Uninorm Based Logic As An Extension of Substructural Logics FL e

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1 Uninorm Based Logic As An Extension of Substructural Logics FL e Osamu WATARI Hokkaido Automotive Engineering College Sapporo , JAPAN watari@haec.ac.jp Mayuka F. KAWAGUCHI Division of Computer Science Hokkaido University Sapporo , JAPAN mayuka@main.ist.hokudai.ac.jp Masaaki MIYAKOSHI Division of Computer Science Hokkaido University Sapporo , JAPAN miyakosi@main.ist.hokudai.ac.jp Abstract This research work focuses on formulating uninorm based logic (UL) as a weaker logic than monoidal t-norm based logic (MTL). It is shown that the system UL is complete with respect to the class of left-continuous conjunctive uninorms, and that it is an extension of intuitionistic substructural logic FL e. Keywords: uninorms, t-norms, fuzzy logics, Substructural logics. 1 Introduction Substructural logics are logics lacking some or all of the structural rules when they are formalized in sequent systems. Ono has pointed out in [11], [12] that monoidal t-norm based logic (MTL), which was formulated in [2] as logic of left-continuous t-norms and their residuals, is an extension of intuitionistic substructural logic FL ew (substructural logic without contraction). In this research work, the authors formulate UL as the logic based on uninorms [1], [2], [4], [13], and UL-algebra as the algebraic interpretation of UL. Moreover, we show that this system is an extension of FL e, which is obtained by removing weakening rule from FL ew, and that UL is complete with respect to any UL-algebra. 2 Substructural Logics and Fuzzy Logics 2.1 Formal Systems of Substructural Logics The formal systems by sequent calculi (e.g. LK and LJ for the classical logic and the intuitionistic logic, respectively) include structural inference rules besides inference rules on logical connectives. Various logics as relevant logic, BCK logic and Lukasiewicz logic etc., can be taken as the systems obtained by restricting some or all of structural inference rules. Such logics lacking structural inference rules have been called substructural logics, and full Lambek calculus was introduced in [9], [10] as the basis of them. The most basic system of substructural logics is called FL, which is obtained by removing all three structural inference rules: contraction, weakening and exchange. Definition 1. The axioms and inference rules of FL is the followings. Axioms (Initial sequents): α α t f Γ,, Γ Inference rules: Γ α, β, Σ ( left) Γα,, βσ,

2 αγ, β ( right) Γ α β Γ α, β, Σ ( left) α, β, Γ, Σ Γα, β ( right) Γ α β Γαβ,,, (left) Γα, β, Γ α β (right) Γ, α β Γα,, ( left 1) Γα, β, Γ, β, ( left 2) Γα, β, Γ α Γ β ( right) Γ α β Γ, α, Γ, β, ( left) Γα, β, Γ α ( right 1) Γ α β Γ β ( right 2) Γ α β Γ α, α, Σ (cut) Γ,, Σ Γ, (-weakening) Γ,, t t Γ ( -weakening) Γ f f Here, Greek capital letters are meta-symbols denoting sequences of formulas (including the empty sequence) and Greek small letters are meta-symbols denoting formulas. Especially, denotes at most one formula. tf,,, are constants, and α α f, α α f. By adding all or some of weakening, contraction and exchange to FL, one obtains the various intuitionistic substructural logics FL e, FL ec, FL ew, FL ecw (=LJ), FL c, FL w, and FL cw. Here, w, c and e denote that the rules: weakening, contraction and exchange are added to FL, respectively. Γαβ,,, (e left) Γ, β, α, Γαα,,, (c left) Γα,, Γ, (w left) Γα,, Γ (w right) Γ α By adding exchange to FL, two implications and coincide. By adding both of contraction and weakening, exchange is derived and two conjunctions and coincide, thus FL cw (=FL ecw ) becomes equivalent to LJ. Also, by modifying the right side of a sequent to be a sequence of over two formulas (and making some slight changes in inference rules), we can obtain classical substructural logics CFL e, CFL ec, CFL ew and CFL ecw (=LK). The classical substructural logics without exchange are not considered here because in such cases many different systems could be constructed depending on the position of the main formula of inference rules. 2.2 Full Lambek Algebra The algebraic interpretation of FL is given as the algebraic structure called residuated lattice. It is called full Lambek algebra (shortly, FLalgebra) and defined as follows [9], [10]. Definition 2. The algebra V,,,,,, 10,,, satisfying the following properties is called a full Lambek algebra. (FL1) V,,,, is a lattice with the largest element and the least element, (FL2) V,, 1 is a monoid of which unit element is 1,

3 (FL3) xyzw,,, V: z ( x y) w = ( z x w) ( z y w), (FL4a) xyz,, V: x y z x y z, (FL4b) xyz,, V: x y z y x z, (FL5) 0 V. 2.3 Monoidal t-norm Based Logic MTL Monoidal t-norm based logic MTL is introduced by Esteva et al. [3] as logic for leftcontinuous t-norms, i.e. MTL is a weaker logic than BL [5] which is a logic for continuous t- norms. Here, it should be noted that the unit element 1 of the monoid and an arbitrary fixed element 0 (not the zero element of ) do not always coincide with the largest element and the least element of the lattice V. Also, an arbitrary element 0 is needed in order to give the interpretation of negation as x x 0 and x x 0. The relationship between two operators expressed in (FL4a, b) is called residuation, such a pair of operators is a residuated pair, and the algebraic structure satisfying (FL1) (FL4) is called a residuated lattice. The structural inference rules in sequent systems correspond to the following properties in algebraic systems, respectively. (FLe) exchange: (FLw) weakening: 0 =, is commutative (FLc) contraction: x x x. x y x, y x x The algebraic interpretations of the sequent systems which are FL with some substructural rules, can be obtained by adding the corresponding properties among (FLe), (FLw) and (FLc) to FL-algebra. In the same manner as the case of the logical systems, by adding (FLe), two implications and coincide with each other, also by adding (FLw), unit element 1 and an arbitrary fixed element 0 coincide with the largest element and the least element, respectively. Definition 3. MTL is the logic with the following axioms and inference rule. Axioms: (A1) ( α β) (( β γ) ( α γ)) (A2) ( α β) α (A3) ( α β) ( β α) (A4) ( α β) α (A5) ( α β) ( β α) (A6) ( α ( α β)) ( α β) (A7a) ( α ( β γ)) (( α β) γ) (A7b) (( α β) γ) ( α ( β γ)) (A8a) ( α β) ((( α β) β) (( β α) α)) (A8b) ((( α β) β) (( β α) α)) ( α β) (A9) (( α β) γ) ((( β α) γ) γ) (A10) α Inference rule: α α β (m.p.) β Definition 4. A two-place function T :[0,1] [0,1] satisfying the following properties is called a t-norm. (T1) Ta (,1) = a (T2) a b T( a, c) T( b, c) (T3) TaTbc (, (, )) = TTab ( (, ), c) (T4) Tab (, ) = Tba (, ) 2

4 When a t-norm T is left-continuous with respect to the second variable, T and the operator I T defined as follows (called the right residual of T ) form a residuated pair. IT ( a, b ) = sup{ c T ( a, c ) b } The definition of MTL-algebra is given as follows by generalizing the domain from the unit real interval [0,1] to a lattice. Definition 5. The algebra V,,,, 10,, satisfying the following properties is called MTL-algebra [3]: (MTL1) V, 10,,, is a lattice with the largest element 1 and the least element 0, (MTL2) V,, 1 is a commutative monoid of which unit element is 1, (MTL3) xyzw,,, V: z ( x y) w = ( z x w) ( z y w), (MTL4) x, y,z V : x y z y x z (MTL5) x, y V : x y x, y x x (MTL6) x,y V : ( x y) ( y x) = 1 An arbitrary MTL-algebra on [0,1] can be expressed as [0,1],max,min, T, I T,1,0 by using a left-continuous t-norm and its right residual. Also, it is clear from (MTL1) (MTL5) that MTL-algebras are FL ew -algebras equipped with the pre-linearity property (MTL6). Thus, we reach the following result. Theorem 1. The following three statements are equivalent [12]: (1) For any MTL-algebra, v( α ) = 1 ; (2) A formula α is provable in MTL; (3) A sequent α is provable in FL ew + ( ( α β) ( β α)) (FL ew equipped with ( α β) ( β α) as an initial sequent). Here, v: F V ( F : a set of formulas in MTL) is a valuation map. The validity of a formula α in MTL-algebra is defined as v( α ) = 1. As shown in Theorem 1, MTL is taken as an extension of FL ew in the framework of substructural logics. 3 Uninorms As the main stage of this work, let us investigate the logical systems in which the weakening rule is restricted from MTL. The key point of such systems is that the unit element of the monoid does not coincide with the largest element of a lattice. Now, let us consider uninorms [1], [2], [4] introduced by Yager [13] as aggregation operators equipped with such a property. Definition 6. A two-place function U :[0,1] [0,1] satisfying the following properties is called a uninorm. (U1) e [0,1] : U( a, e) = a (U2) a b U( a, c) U( b, c) (U3) U( a, U( b, c)) = U( U( a, b), c) (U4) U( a, b) = U( b, a) A uninorm becomes a t-norm when e = 1, and becomes a t-conorm when e = 0. A uninorm is called conjunctive if it satisfies U(0,1) = U(1,0) = 0, and disjunctive if it satisfies U(0,1) = U(1,0) = 1. 4 Uninorm Based Logic UL Let us consider the operator defined as IU ( a, b ) sup{ cu( ac, ) b} for a uninorm U, in the same way as the case of a t-norm. When U is left-continuous and conjunctive, U and I U form a residuated pair [2], [6]. Now, the authors introduce the definition of UL (uninorm based logic)-algebra. 2

5 Definition 7. The algebra V,,,,, e, f, 10, satisfying the following properties is called UL-algebra. (UL1) V, 10,,, is a lattice with the largest element 1 and the least element 0, (UL2) V,, e is a monoid of which unit element is e, (UL3) xyzw,,, V: z ( x y) w = ( z x w) ( z y w), (UL4) xyz,, V: x y z y x z, (UL5) f V, (UL6) x, y V : ( x y) ( y x) e. An arbitrary UL-algebra on [0,1] can be expressed as [0,1], max, min, U, IU, e, f,1,0 by using a left-continuous conjunctive uninorm and its right residual IU ( a, b) = sup{ cu( ac, ) b}. Clearly from (UL1) (UL5), the class of ULalgebras is a subclass of FL e -algebras. Thus, uninorm based logic UL defined as the complete logic with respect to UL-algebra becomes an extension of substructural logic FL e. The authors give the definition of UL below. Definition 8. UL is the logic with the following axioms and inference rule. Axioms: (A1) ( α β) (( β γ) ( α γ)) (A3) ( α β) ( β α) (A4) ( α β) α (A5) ( α β) ( β α) (A6 ' ) (( α β) ( α γ)) ( α ( β γ)) (A7a) ( α ( β γ)) (( α β) γ) (A7b) (( α β) γ) ( α ( β γ)) (A8a) ( α β) ((( α β) β) (( β α) α)) (A8b) ((( α β) β) (( β α) α)) ( α β) (A9) (( α β) γ) ((( β α) γ) γ) (A10) α Inference rule: α α β (m.p.) β The following theorem holds among UL-algebra, UL and FL e. Theorem 2. The following three statements are equivalent: (1) For any UL-algebra, v( α) e; (2) A formula α is provable in UL; (3) A sequent α is provable in FL e + ( ( α β) ( β α)) (FL e equipped with ( α β) ( β α) as an initial sequent). Here, it should be noted that the definition of the validity is different from the case of Theorem 1. The validity of a formula α in UL-algebra is defined as v( α) e. The above theorem can be proven in the usual manner to prove the soundness and the completeness. It should be shown that each axiom is valid, each inference rule keeps the validity, and then, the Lindenbaum algebra constructed for each system forms UL-algebra. 5 Concluding Remarks A uninorm based logic UL has been formulated by analogy with the relation between MTL and substructural logics. Furthermore, it has been shown that UL is complete with respect to ULalgebra defined as the class of algebras equipped with left-continuous conjunctive uninorms, and UL is an extension of intuitionistic substructural logic FL e. The authors have already introduced further weak logics: psul satisfying the prelinearity property in the framework of FL (i.e. exchange is removed from FL e ) [7]. The notion of uninorm based logics has been investigated by Metcalfe [8], too. He has given very similar definition of uninorm based logic to

6 that of ours. For example, our axiom of the prelinearity property (i.e. (UL6) and ( α β) ( β α) ) is slightly weaker than that of Metcalfe s logic. It is still an open problem if our weaker definition of uninorm based logic could support the soundness and the completeness for UL with respect to linearly ordered UL-algebra. As further steps of this research, the authors intend to investigate the effectiveness and the possible applications of uninorm based logics from the viewpoint of fuzzy logics. References [1] B. De Baets, Idempotent uninorms, European J. Operational Research, 118 (1999) pp [2] B. De Baets and J.C. Fodor: Residual operators of uninorms, Soft Computing, 3 (1999) pp [3] F. Esteva and L.Godo, Monoidal t-norm based logic: towards a logic for leftcontinuous t-norms, Fuzzy Sets and Systems, 124 (2001) pp [4] J.C. Fodor, R.R. Yager and A. Rybalov, Structure of uninorms, Int. J. of Uncertainty, Fuzziness and Knowledge- Based Systems, 5 (1997) pp [5] P. Hájek, Metamathematics of Fuzzy Logic, Trends in Logic 4, Kluwer Academic Publishers (1998). [6] M.F. Kawaguchi, O. Watari and M. Miyakoshi, Involutions in fuzzy logics based on t-norms and uninorms, Note on Multiple-Valued Logic in Japan, 27 (2004) pp.10_1-10_7, in Japanese. [7] M.F. Kawaguchi, O. Watari and M. Miyakoshi, Fuzzy logics and substructural logics without exchange, Proc. of EUSFLAT-LFA 2005, Barcelona, Spain (2005) pp [8] G. Metcalfe, Uninorm based logics, Proc. of EUROFUSE 2004, Warsaw, Poland (2004) pp [9] H. Ono, Semantics for substructural logics, Substructural Logics (K. Došen and P. Shroeder- Heister eds.), Oxford Univ. Press (1993) pp [10] H. Ono, Proof-theoretic methods in non-classical logic an introduction, Theories of Types and Proofs, MSJ Memoirs 2, Mathematical Society of Japan (1998) pp [11] H. Ono, Residuation theory and substructural logics, Proc. of 34th MLG meeting, Echigo-Yuzawa, Japan (2001) pp [12] H. Ono, Substructural logics and residuated lattices an introduction, Trends in Logic 50 Years of Studia Logica (V.F. Hendrics and J. Malinowski eds.), Trends in Logic 21, Kluwer Academic Publishers (2003). [13] R.R. Yager and A. Rybalov, Uninorm aggregation operators, Fuzzy Sets and Systems, 80 (1996) pp