FFuzzy Logic in Narrow Sense withHedges
Van-Hung LeFaculty of Information TechnologyHanoi University of Mining and Geology, Vietnam [email protected] A BSTRACT
Classical logic has a serious limitation in that it cannot cope with the issues of vagueness anduncertainty into which fall most modes of human reasoning. In order to provide a foundationfor human knowledge representation and reasoning in the presence of vagueness, imprecision, anduncertainty, fuzzy logic should have the ability to deal with linguistic hedges, which play a veryimportant role in the modification of fuzzy predicates. In this paper, we extend fuzzy logic innarrow sense with graded syntax, introduced by Nov´ak et al., with many hedge connectives. In onecase, each hedge does not have any dual one. In the other case, each hedge can have its own dualone. The resulting logics are shown to also have the Pavelka-style completeness. K EYWORDS
Fuzzy Logic in Narrow Sense, Hedge Connective, First-Order Logic, Pavelka-Style Completeness
1. Introduction
Extending logical systems of mathematical fuzzy logic (MFL) with hedges is axiomatizedby H´ajek [1], Vychodil [2], Esteva et al. [3], among others. Hedges are called truth-stressing or truth-depressing if they, respectively, strengthen or weaken the meaning ofthe applied proposition. Intuitively, on a chain of truth values, the truth function ofa truth-depressing (resp., truth-stressing) hedge (connective) is a superdiagonal (resp.,subdiagonal) non-decreasing function preserving 0 and 1. In [1, 2, 3], logical systems ofMFL are extended by a truth-stressing hedge and/or a truth-depressing one.Nevertheless, in the real world, humans often use many hedges, e.g., very , highly , rather ,and slightly , simultaneously to express different levels of emphasis. Furthermore, a hedgemay or may not have a dual one, e.g., slightly (resp., rather ) can be seen as a dual hedgeof very (resp., highly ). Therefore, in [4, 5], Le et al. propose two axiomatizations forpropositional logical systems of MFL with many hedges. In the axiomatization in [5],each hedge does not have any dual one whereas in the axiomatization in [4], each hedgecan have its own dual one. In [5, 6], logical systems with many hedges for representingand reasoning with linguistically-expressed human knowledge are also proposed.Fuzzy logic in narrow sense with graded syntax (FLn) is introduced by Nov´ak et al. in[7]. In FLn, both syntax and semantics are evaluated by degrees. The graded approach tosyntax can be seen as an elegant and natural generalization of classical logic for inferenceunder vagueness since it allows one to explicitly represent and reason with partial truth,i.e., proving partially true conclusions from partially true premises, and it enjoys thePavelka-style completeness. In this paper, we extend FLn with many hedges in order toprovide a foundation for human knowledge representation and reasoning in the presence ofvagueness since linguistic hedges are very often used by humans and play a very important a r X i v : . [ c s . A I] A ug ole in the modification of fuzzy predicates. FLn is extended in two cases: (i) each hedgedoes not have a dual one, and (ii) each hedge can have its own dual one. We show thatthe resulting logics also have the Pavelka-style completeness.The remainder of the paper is organized as follows. Section 2 gives an overview of notionsand results of FLn. Section 3 presents two extensions of FLn with many hedges. In onecase, each hedge does not have any dual one, and in the other, each hedge can have itsown dual one. The resulting logics are also shown to have the Pavelka-style completeness.Section 4 concludes the paper.
2. Fuzzy Logic in Narrow Sense
FLn [7] is truth functional . A compound formula is built from its constituents using alogical connective. The truth value of a compound formula is a function of the truthvalues of its constituents. The function is called the truth function of the connective.The set of truth values forms a residuated lattice, and more precisely, a (cid:32)Lukasiewicz algebra(or MV-algebra) L = (cid:104) L, ∨ , ∧ , ⊗ , ⇒ , , (cid:105) , where L = [0 , (cid:104) L, ∨ , ∧ , , (cid:105) is a lattice with the ordering ≤ defined using the operations ∨ (supremum), ∧ (infimum) as usual, and 0 , ⊗ and ⇒ are (cid:32)Lukasiewicz conjunction and implication defined by a ⊗ b = 0 ∨ ( a + b −
1) and a ⇒ b = 1 ∧ (1 − a + b ), respectively. Thus, a ≤ b iff a ⇒ b = 1.They satisfy the residuation property [8]: a ⊗ b ≤ c iff a ≤ b ⇒ c .The language J of FLn consists of: ( a ) a countable set V ar of object variables x, y, . . . ; ( b )a finite or countable set of object constants u , u , . . . ; ( c ) a finite or countable set F unc of functional symbols f, g, . . . ; ( d ) a nonempty finite or countable set P red of predicatesymbols
P, Q, . . . ; ( e ) logical constants a for all a ∈ L ; ( f ) implication connective → ; ( g )general quantifier ∀ ; and ( h ) various types of brackets as auxiliary symbols. Terms are defined as follows: ( i ) a variable x or constant u is a (atomic) term; ( ii ) if f be an n -ary functional symbol and t l , . . . , t n terms, then f ( t l , . . . , t n ) is a term. Formulae are defined as follows: (a) logical constants a are a formula; (b) if P is an n -arypredicate symbol, and t l , . . . , t n are terms, then P ( t l , . . . , t n ) is a formula; (c) if A, B areformulae, then A → B is a formula; and (d) if x is a variable, and A is a formula, then( ∀ x ) A is a formula. Other connectives and formulae are defined as follows: ¬ A ≡ A → A & B ≡ ¬ ( A → ¬ B ) ((cid:32)Lukasiewicz conjunction); A ∨ B ≡ ( B → A ) → A (disjunction); A (cid:79) B ≡ ¬ ( ¬ A & ¬ B ) ((cid:32)Lukasiewicz disjunction); A ∧ B ≡ ¬ (( B → A ) → ¬ B ) (conjunction); A ↔ B ≡ ( A → B ) ∧ ( B → A ) (equivalence); A n ≡ A & A & . . . & A (cid:124) (cid:123)(cid:122) (cid:125) n − times (n-fold conjunction); nA ≡ A (cid:79) A (cid:79) . . . (cid:79) A (cid:124) (cid:123)(cid:122) (cid:125) n − times (n-fold disjunction);( ∃ x ) A ≡ ¬ ( ∀ x ) ¬ A (existential quantifier) . The set of all formulae of J is denoted by F J .An evaluated formula is a pair a/A , where A ∈ F J and a ∈ L is its syntactic evalua-tion. Axioms are sets of evaluated formulae. Since the evaluations can be interpreted asmembership degrees in the fuzzy set, axioms can be seen as fuzzy sets of formulae. A (cid:98) U denotes that A is a fuzzy set on a universe U . The set of all fuzzy sets on U isdenoted by F ( U ) = { A | A (cid:98) U } . F ( U ) also contains all ordinary subsets of U .ince a fuzzy set can be considered as a function, if we have a function V : F J ⇒ L and afuzzy set W (cid:98) F J , then V ≤ W means the ordering of functions. Definition 1 [7] An n -ary inference rule r in FLn is of the form: r : a /A , . . . , a n /A n r evl ( a , . . . , a n ) /r syn ( A , . . . , A n ) (1) which means from a /A , . . . , a n /A n infer r evl ( a , . . . , a n ) /r syn ( A , . . . , A n ) , where r syn is a partial n -ary syntactic operation on F J , and r evl is an n -ary lower semicontinuous evaluation operation on L (i.e., it preserves arbitrary suprema in all variables). A fuzzy set V (cid:98) F J is closed w.r.t. r if V ( r syn ( A , . . . , A n )) ≥ r evl ( V ( A ) , . . . , V ( A n ))holds for all formulae A , . . . , A n ∈ Dom ( r syn ).The set R of inference rules of FLn consists of the following:Modus ponens: r MP : a/A, b/A → Ba ⊗ b/B ; Generalization: r G : a/Aa/ ( ∀ x ) A Logical constant introduction: r LC : a/Aa ⇒ a/a → A Note that the evaluation operation r evlLC ( x ) of r LC is a ⇒ x .An proof of A ∈ F J from a fuzzy set X (cid:98) F J is a finite sequence of evaluated formulae w := a /A , a /A , . . . , a n /A n , whose each member is either a member of X , i.e., a i /A i := X ( A i ) /A i , or follows from some preceding members of the sequence using an inferencerule of FLn, and the last member is a n /A n := a/A . The evaluation a is called the value of the proof w , denoted V al ( w ). A proof w of a formula A can be denoted w A .A graded consequence operation is a closure operation C : F ( F J ) ⇒ F ( F J ) assigning to afuzzy set X (cid:98) F J a fuzzy set C ( X ) (cid:98) F J and fulfilling C ( X ) = C ( C ( X )). Definition 2 [7] Let R be the set of inference rules. The fuzzy set of syntactic conse-quences of a fuzzy set X (cid:98) F J is the following membership function, for all A ∈ F J : C syn ( X )( A ) = (cid:94) { V ( A ) | V (cid:98) F J , X ≤ V and V is closed w.r.t. all r ∈ R } (2) Theorem 1 [7] Let X (cid:98) F J . Then, C syn ( X )( A ) = (cid:87) { V al ( w ) | w is a proof of A from X } . A structure for the language J of FLn is D = (cid:104) D, ( P D ) P ∈ P red , ( f D ) f ∈ F unc , u , u , . . . (cid:105) ,where D is a non-empty domain (set); ( P D ) (cid:98) D assigns to each n -ary predicate symbol P ∈ P red an n -ary fuzzy relation P D on D ; ( f D ) assigns to each n -ary functional symbol f ∈ F unc an n -ary function f D on D ; u , u · · · ∈ D are designated elements which areassigned to each constant u , u , . . . of the language J , respectively.A truth valuation of formulae in a structure D is a function (also denoted by) D : F J → L defined by means of interpretation . Let D be a structure for the language J . The language J ( D ) is obtained from J by adding new constants being names for all elements from D ,i.e., J ( D ) = J ∪ { d | d ∈ D } .Let A ( x ) be a formula and t a term. A x [ t ] denotes a formula obtained from A by replacingall free occurrences of the variable x with t . Interpretations of closed terms and formulae are defined as follows:(i)
Interpretation of closed terms: D ( u i ) = u i if u i ∈ J and u i ∈ D ; D ( d ) = d if d ∈ D ; D ( f ( t , . . . , t n )) = f D ( D ( t ) , . . . , D ( t n )).ii) Interpretation of closed formulae (where t , . . . , t n are closed terms): D ( a ) = a for all a ∈ L ; D ( P ( t , . . . , t n )) = P D ( D ( t ) , . . . , D ( t n )); D (( ∀ x ) A ) = (cid:86) {D ( A x [ d ]) | d ∈ D } .(iii) Interpretation of the derived connectives: D ( ¬ A ) = ¬D ( A ) D ( A ∧ B ) = D ( A ) ∧ D ( B ) D ( A & B ) = D ( A ) ⊗ D ( B ) D ( A ∨ B ) = D ( A ) ∨ D ( B ) D ( A (cid:79) B ) = D ( A ) ⊕ D ( B ) D ( A ↔ B ) = D ( A ) ⇔ D ( B ) D (( ∃ x ) A ) = (cid:87) {D ( A x [ d ]) | d ∈ D } where ¬ and ⊕ are (cid:32)Lukasiewicz negation and disjunction defined by ¬ a = 1 − a and a ⊕ b = 1 ∧ ( a + b ), respectively.If A ( x , . . . , x n ) is not a closed formula, an evaluation of its free variables x , . . . , x n isfirst defined such that e ( x ) = d , . . . , e ( x n ) = d n . Then, the interpretation of A is theinterpretation of A x ,...,x n [ d , . . . , d n ], which is a closed formula. Definition 3 [7] Let X (cid:98) F J be a fuzzy set of formulae. Then the fuzzy set of its semantic consequences is the following membership function: C sem ( X )( A ) = (cid:94) {D ( A ) | for all structure D , X ≤ D} . (3)A formula A is an a-tautology (tautology in the degree a ) if a = C sem ( ∅ )( A ), and it isdenoted by | = a A . If a = 1, it is simply written by | = A , and A is called a tautology .The following lemma gives simple rules how to verify tautologies in FLn. Lemma 1 [7] Let
A, B be formulae in the language J .(a) | = A → B iff D ( A ) ≤ D ( B ) holds for every structure D for the language J .(b) | = A ↔ B iff D ( A ) = D ( B ) holds for every structure D for the language J . Definition 4 [7] Let (R1) A → ( B → A ) ; (R2) ( A → B ) → (( B → C ) → ( A → C )) ;(R3) ( ¬ B → ¬ A ) → ( A → B ) ; (R4) (( A → B ) → B ) → (( B → A ) → A ) ;(B1) ( a → b ) ↔ ( a ⇒ b ) , where ( a ⇒ b ) denotes the logical constant for the value a ⇒ b when a and b are given. This is called book-keeping axiom ;(T1) ( ∀ x ) A → A x [ t ] for any substitutible term t . This is called substitution axiom ;(T2) ( ∀ x )( A → B ) → ( A → ( ∀ x ) B ) provided that x is not free in A . Using Lemma 1, it can be verified that (R1)-(R4), (B1) and (T1)-(T2) are (1-)tautologies.The fuzzy set
LAx of logical axioms of FLn is as follows: LAx ( F ) = 1 if F is one of theforms (R1)-(R4), (B1) and (T1)-(T2) ; LAx ( F ) = a if F = a ; and LAx ( F ) = 0 otherwise. Definition 5 [7] A fuzzy theory (or theory for short) T in the language J of FLn is atriple T = (cid:104) LAx, SAx, R (cid:105) , where
LAx is the fuzzy set of logical axioms,
SAx (cid:98) F J is afuzzy set of special axioms , and R is the set of inference rules. A theory can be viewed as a fuzzy set T = C syn ( LAx ∪ SAx ) (cid:98) F J . Definition 6 [7] Let T be a theory and A ∈ F J a formula.(i) If C syn ( LAx ∪ SAx )( A ) = a , it is denoted by T (cid:96) a A , and A is said to be a theorem or provable in the degree a in T . The value a is called the provability degree of A over T .(ii) If C sem ( LAx ∪ SAx )( A ) = a , it is denoted by T (cid:15) a A , and A is said to be true in thedegree a in T . The value a is called the truth degree of A over T .iii) Let D be a truth valuation of formulae. Then, it is a model of T , denoted D (cid:15) T , if SAx ( A ) ≤ D ( A ) holds for all formulae A ∈ F J . Therefore, Theorem 1 can be restated as follows: T (cid:96) a A iff a = (cid:95) { V al ( w ) | w is a proof of A from LAx ∪ SAx } (4)Also, due to the assumption made on LAx, we have LAx ( A ) ≤ D ( A ) holds for every truthvaluation D of formulae. Thus, T (cid:15) a A iff a = (cid:94) {D ( A ) |D (cid:15) T } . (5) Definition 7 [7] A theory T is contradictory if there exists a formula A , and there areproofs w A and w ¬ A of A and ¬ A from T , respectively, such that V al ( w A ) ⊗ V al ( w ¬ A ) > .Otherwise, it is consistent . Theorem 2 [7] A fuzzy theory T is consistent iff it has a model. Theorem 3 (Completeness) [7] T (cid:96) a A iff T | = a A holds for every formula A ∈ F J and every consistent fuzzy theory T . This means that the provability degree of A in T coincides with its truth degree over T .This is usually referred to as the Pavelka-style completeness [9, 8].
3. Fuzzy Logic in Narrow Sense with Hedges
In order to provide a foundation for a computational approach to human reasoning in thepresence of vagueness, imprecision, and uncertainty, fuzzy logic should have the ability todeal with linguistic hedges, which play a very important role in the modification of fuzzypredicates. Therefore, in this section, we will extend FLn with hedges connectives in orderto model human knowledge and reasoning. In addition to extending the language and thedefinition of formulae, new logical axioms characterizing properties of the new connectivesare added. One of the most important properties should be preservation of the logicalequivalence.
Definition 8 [7] Let (cid:5) : L n → L be an n-ary operation. It is called logically fitting if itsatisfies the following condition: there are natural numbers k > , . . . , k n > such that ( a ⇔ b ) k ⊗ · · · ⊗ ( a n ⇔ b n ) k n ≤ (cid:5) ( a , . . . , a n ) ⇔ (cid:5) ( b , . . . , b n ) holds for all a , . . . , a n , b , . . . , b n ∈ L , and the power is taken w.r.t. the operation ⊗ , e.g., ( a ⇔ b ) k = ( a ⇔ b ) ⊗ · · · ⊗ ( a ⇔ b ) (cid:124) (cid:123)(cid:122) (cid:125) k − times Hence, logically fitting operations preserve the logical equivalence. It can be proved that allthe basic operations ∨ , ∧ , ⊗ , ⇒ are logically fitting, and any composite operation obtainedfrom logically fitting operations is also logically fitting.A connective is logically fitting if it is assigned a logically fitting truth function (operation). In this subsection, FLn is extended with many hedges, in which each hedge does not haveany dual one. To ease the presentation, we let s , d denote the identity connective, i.e.,for all formula A , A ≡ s A ≡ d A , and their truth functions s • and d • are the identity. efinition 9 (FLn with many hedges) On the syntactic aspect, FLn is extended asfollows (where p, q are positive integers):(i) The language J is extended into a language J h by a finite set H = { s , . . . , s p , d , . . . , d q } of additional unary connectives, where s i ’s are truth-stressing hedges, and d j ’s are truth-depressing ones.(ii) The definition of formulae is extended by adding the following: If A is a formula and h ∈ H , hA is a formula.(iii) The fuzzy set LAx of logical axioms is extended by the following axioms: / ( A → B ) → ( hA → hB ) , for all h ∈ H (6)1 /s i A → s i − A, for i = 1 , . . . , p (7)1 /s p /d j − A → d j A, for j = 1 , . . . , q (9)1 / ¬ d q A implies B , then very (resp., slightly ) A implies very (resp., slightly ) B . Axiom (8) (resp., (10)) says that the truth function s • p (resp., d • q ) preserves 1(resp., 0). Axiom (7) (resp., (9)) expresses that s i (resp., d j ) modifies truth more than s i − (resp., d j − ), for i = 2 , . . . , p (resp., j = 2 , . . . , q ). For example, slightly (resp., very ) mod-ifies truth more than rather (resp., highly ) since slightly true < rather true < true (resp., true < highly true < very true ). Also, for instance, let A = young ( x ) , s = highly , s = very , d = rather , d = slightly , by (7), we have very young ( x ) → highly young ( x ) and highly young ( x ) → young ( x ). Moreover, by (9), we have young ( x ) → rather young ( x ) and rather young ( x ) → slightly young ( x ). Therefore, we have: very young ( x ) → highly young ( x ) → young ( x ) → rather young ( x ) → slightly young ( x )This is also in accordance with fuzzy-set-based interpretations of hedges [10, 11, 12], inwhich very and highly are called intensifying modifiers while rather and slightly are called weakening modifiers , and they satisfy the so-called semantic entailment property: x is very A ⇒ x is highly A ⇒ x is A ⇒ x is rather A ⇒ x is slightly A where A is a fuzzy predicate. Note that, according to [10], if A is represented by afuzzy set with a membership function µ A ( x ), the membership function of very A can be µ very A ( x ) = µ A ( x ). Since for all x, ≤ µ A ( x ) ≤
1, we have for all x, µ very A ( x ) ≤ µ A ( x ).By fuzzy set inclusion, it is said that very A is included by A , denoted very A ⊆ A . Sincethe degree of membership of x to A is regarded as the truth value of “ x is A ”, the truthvalue of “ x is very A ” is less than or equal to that of “ x is A ”. Theorem 4
For every hedge connective h ∈ H , its truth function h • is non-decreasingand preserves 0 and 1.Proof. By (6), for all structure D of the language J h , we have D ( A → B ) ≤ D ( hA → hB ).Hence, D ( A ) ⇒ D ( B ) ≤ h • ( D ( A )) ⇒ h • ( D ( B )). Let a, b ∈ L and a ≤ b . Since D ( a ) ⇒D ( b ) = a ⇒ b = 1, we have h • ( D ( a )) ⇒ h • ( D ( b )) = 1, i.e., h • ( a ) ≤ h • ( b ). Therefore, thetruth function h • of any hedge connective h ∈ H is non-decreasing.By (8), we have s • p (1) = 1. Using (7) and taking A = 1 , i = p , we have, for all structure D of the language J h , D ( s p ≤ D ( s p − s • p (1) ≤ s • p − (1). Hence, s • p − (1) = 1.imilarly, we have s • i (1) = 1 for all i = p − , . . . ,
1. Also, using (7) and taking A = 0 , i = 1,we have, for all structure D of the language J h , D ( s ≤ D ( s s • (0) ≤ s • (0) = 0.Hence, s • (0) = 0. Similarly, we have s • i (0) = 0 for all i = 2 , . . . , p . Therefore, all s • i preserve 0 and 1.By (10), we have d • q (0) = 0. Using (9) and taking A = 0 , j = q , we have, for all structure D of the language J h , D ( d q − ≤ D ( d q d • q − (0) ≤ d • q (0) = 0. Hence, d • q − (0) = 0.Similarly, we have d • j (0) = 0 for all j = q − , . . . ,
1. Also, using (9) and taking A = 1 , j = 1,we have, for all structure D of the language J h , D ( d ≤ D ( d d • (1) ≤ d • (1).Hence, d • (1) = 1. Similarly, we have d • j (1) = 1 for all j = 2 , . . . , q . Therefore, all d • j preserve 0 and 1. (cid:3) Theorem 5
For every truth-stresser s i ∈ H , i = 1 , p , its truth function s • i is subdiagonal.Proof. For any a ∈ L , using (7) and taking A = a, i = 1, we have, for all structure D of the language J h , s • ( a ) = D ( s a ) ≤ D ( s a ) = s • ( a ) = a . Thus, s • ( a ) ≤ a forall a ∈ L , i.e., s • is subdiagonal. Again, using (7) and taking A = a, i = 2, we have s • ( a ) = D ( s a ) ≤ D ( s a ) = s • ( a ) ≤ a . Similarly, s • i is subdiagonal for all i = 3 , . . . , p . (cid:3) Theorem 6
For every truth-depresser d j ∈ H , j = 1 , q , its truth function d • j is superdiag-onal.Proof. For any a ∈ L , using (9) and taking A = a, j = 1, we have, for all structure D of the language J h , a = d • ( a ) = D ( d a ) ≤ D ( d a ) = d • ( a ). Thus, d • ( a ) ≥ a for all a ∈ L , i.e., d • is superdiagonal. Again, using (9) and taking A = a, j = 2, we have a ≤ d • ( a ) = D ( d a ) ≤ D ( d a ) = d • ( a ). Similarly, d • j is superdiagonal for all j = 3 , . . . , q . (cid:3) The following theorem shows that the additional hedge connectives are logically fitting.
Theorem 7
For every hedge connective h ∈ H , its truth function h • is logically fitting.Proof. Given a, b ∈ L , using (6) and taking A = a, B = b , we have, for all structure D of the language J h , D ( a → b ) ≤ D ( ha → hb ). Hence, a ⇒ b ≤ h • ( a ) ⇒ h • ( b ).Again, using (6) and taking A = b, B = a , we have b ⇒ a ≤ h • ( b ) ⇒ h • ( a ). Therefore, a ⇔ b = ( a ⇒ b ) ∧ ( a ⇐ b ) ≤ ( h • ( a ) ⇒ h • ( b )) ∧ ( h • ( a ) ⇐ h • ( b )) = h • ( a ) ⇔ h • ( b ), i.e., h • is logically fitting according to Definition 8. (cid:3) We can also say that all the hedge connectives h ∈ H are logically fitting.Since it is shown in [7] that introducing logically fitting operations have no influence on thealgebraic proof of the completeness theorem of FLn (Theorem 3), we have the followingcompleteness theorem for FLn with many hedges. Theorem 8
Let T be a consistent fuzzy theory in the extended language J h . Then T (cid:96) a A iff T | = a A holds for every formula A ∈ F J h . That means FLn with many hedges also has the Pavelka-style completeness.
It can be observed that each hedge can have a dual one, e.g., slightly and rather canbe seen as a dual hedge of very and highly , respectively. Thus, there might be axiomsexpressing dual relations of hedges in addition to axioms expressing their comparativetruth modification strength.n this subsection, FLn is extended with many dual hedges, in which each hedge has itsown dual one.
Definition 10 (FLn with many dual hedges)
On the syntactic aspect, FLn is extendedas follows (where n is a positive integer):(i) The language J is extended into a language J dh by a finite set H = { s , . . . , s n , d , . . . , d n } of additional unary connectives, where s i ’s are truth-stressing hedges, and d i ’s are truth-depressing ones.(ii) The definition of formulae is extended by adding the following: If A is a formula and h ∈ H , hA is a formula.(iii) The fuzzy set LAx of logical axioms is extended by the following axioms: / ( A → B ) → ( hA → hB ) , for all h ∈ H (11)1 /s i A → s i − A, for i = 1 , . . . , n (12)1 /s n /d i − A → d i A, for i = 1 , . . . , n (14)1 /d i A → ¬ s i ¬ A (15)The main differences between the fuzzy set LAx of FLn with many dual hedges and thatof FLn with many hedges are that we have the axiom (15) expressing the duality betweenhedges s i and d i , and we do not need an axiom similar to (10).Concerning Axiom (15), for instance, let A = young ( x ) , s i = very, d i = slightly , thenit means “ slightly young implies not very old ”. Using (15) and taking A = a for any a ∈ L , we have D ( d i a ) ≤ D ( ¬ s i ¬ a ) for all structure D of the language J dh . Thus, d • i ( a ) ≤ − s • i (1 − a ), then s • i (1 − a ) ≤ − d • i ( a ). Let b = 1 − a = ¬ a , hence a = 1 − b = ¬ b .We have s • i ( b ) ≤ − d • i ( ¬ b ). Let B be a formula whose truth valuation is b , i.e., D ( B ) = b .We have D ( s i B ) = s • i ( b ) ≤ − d • i ( ¬ b ) = D ( ¬ d i ¬ B ). Therefore, we have a tautology1 /s i B → ¬ d i ¬ B . This means, for instance, “ very old implies not slightly young ” as well.Since axioms (12) and (13) are respectively similar to (7) and (8), we also have that all s • i , i = 1 , n, preserve 1. Using (15) and taking A = 0, we have D ( d i ≤ D ( ¬ s i ¬
0) for allstructure D of the language J dh . Hence, d • i (0) ≤ − s • i (1) = 0. Therefore, d • i (0) = 0 forall i = 1 , n . That means we do not need an axiom similar to (10).Now, given a hedge function s • n , which is non-decreasing, subdiagonal, and preserves 0and 1, the boundaries for the other hedge functions are one by one determined as follows.Using (12) and taking A = a for any a ∈ L and i = n , we have s • n ( a ) ≤ s • n − ( a ).Thus, the lower boundary of s • n − ( a ) is s • n ( a ), and its upper boundary is a . Similarly,we have s • n − ( a ) ≤ s • n − ( a ) ≤ a , and finally, s • ( a ) ≤ s • ( a ) ≤ a . Then, by (15), wehave a ≤ d • i ( a ) ≤ ¬ s • i ( ¬ a ) for all i = 1 , n , thus, a and ¬ s • ( ¬ a ) are the lower boundaryand upper boundary of d • ( a ), respectively (note that since s • (1 − a ) ≤ − a , we have ¬ s • ( ¬ a ) = 1 − s • (1 − a ) ≥ − (1 − a ) = a ). Using (14) and taking A = a, i = 2, we have d • ( a ) ≤ d • ( a ), thus, d • ( a ) and ¬ s • ( ¬ a ) are the lower boundary and upper boundary of d • ( a ), respectively. Similarly, we have d • i − ( a ) and ¬ s • i ( ¬ a ) are the lower boundary andupper boundary of d • i ( a ), respectively, for all i = 3 , n . In summary, the boundaries areshown in Table 1.We also have the following completeness theorem for FLn with many dual hedges. Theorem 9
Let T be a consistent fuzzy theory in the extended language J dh . Then T (cid:96) a A iff T | = a A able 1: Boundaries of hedge functionsHedge function Lower boundary Upper boundary s • n − ( x ) s • n ( x ) x . . . s • ( x ) s • ( x ) xd • ( x ) x − s • ( − x ) d • ( x ) d • ( x ) − s • ( − x ). . . d • n ( x ) d • n − ( x ) − s • n ( − x ) holds for every formula A ∈ F J dh . It can be seen that in a case when we want to extend the above logic with one truth-stressing (resp., truth-depressing) hedge without a dual one, we only need to add axiomsexpressing its relations to the existing truth-stressing (resp., truth-depressing) hedges ac-cording to their comparative truth modification strength.
4. Conclusion
In this paper, we extend fuzzy logic in narrow sense with many hedge connectives in orderto provide a foundation for human knowledge representation and reasoning. In fuzzy logicin narrow sense, both syntax and semantics are evaluated by degrees in [0,1]. The gradedapproach to syntax can be seen as an elegant and natural generalization of classical logicfor inference under vagueness since it allows one to explicitly represent and reason withpartial truth, i.e., proving partially true conclusions from partially true premises, and itenjoys the Pavelka-style completeness. FLn is extended in two cases: (i) each hedge doesnot have a dual one, and (ii) each hedge can have its own dual one. In addition to extendingthe language and the definition of formulae, new logical axioms characterizing propertiesof the hedge connectives are added. The truth function of a truth-depressing (resp., truth-stressing) hedge connective is a superdiagonal (resp., subdiagonal) non-decreasing functionpreserving 0 and 1. The resulting logics are shown to have the Pavelka-style completeness.
5. Acknowledgments
Funding from HUMG under grant number T16-02 is acknowledged.
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