ) → ’ (’1 ; : : : ; ’t ) ≈T ’ ( 1 ; : : : ; t ) → ’ ( 1 ; : : : ; t ), i.e., ’ ≈T ’( 1 ; : : : ; t ). Hence by the principle of mathematical induction the theorem holds whatever the number of connectives in building up ’ from ’1 ; ’2 ; : : : ; ’t . The above proof is analogous to that of Theorem 3.2.23 on p. 79 of [10]. Proposition 2.9. The following is a theorem of L∗ : (D15 ) (’ → (’ → (’ → ))) ≈ (’ → (’ → )) Proof. (1◦ ) (’ → + ’) → ((’ → (’ → + ’)) → (’ → + ’)) by (A1 ), (2◦ ) ((’ → + ’) → + ’) → ((’ → (’ → + ’)) → (’ → + ’)) by (A4 ), (3◦ ) (’ → + ’) ∨ ((’ → + ’) → + ’) by (A10 ); (D12 ) and (Th: 2:8), (4◦ ) (’ → + ’) ∨ ((’ → + ’) → + ’) → ((’ → (’ → + ’)) → (’ → + ’)) by (1◦ ); (2◦ ); (D8 ); (D12 ) and (Th: 2:8), (5◦ ) (’ → (’ → + ’)) → (’ → + ’) by (3◦ ); (4◦ ) and (MP), (6◦ ) (’ → (’ → + ’)) ≈ (’ → + ’) by (5◦ ) and (A1 ), (7◦ ) (’ → (’ → (’ → ))) ≈ (’ → (’ → (+ → + ’))) by (D0 ); (D13 ) and (Th: 2:8), (8◦ ) (’ → (’ → (’ → ))) ≈ (+ → (’ → (’ → + ’))) by (7◦ ); (D14 ) and (Th: 2:8), (9◦ ) (’ → (’ → (’ → ))) ≈ (+ → (’ → + ’)) by (8◦ ); (6◦ ) and (Th: 2:8), (10◦ ) (’ → (’ → (’ → ))) ≈ (’ → (’ → )) by (9◦ ); (D14 ); (D13 ) and (Th: 2:8). Theorem 2.10 (the generalized deduction theorem). Let T be a theory of L∗ and ’ and and T ∪ {’} . Then T ’ → (’ → ).
w:f:’s
Proof. The proof is by induction on the number n of w:f:’s in the sequence forming the deduction of from T ∪ {’}. For the base step suppose that this sequence has one member. That member must be itself, and so either is an axiom of L∗ or is a member of T ∪ {’}. Case 1: is an axiom of L∗ or ∈ T . The following is then a deduction of ’ → (’ → ) from T : (1◦ ) axiom of L∗ or a member of T , (2◦ ) → (’ → ); (’ → ) → (’ → (’ → )) by (A1 ), (3◦ ) ’ → (’ → ) by (1◦ ); (2◦ ) and (MP).
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Case 2: is ’. By D0 we can easily get its proof. This ends the base step. Now suppose that the theorem holds if n¡k, where k is an integer and k¿1. Then, similarly, we can get the theorem hold if is an axiom of L∗ , or ∈ T , or = ’. Now suppose that is obtained from two previous w:f:’s in the deduction by an application of MP. These two w:f:’s must have the forms & and & → . By the hypothesis of induction, we get T ’ → (’ → &) and T ’ → (’ → (& → )). The required deduction of ’ → (’ → ) from T may now be given as follows: (1◦ ) (2◦ ) (3◦ ) (4◦ ) (5◦ ) (6◦ ) (7◦ )
’ → (’ → &) Assumption, ’ → (’ → (& → )) Assumption, & → (’ → (’ → )) by (2◦ ) and (D14 ), (& → (’ → (’ → ))) → ((’ → &) → (’ → (’ → (’ → )))) by (A4 ), (’ → &) → (’ → (’ → )) by (3◦ ); (4◦ ); (MP); (D15 ) and (Th: 2:8), ’ → (’ → (’ → )) by (1◦ ); (5◦ ) and (D2 ), ’ → (’ → ) by (6◦ ); (D15 ) and (Th: 2:8).
Thus, the theorem holds if n = k. Hence by the principle of mathematical induction the theorem holds whatever the number of w:f:’s in the deduction of from T ∪ {’}. Proposition 2.11 (see Theorem 2.3 [4]). Let {’ ∨ } &.
T1 ∪ {’} &
and
T2 ∪ { } &,
then
T1 ∪ T2 ∪
Proof. By Theorem 2.10 and T1 ∪ {’} & we can get T1 ’ → (’ → &) then T1 ∪ T2 ∪ {’ ∨ } ’ → (’ → &). Similarly, T1 ∪ T2 ∪ {’ ∨ } → ( → &). Now we prove {’ → (’ → &); → ( → &); ’ ∨ } &. (1◦ ) (2◦ ) (3◦ ) (4◦ ) (5◦ ) (6◦ ) (7◦ ) (8◦ ) (9◦ ) (10◦ ) (11◦ ) (12◦ ) (13◦ ) (14◦ ) (15◦ ) (16◦ ) (17◦ ) (18◦ ) (19◦ )
’ → (’ → &) Assumption, → ( → &) Assumption, ’∨ Assumption, → (’ → (’ → &)) by (1◦ ); (A1 ) and (MP), ’ → (’ → ( → &)) by (4◦ ); (D14 ) and (Th: 2:8), ’ → ( → ( → &)) by (2◦ ); (A1 ) and (MP), → (’ → ( → &)) by (6◦ ); (D14 ) and (Th: 2:8), ’ ∨ → (’ → ( → &)) by (5◦ ); (7◦ ); (D8 ); (D12 ) and (Th: 2:8), ’ → ( → &) by (3◦ ); (8◦ ) and (MP), ’ ∨ → ( → &) by (2◦ ); (9◦ ); (D8 ); (D12 ) and (Th: 2:8), → & by (3◦ ); (10◦ ) and (MP), → ( → (’ → &)) by (7◦ ); (D14 ) and (Th: 2:8), ’ → ( → (’ → &)) by (4◦ ); (D14 ) and (Th: 2:8), ∨ ’ → ( → (’ → &)) by (12◦ ); (13◦ ); (D8 ); (D12 ) and (Th: 2:8), → (’ → &) (3◦ ); (14◦ ) and (MP), ’ ∨ → (’ → &) (1◦ ); (15◦ ); (D8 ); (D12 ) and (Th: 2:8), ’ → & by (3◦ ); (16◦ ) and (MP), ’ ∨ → & by (11◦ ); (17◦ ); (D8 ); (D12 ) and (Th: 2:8), & by (3◦ ); (18◦ ) and (MP).
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Theorem 2.12 (the soundness theorem of L∗ ). Let T be a theory and ’ be a w.f. If T ’, then T |= ’. Proof (Wang [10]): The proof is by induction on the number of w:f:’s in a sequence of w:f:’s which constitutes a proof of ’ in T . We can easily get that all axioms are tautologies and MP keeps tautologies, i.e., if ’ and ’ → are tautologies then is a tautology. By these we can get the proof of this theorem. 3. The completeness theorem for L∗ 8
8
Now we introduce the truth constant 12 into the language of L∗ such that e( 12 ) = 12 for any W8 -evaluation e of L+∗ and denote the extended set of w:f:’s by F(S + ). Accordingly, we introduce 8 8 8 8 the axiom ( 12 → + 21 ) ∧ (+ 21 → 12 ) into the set of axioms of L∗ and denote the new formal system by L+∗ . And if ’ is provable in T over L+∗ we denote it by T + ’. Denition 3.1 (Menu and Pavelka [5], Wang [10]). Let M be an algebra of type (+ →): M is said to be a linearly ordered R0 -algebra if there is a linearly ordering relation 6 on M such that (i) M is bounded with respect to 6 and 0M and 1M stand for the least element in M and the greatest one, respectively; (ii) + is an order-reversing involution on M with respect to 6 and there is an element denoted by 1=2M ; s:t: + 1=2M = 1=2M ; (iii) ∀x; y ∈ M; x → y = 1 if x6y; x → y = max(+ x; y) otherwise. Let M be a linearly ordered R0 -algebra. We can deGne the following operations on M: ∀x; y ∈ M; x ⊗ y = +(x → + y); x ∨ y = max(x; y); x ∧ y = min(x; y). Easily, we can get a linearly ordered R0 -algebra is a residuated lattice in which the adjoint couple is ⊗ and →. But it is not a BL-algebra since the following does not hold: x ∧ y = x ⊗ (x → y) which was given in [9]. Denition 3.2. Let T be a theory over L∗ . (i) T is consistent if there is no a w:f: ’ such that T ’ and T + ’, (ii) T is complete if for any ’; ∈ F(S); T ’ → or T → ’. Proposition 3.3. Let T be a theory. Then T is consistent if and only if there is a w.f. ’ unprovable in T. Proof. We only prove the suOciency. Assume T is inconsistent. Then there is a w:f: such that T and + . By (A1 ), we get → (+ ’ → ), then T + ’ → , then + → ’ by (D3 ); (D1 ) and (D2 ), then ’. A contradiction to ’ being unprovable in T ! Hence T is consistent. Proposition 3.4. Let T be a theory over L∗ and ’ unprovable in T. Then there is a complete consistent theory T8 over L+∗ such that T ⊆ T8 and ’ is unprovable in T8 .
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Proof. Let {((n ; n )} be the sequence of all w:f: pairs in F(S + ). We construct inductively the sequence {Tn } of theories as follows: Let T0 = T . Then ’ is unprovable in T0 over L+∗ . Assume Tn−1 has been constructed such that T0 ⊆ Tn−1 and ’ is unprovable in Tn−1 over L+∗ , then Tn is decided by Tn−1 and ((n ; n ). If ’ is unprovable in Tn−1 ∪ {(n → n } over L+∗ , put Tn = Tn−1 ∪ {(n → n }; otherwise Tn = Tn−1 ∪ { n → (n }, in which case, assume Tn + ’. Then Tn−1 + ’ by (D10 ) and Theorem 2.11. A contradiction! Thus, ’ is unprovable in Tn over L+∗ . Now let T8 be the union of all Tn . Then clearly T8 is complete and Tn ⊆ Tm if 06n6m for any integers n and m. Assume T8 + ’, then there is an integer n such that Tn + ’. A contradiction! Thus, ’ is unprovable in T8 over L+∗ : T8 is also consistent by Proposition 3.3. Proposition 3.5 (Wang [10]). Let T8 be a complete consistent theory over L+∗ and ≈+ be T8 + + + + + 8 T -provably equivalent relation on F(S ). Let [’]T8 = { ∈ F(S ) | ≈T8 ’} for any ’ ∈ F(S ) and [F]+ = {[’]+ | ’ ∈ F(S + )}. Then: T8 T8 + (i) ≈T8 is a congruent relation on F(S + ). (ii) The quotient algebra [F]+ given by the congruence ≈+ on F(S + ) is a linearly ordered set in T8 T8 which the order relation 6 is de8ned as follows: ∀’; ∈ F(S + ); [’]+ 6[ ]+ if and only if T8 T8 8 T + ’ → . 8 +8 ; 1[F]+ = [1] 8 +8 ; (iii) + is an order-reversing involution on [F]+ with respect to 6 and 0[F]+8 = [0] T8 T T T T8 + + + [’]+ ∨ [ ] = max([’] ; [ ] ). T8 T8 T8 T8 + + + (iv) [’]+ → [ ] = 1 if [’] 6[ ] ; [’]+ → [ ]+ = max(+ [’]+ ; [ ]+ ) otherwise. 8 8 8 T T T T8 T8 T8 T8 T8 + + 8 8 if and only if T8 + ’. (v) [’]T8 = [1] T 8 with respect to 6. (vi) 12 [F]+ = [ 12 ]+ T8 T8
is an equivalent relation on F(S + ). By (D4 ) and (D13 ) Proof. (i) By (D0 ) and (D2 ) we can get ≈+ T8 we can get it is congruent. (ii) We deGne 6 on [F]+ as follows: ∀’; ∈ F(S + ); [’]+ 6[ ]+ if and only if T8 + ’ → . T8 T8 T8 Easily, we can get 6 is partially ordered by (D0 ) and (D2 ). And it is linearly ordered since T8 is complete. 8 +8 , i.e. 1[F]+ = [1] 8 +8 . Similarly, we (iii) ∀’ ∈ F(S + ); T8 + ’ → 18 by (D0 ) and (A1 ). Then [’]+ 6[1] T8 T T T8 8 +8 by (D0 ); (D13 ) and (A1 ); + is an involution on [F]+8 by (D1 ) and (A5 ) and can get 08 [F]+ = [0] T8
T
T
it is order-reversing with respect to 6 by (D3 ); [’]+ ∨ [ ]+ = max([’]+ ; [ ]+ ) holds by (A6 ) and T8 T8 T8 T8 (D8 ). (iv) ∀’; ∈ F(S + ); [’]+ 18 if and 6[ ]+ if and only if T8 + ’ → if and only if ’ → ≈+ T8 T8 T8 + + + + + only if [’ → ]T8 = 1[F]+8 if and only if [’]T8 → [ ]T8 = 1[F]+8 . If [’]T8 [ ]T8 , then [’ → ]+ ¡1[F]+8 T8 T T T + + + and [’ → ]T8 ∨ [(’ → ) → + ’ ∨ ]T8 = 1[F]+8 by (A10 ). Hence [(’ → ) → + ’ ∨ ]T8 = 1[F]+8 . Then T T [’ → ]+ 6[+ ’ ∨ ]+ and [+ ’ ∨ ]+ 6[’ → ]+ by (D11 ). Thus [’]+ → [ ]+ = [’ → ]+ = [+ ’ T8 T8 T8 T8 T8 T8 T8 ∨ ]+ = max(+ [’]+ ; [ ]+ ). T8 T8 T8 (v) and (vi) are trivial and their proof are omitted. These show [F]+ is a linearly ordered R0 T8 algebra.
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Proposition 3.6. Let M be a countable linearly ordered R0 -algebra. Then there is a countable densely linearly ordered R0 -algebra M such that there is an isomorphic embedding i : M → M of type (+; →). Proof. (i) ∀u ∈ M , put Cu = {(u; 0)} if u has no successor in M; Cu = {(u; r) | 06r¡1; r ∈ Q} otherwise. Let M = u∈M Cu and for (u; r); (v; s) ∈ M put (u; r)6(v; s) if and only if u¡M v or (u = v and r6s) (lexicographic product). Clearly (M ; 6) is a bounded and densely ordered chain in which the least and the greatest elements are (0; 0) and (1; 0), respectively. (ii) ∀(u; r) ∈ M , put + (u; 0) = (+ u; 0) if r = 0; + (u; r) = (+ u ; 1 − r) otherwise, where u is the successor of u in M . Trivially, we get + is an order-reversing involution on M and +(1=2M ; 0) = (+ 1=2M ; 0) = (1=2M ; 0). (iii) ∀(u; r); (v; s) ∈ M , put (u; r) → (v; s) = (1; 0) if (u; r)6(v; s); (u; r) → (v; s) = max(+ (u; r); (v; s)) otherwise. Then M is a densely linearly ordered R0 -algebra by (i)–(iii). (vi) Trivially, we can get the mapping f(u) = (u; 0) embeds M isomorphically to M . Proposition 3.7. Let M be a countable densely linearly ordered R0 -algebra. Then there is an isomorphic embedding f of M into W8 . Proof. Let M1 = {x ∈ M | x¡1=2M }; M2 = {x ∈ M | x¿1=2M }; W1 = {x ∈ W | x¡ 12 } and W2 = {x ∈ W | x¿ 12 }. Then M1 ; M2 ; W1 and W2 are countable densely linearly ordered sets. It is well known that any two countable densely linearly ordered sets with the greatest and least elements are orderisomorphic. Let f be an order isomorphism of M1 onto W1 . Now we construct f8 : M → W , put 8 = f(x) if x ∈ M1 ; f(x) 8 = + f(+ x) if x ∈ M2 and f( 1 ) = 1 . Trivially, we get f8 is an algebraic f(x) 2M 2 isomorphism of M onto W and, the identical mapping i : W → W8 is an isomorphic embedding. Thus i ◦ f8 is an isomorphic embedding of M into W8 . Theorem 3.8 (the strong completeness theorem for L∗ ). Let T be a consistent theory over L∗ and ’ ∈ F(S) and T |= ’, then T ’. Proof. Assume ’ be unprovable in T over L∗ . By Proposition 3.4, there is a complete consistent theory T8 over L+∗ such that T ⊆ T8 and ’ is unprovable in T8 over L+∗ . By Proposition 3.5, [F]+ T8 is a linearly ordered R0 -algebra. Let e be the natural homomorphism from F(S + ) onto [F]+ , i.e. T8 + + 8 8 + if and only if T for any ∈ F(S ). Then e( ) = [ 1] = 1 . Thus e(’)¡1. By e( ) = [ ]+ + [F]T8 T8 T8 + 8 Propositions 3.6 and 3.7, there is an isomorphically embedding f from [F]T8 into W . Thus, f ◦ e is 8 ) = f ◦ e( ) for any ∈ F(S). Trivially, e8 is a W8 -evaluation of L∗ a W8 -evaluation of L+∗ . Put e( and a model of T . And e(’) 8 = f ◦ e(’) = f(e(’))¡1. A contradiction to T |= ’. Thus T ’. In the preceding theorem, put T = ∅, we get Theorem 3.9 (the completeness theorem for L∗ ). Let w:f: ’ be a W8 -tautology of L∗ . Then ’ is a theorem of L∗ , i.e. |= ’ implies ’.
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4. The completeness theorem for Ln∗ In this section, we will consider the axiomatization of the propositional calculi PC(Wn ) given by Wn . Denition 4.1. Let n be an integer and n¿4. The formal system Ln∗ of PC(Wn ) is the schematic extension (see [2, 2.3.20]) of L∗ by adding the following to its axiom schemes: (A13 ) /(p1 ; : : : ; pn ) where w:f: /(p1 ; : : : ; pn )(/ for short) is deGned as follows: (1◦ ) n = 2m. Let /0 = + ((pn → + pn ) → + (+ pn → pn )) → pn ∧ + pn and let /1 = p1 ∨ + p1 and let k be an integer. If 26k6m, then /k = (pk → /k −1 ) ∨ pk . If m + 16k62m − 1, then /k = /k −1 ∧ (pk → pk ). Then /(p1 ; : : : ; pn ) = /2m−1 ∧ /0 ; (2◦ ) n = 2m + 1. Let /2 = p1 ∨ (p1 → + p2 ) ∨ (+ p2 → p2 ) and let k be an integer. If 36k6m, then /k = (pk → /k −1 ) ∨ pk . If m+16k62m+1; /k = /k −1 ∧ (pk → pk ). Then /n = /(p1 ; : : : ; pn ). Let T be a theory and ’ w:f: T ∪ {/} ’ is denoted by T n ’. For any w:f:’s ’ and , if n ’ → short).
and n
→ ’ we say ’ is T -provably equivalent to
Ln∗
and write ’ ≈ T
(’ ≈nT
for
Denition 4.2. (i) A Wn -evaluation of Ln∗ is a mapping e : F(S) → Wn such that e(+ ’) = + e(’) and e(’ → ) = e(’) → e( ) for any ’; ∈ F(S). The set of all Wn -evaluation is denoted by #(Wn ) and called Wn -semantic of Ln∗ ; (ii) a w:f: ’ is a Wn -tautology of Ln∗ (denoted by |=n ’) if e(’) = 1 for each e ∈ #(Wn ), and the set of all Wn -tautologies of Ln∗ is denoted by T (Wn ); (iii) let T be a theory and e ∈ #(Wn ): e is a Wn -model of T if e(’) = 1 for any ’ ∈ T ; (iv) if e(’) = 1 for each Wn -model e of the theory T , then we say T yields semantically ’ and write T |=n ’. Denition 4.3 (Wang [11]). (i) Let 0 ∈ Wn . If e(’)¿0 for each Wn -evaluation e, we say ’ is an 0-tautology of Ln∗ . The set of all 0-tautologies of Ln∗ is denoted by 0 − T (Wn ). (ii) Let ’ ∈ 0 − T (Wn ). If there is a Wn -evaluation e such that e(’) = 0, we say 0 is an exact 0-tautology. The set of all exact 0-tautologies is denoted by [0] − T (Wn ). Proposition 4.4. Let 12 60¡1 and let ’ = ’(p1 ; : : : ; pt ) be a w.f. If ’ ∈ [0]−T (Wn ) and p ∈ {p1 ; : : : ; pt }, then (p → ’) ∨ p ∈ [0 + 1=n − 1] − T (Wn ). We can use the proof given for Theorem 9 of [11]. Proposition 4.5. (i) Let ∅ = M ⊆ Wn . Then M is a subalgebra of Wn if and only if {0; 1} ⊆ M and x ∈ M implies +x ∈ M for any x ∈ Wn . (ii) Let n be an odd integer, n¿3. For any integer k; Wk is isomorphic to a subalgebra of Wn if and only if 26k6n.
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(iii) let n be even integer and n¿2, then for any integer k; Wk is isomorphic to a subalgebra of Wn if and only if 26k6n and k is an even one. (iv) If M is isomorphic to a subalgebra of Wn , then T (Wn ) ⊆ T (M ). Proof. (i) Necessity is proved by 0 = + (x → x) and 1 = x → x and suOciency by x → y ∈ {1; + x; y}. (ii) Necessity is trivial and omitted. SuOciency is proved as follows: Case 1. (k is an even integer). We construct an isomorphically embedding f from Wk to Wn by letting f(i=k − 1) = i=n − 1 if 06i6k=2 − 1 and f(i=k − 1) = 1 − f(1 − i=k − 1) otherwise. Case 2. (k is an odd integer). An isomorphically embedding f from Wk to Wn is constructed by letting f(i=k − 1) = i=n − 1 if 06i¡k − 1=2 and f(1=2) = 1=2 and f(i=k − 1) = 1 − f(1 − i=k − 1) otherwise. (iii) is similar to (ii) and omitted. (iv) Let i be an isomorphic embedding from M to Wn and ’ ∈ T (Wn ): ∀e ∈ #(M ); i ◦ e ∈ #(Wn ). Then i ◦ e(’) = i(e(’)) = 1. Thus e(’) = 1. Then ’ ∈ T (M ). Hence T (Wn ) ⊆ T (M ). Proposition 4.6. (i) If n is an odd integer and k is an integer, then / ∈ T (Wk ) if and only if 26k6n. (ii) If n is an even integer and k is an integer, then / ∈ T (Wk ) if and only if 26k6n and k is an even integer. Proof. (i) Let n = 2m + 1. By trivial calculation, we can get /2 ∈ [m + 2=2m] − T (W2m+1 ) ∩ [m + 2=2m+1]−T (W2m+2 ). By Proposition 4.4, we can get /3 ∈ [m+3=2m]−T (W2m+1 ) ∩ [m+3=2m+1]− T (W2m+2 ); : : :, and /m ∈ [2m=2m] − T (W2m+1 ) ∩ [2m=2m + 1] − T (W2m+2 ). Thus /m ∈ T (W2m+1 ) and /m ∈ T (W2m+2 ). Trivially, we can get / ∈ T (W2m+1 ) and / ∈ T (W2m+2 ). Since W2m+2 is isomorphic to a subalgebra of Wk for k ¿ 2m+2; T (Wk ) ⊆ T (W2m+2 ) by Proposition 4.5. Therefore, / ∈ T (Wk ) for k¿2m + 2. For 26k62m + 1; Wk is isomorphic to a subalgebra of W2m+1 . Then T (W2m+1 ) ⊆ T (Wk ). Therefore, / ∈ T (Wk ) for 26k6n. (ii) Let n = 2m. By trivial calculation, we can get /1 ∈ [m=2m − 1] − T (W2m ) ∩ [m + 1=2m + 1] − T (W2m+2 ); /0 ∈ T (W2k ) and /0 ∈ T (W2k+1 ) for any integer k¿1. By Proposition 4.4, we can get /2 ∈ [m + 1=2m − 1] − T (W2m ) ∩ [m + 2=2m + 1] − T (W2m+2 ); : : :, and /m ∈ [2m − 1=2m − 1] − T (W2m ) ∩ [2m=2m + 1] − T (W2m+2 ). Then /m ∈ T (W2m ) and /m ∈ T (W2m+2 ). Trivially, we can get / ∈ T (Wk ) if and only if k is an even integer and 26k6n. Proposition 4.7. Let k be an integer and k¿2. Then / ∈ T (Wk ) implies there is an isomorphically embedding i : Wk → Wn of type (+; →). We can prove it by Propositions 4.6 and 4.5. Theorem 4.8 (the soundness theorem for Ln∗ ). Let T be a theory and ’ ∈ F(S). Then T n ’ implies T |=n ’. We can use the proof given for Theorem 2.12 and / ∈ T (Wn ).
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Denition 4.9. Let M be an algebra of type (+; →). M is said to be a linearly ordered Rn -algebra if there is a linearly ordering relation 6 on M such that (i) M is bounded with respect to 6 and, 0M and 1M stand for the least element in M and the greatest, respectively; (ii) + is an order-reversing involution on M with respect to 6; (iii) ∀x; y ∈ M; x → y = 1 if x6y; x → y = max(+ x; y) otherwise; 8 1 ; : : : ; xn ) = 1, where 1(x 8 1 ; : : : ; xn ) is the truth function corresponding to the 13th axiom (iv) 1(x scheme /(p1 ; : : : ; pn ) of Ln∗ and x1 ; : : : ; xn are arbitrary elements in M . Proposition 4.10. Let M be a linearly ordered Rn -algebra. Then there is an isomorphic embedding i : M → Wn of type (+; →). Proof. (i) Let |M | = k + 1¡∞ and M = {x0 ; x1 ; : : : ; xk }, where 0 = x0 ¡x1 ¡ · · · ¡xk = 1. We can deGne a mapping f : M → Wk+1 by letting f(xi ) = i=k for any xi ∈ M . Clearly, f is an isomor8 1 =k; : : : ; in =k) = 1(f(x 8 8 phism of type (+; →). Since 1(i i1 ); : : : ; f(xin )) = f(1(xi1 ; : : : ; xin )) = 1 for any n (i1 =k; : : : ; in =k) ∈ Wk+1 , Wk+1 is a linearly ordered Rn -algebra and / ∈ T (Wk+1 ). By Proposition 4.7, there is an isomorphic embedding g : Wk+1 → Wn of type (+; →). Hence i = g ◦ f is an isomorphic embedding from M to Wn of type (+; →). (ii) Assume |M | = ∞. Let {x1 ; : : : ; xn } ⊆ M , where x1 ¡x2 ¡ · · · ¡xn ¡1. Trivially, we can get {x1 ; : : : ; xn } ∪ {+ x1 ; : : : ; + xn } ∪ {0; 1} is a linearly ordered Rn -subalgebra of M and its cardinal number is ¿n, a contradiction to (i). Hence |M |¡∞. Proposition 4.11. (i) Let T be a theory and ’ unprovable in T. Then there is a complete consistent theory T8 such that T ⊆ T8 and ’ is unprovable in T8 . (ii) Let T8 be a complete consistent theory over Ln∗ and ≈nT8 be T8 -provably equivalent relation on F(S). Let [’]nT8 = { ∈ F(S) | ≈nT8 ’} for any ’ ∈ F(S) and [F]nT8 = {[’]nT8 | ’ ∈ F(S)}. Then (1◦ ) ≈nT8 is a congruent relation on F(S). (2◦ ) The quotient algebra [F]nT8 given by ≈nT8 is a linearly ordered Rn -algebra in which the order relation 6 is de8ned as follows: ∀’; ∈ F(S); [’]nT8 6[ ]nT8 if and only if T8 n ’ → . 8 n8 if and only if T8 n ’. (3◦ ) [’]nT8 = [1] T The proofs of (i) and (ii) are similar to that of Propositions 3.4 and 3.5, respectively. Theorem 4.12 (the strong completeness theorem for Ln∗ ). Let T be a consistent theory over Ln∗ and ’ ∈ F(S). If T |=n ’, then T n ’. The proof is similar to that of Theorem 3.8. Theorem 4.13 (the completeness theorem for Ln∗ ). Let w:f: ’ be a Wn -tautology of Ln∗ . Then ’ is a theorem of Ln∗ , i.e. if |=n ’. Then n ’.
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5. Concluding remarks Since Wang introduced the formal deductive system L∗ for the fuzzy propositional calculus in 1997, more and more studies show it to have many good properties and important applications. Firstly, since the nilpotent minimum and the corresponding implication, as pointed out in Ref. [1], combine advantageous properties of Lukasiewicz-like t-norms (e.g. the law of contradiction holds, the corresponding R- and S-implication coincide) and those of minimum itself (e.g. easy usage of 0-cuts in practice), L∗ based on them combines advantageous properties of the G6odel R-fuzzy logic RG (e.g. the deduction theorem is analogous to the classical one) and the Lukasiewicz R-fuzzy logic RL (e.g. +; ∨, and ∧ are the standard fuzzy negation, disjunction and conjunction). Secondly, the important issues of compactness [13], deduction, soundness and completeness for L∗ have been studied and very nice and deep logical results have been proved, which lay a Grm foundation for the investigations of practical problems. Thirdly, the formal system KL∗ of the fuzzy predicate calculus corresponding to L∗ has been constructed and the completeness for KL∗ with respect to W8 -interpretation has been proved. It should be notable that Ref. [12] shows L∗ as well as KL∗ to be powerful tools in analyzing topics of Zadeh’s agenda, such as fuzzy modus ponens, composition al rule of inference, fuzzy functions and fuzzy control, etc. Finally, the completeness theorem for Ln∗ with respect to Wn -semantic has practical signiGcance since no computer could deal with inGnite data. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
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