Uncertain Second-order Logic

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1 Uncertain Second-order Logic Zixiong Peng, Samarjit Kar Department of Mathematical Sciences, Tsinghua University, Beijing , China Department of Mathematics, National Institute of Technology, Durgapur , India kar s k@yahoo.com Abstract: Uncertain second-order logic is introduced as an extension of uncertain propositional logic and uncertain predicate logic, which are branches of multi-valued logic for dealing with uncertain knowledge. In this paper, some concepts of uncertain second-order logic are proposed. The definition of truth values for uncertain second-order formulas is proposed. Finally, some theorems of uncertain second-order logic is given. Keywords: Uncertainty theory, uncertain logic, uncertain second-order logic 1 Introduction The ability to reason is a marvel of human nature. methods of improving our use of reason have arisen the intellectual discipline known as logic, logics help people to make conclusions from what is known. In practical cases, people are likely to make some conclusions from something which are not so surely known. For this reason, classic logic was extended to many kinds of logics. As early as in 1920, multivalued logic by Lukasiewicz. After a long time, Zadeh [18] proposed the notion of fuzzy set in 1965, and Zadeh [19] proposed the fuzzy logic in 1975, which is a logic that handles vague statements. Different from fuzzy logic, probabilistic logic was proposed by Nilsson [15] via the theory of probability in While in 2009, Li and Liu [5] proposed credibilistic logic via credibility theory. However, randomness and fuzziness are not all the uncertainty in the world. In order to model the uncertainty which are not randomness and fuzziness, uncertainty theory was founded by Liu [10] in 2007 and refined by Liu [11] in 2010 and became a branch of mathematics based on the normality, monotonicity, self-duality, countable subadditivity, and product measure axioms. It is a new tool to study subjective uncertainty. Based on the uncertainty theory, some theoretical work of uncertainty theory such as uncertain process [7], uncertain calculus [8], uncertain differential equation[2] [8], uncertain logic [6] and uncertain inference [13] have been established. As an application of uncertainty theory, Liu [9] proposed a spectrum of uncertain programming which is mathematical Proceedings of the First International Conference on Uncertainty Theory, Urumchi, China, August 11-19, 2010, pp programming involving uncertain variables. In 2009, Li and Liu [6] proposed uncertain propositional logic based on uncertainty theory, which explains formula as uncertain variable and defines its truth value as uncertain measure that formula is true. Following that, Chen and Ralescu [1] gave a truth value theorem for computing the true value of uncertain formula. Furthermore, uncertain entailment was developed by Liu [12] as a methodology for calculating the truth value of an uncertain propositional formula via the maximum uncertainty principle. After that, Gao [3] discussed some inference rule for uncertain systems. Other references related to uncertainty theory are Gao [4], You [17], Liu [14] and Peng and Iwamura [16], etc. For exploring the recent developments of uncertainty theory, the readers may consult Liu [11]. In this paper, uncertain second-order logic is introduced as a new branch of logics via uncertainty theory for dealing with uncertain knowledge. The rest of this paper is organized as follows. Some basic concepts of uncertainty theory are recalled in Section 2. The concepts of uncertain second-order logic are introduced in Section 3. The truth value of uncertain second-order formula is discussed in Section 4. Some basic properties of uncertain second-order logic is shown in Section 5. At the end of this paper, a brief summary is given. 2 Preliminary In this section, we will introduce some useful definitions about uncertain measure, uncertain variables, uncertain logic and so on. Let Γ be a nonempty set, and L be a σ-algebra over Γ. Each element Λ L is called an event. A number M(Λ) indicates the level that Λ will occur. Uncertain measure M was introduced as a set function satisfying the following four axioms (Liu [10]): Axiom 1. M{Γ} = 1. Axiom 2. M{Λ 1 } M{Λ 2 } whenever Λ 1 Λ 2. Axiom 3. M{Λ} + M{Λ c } = 1 for any event Λ. Axiom 4. For every countable sequence of events {Λ i }, we have { } Λ i M{Λ i }. M i=1 i=1

2 260 ZIXIONG PENG & SAMARJIT KAR Liu [8] presented the product measure axiom of uncertainty theory in 2009 as follows. Axiom 5. Let Γ k be nonempty sets on which M k are uncertain measures, k = 1, 2,, n, respectively. Then the product uncertain measure M is an uncertain measure on the product σ-algebra L 1 L 2 L n satisfying { n } M Λ k = min M k{λ k } 1 k n k=1 where Λ k L k, k = 1, 2,, n. The concept of uncertain variable was introduced by Liu [10] as a measurable function from an uncertainty space (Γ,L,M) to the set of real numbers. Uncertain Propositional Logic Uncertain propositional logic was designed by Li and Liu [6] as a generalization of classical logic. Definition 1. (Li and Liu [6]) An uncertain proposition is a statement whose truth value is quantified by an uncertain measure. In fact, an uncertain proposition X is essentially an uncertain variable taking values 0 or 1, where X = 1 means X is true and X = 0 means X is false. That is an uncertain variable ξ satisfies { 0, with uncertain measure u ξ(γ) = 1, with uncertain measure 1 u where u is a real number and u [0, 1]. Definition 2. (Li and Liu [6]) An uncertain formula is defined as a member of the minimal set S of finite sequence of primitive symbols satisfying: (a). ξ S for each uncertain proposition ξ. (b). if X S, then X S. (c). if X S and Y S, then X Y S. Notice that, an uncertain proposition ξ is indeed a {0, 1}- valued uncertain variable, which can be measured by an uncertain measure. Truth value is a key concept in uncertain logic and is defined as the uncertain measure that the uncertain formula is true. Definition 3. (Li and Liu [6]) Let X be an uncertain formula. Then the truth value of X is defined as the uncertain measure that the uncertain formula X is true, i.e., T (X) = M{X = 1}. Uncertain Predicate Logic Uncertain predicate logic was designed by Zhang and Peng [20] as a generalization of uncertain propositional logic. Definition 4. (Zhang and Peng [20]) Uncertain predicate proposition is a sequence of uncertain propositions indexed by one or more parameters. For example, let ξ(a) be an uncertain predicate proposition, then for any a D, ξ(a) is a {0, 1}-valued uncertain variable. D is the domain of discourse, and a is called a variable symbol. Definition 5. (Zhang and Peng [20]) An uncertain predicate formula is defined as a member of the minimal set S of finite sequence of primitive symbols satisfying: (a ). ξ(a 1, a 2,, a m ) S for each uncertain predicate proposition ξ(a 1, a 2,, a m ). (b ). if X S, then X S. (c ). if X S and Y S, then (X Y ) S. (d ). if X S, then ( a D)X S, where D is the domain of discourse, and a is an arbitrary variable symbol. If the related D is unique, then we write the formula ( a D)X as ax. For example, aξ(a) is an uncertain predicate formula but not an uncertain propositional formula. The truth value of an uncertain predicate formula is defined as Definition 3. In the following section of this paper, an uncertain variable always means a {0, 1}-valued uncertain variable. 3 Uncertain Second-order Logic In this section, we introduce some concepts and symbols of uncertain second-order logic. Firstly, let U be a set of some {0, 1}-valued uncertain variables satisfies (1). 0 U. (2). if ξ U, then 1 ξ U. (3). if ξ, δ U, then ξ δ U. Here, all the uncertain variables ξ in U is defined on an uncertainty space (Γ,L,M), the symbol 0 U stands for the uncertain variable 0, such that, 0(γ) = 0, for all γ Γ. first of all, we have three nonempty sets which are a discourse of universe D, a predicate constant set F and a second-order predicate constant set S. Definition 6. An uncertain second-order logic proposition is a sequence of uncertain propositions indexed by one or more parameters, which are in the following form: (1 ). for any ξ F, F is the predicate constant set, ξ is a map from some product spaces of D to U, that is, ξ(a 1, a 2,, a m ) is an uncertain variable in U for any a 1, a 2,, a m D. (2 ). for any A S, S is the second-order predicate constant set, A is a map from some product spaces of F to D, that is, A(ξ 1, ξ 2,, ξ m ) is an uncertain variable in U for any ξ 1, ξ 2,, ξ m F.

3 UNCERTAIN SECOND-ORDER LOGIC 261 For example, A(ξ) is an uncertain second-order proposition, while ξ(a) is also an uncertain second-order proposition. which are uncertain variables in U. An element in F is called an uncertain predicate constant and an element in S is called an uncertain second-order predicate constant. Definition 7. An uncertain second-order formula defined as a member of the minimal set S of finite sequence of primitive symbols satisfying: (a ). ξ(a 1, a 2,, a m ) S and A(ξ 1, ξ 2,, ξ m ) S for each uncertain predicate proposition. (b ). if X S, then X S. (c ). if X S and Y S, then (X Y ) S. (d ). if X S, then ( x D)X S, where D is the domain of discourse, and x is an arbitrary variable symbol. (e ). if X S, then ( δ F )X S, where F is the predicate constant set and δ is a variable symbol. The symbol means not, if the uncertain second-order formula ξ(a) stands for Beijing is a big city, then ξ(a) means Beijing is not a big city. The symbol means or, if ξ(b) stands for Shanghai is a big city, then ξ(a) ξ(b) means Beijing is a big city or Shanghai is a big city. The symbol means for any or all the, if ξ(x) stands for x is a big city, then xξ(x) means for any x, x is a big city. We next introduce several abbreviations and some new symbols. Definition 8. Let X and Y be uncertain second-order formulas, x is a variable symbol, the connectives symbols,, are defined as follow. (X Y ) stands for ( X Y ) (X Y ) stands for ( X Y ) ( x)a stands for ( x) A. The symbol means and, the uncertain second-order formula ξ(a) ξ(b) means Beijing is a big city and Shanghai is a big city. The symbol means if...then..., if ξ(b) ξ(a) means If Shanghai is a big city, then Beijing is a big city. The symbol means There is a, if ξ(x) means x is a big city, then xξ(x) stands for there is an x, x is a big city. By now, uncertain second-order formula is given. 4 The Truth Value of Uncertain Second-order Formula Definition 9. Let X be an uncertain second-order formula. The truth value of the uncertain second-order formula X is defined as T (X) = M{X = 1}, where an uncertain second-order formula is an uncertain variable in U, which can be derived from following rules: 1. For any uncertain second-order propositions ξ(a 1, a 2,, a m ) and A(ξ 1, ξ 2,, ξ m ), are uncertain variables in U. 2. if X S is an uncertain variable, then formula X S is the uncertain variable 1 X U. 3. if X S and Y S are uncertain variables, then X Y is the uncertain variable max{x, Y } U. 4. if for all x D, A(x) is an uncertain variable, then (x D)A(x) is the uncertain variable min x D {A(x)} D. 5. if for all δ F, A(δ) is an uncertain variable, then (δ F )A(δ) is the uncertain variable min {A(δ)}. One can check that, every uncertain second-order formula is an uncertain variable the definition T (X) = M{X = 1}. Theorem 1. Let X and Y be uncertain second-order formulas. The corresponding uncertain variables of some formulas are shown as follow. (X Y ) is min{x, Y } ( x D)Y is max{y }. x D ( δ F )Y is max{y }. Proof. The uncertain second-order formula (X Y ) is the abbreviation of ( X Y ), then we have (X Y ) is an uncertain variable 1 max{1 X, 1 Y } = min{x, Y }. The uncertain second-order formula ( x D)Y is the abbreviation of ( x D) Y, then we have ( x D)Y is an uncertain variable 1 min{(1 Y )} = max{y }. x D x D The left part may be proved by a similar way. The theorem is proved. 5 Some Basic Laws in Second-order Logic Theorem 2. For any uncertain second-order formula A, we have (i) (Law of Excluded Middle) T (A A) = 1. (ii) (Law of Contradiction) T (A A) = 0. (iii) (Law of Truth Conservation) T (A) + T ( A) = 1. Proof. (i) Note that A is an uncertain variable A is also an uncertain variable satisfies A = 1 if and only if A = 0. From the definition of truth value and the basic property of uncertain measure, we have T ((A A) = M{(A A = 1} = M{max{A, 1 A} = 1} = M{{A = 1} {A = 0}} = M{Γ} = 1,

4 262 ZIXIONG PENG & SAMARJIT KAR where Γ denotes the universal set of uncertainty space. For example, an uncertain second-order formula like P ap (a). The Law of Excluded Middle is written as: Theorem 4. For any uncertain second-order formulas ( δ F )δ(a) and ( X S )X (ξ) (ii) We have T ( P ap (a) P ap (a)) = 1. we have and T (ξ(a) ( δ F )δ(a)) = 1 T (A(ξ) ( X S )X (ξ)) = 1. (iii) We have T (A A) = M{(A A = 1} The theorem is proved. = M{min{A, 1 A} = 1} = M{ } = 0. T ( A) = M{ A = 1} = M{A = 0} = 1 M{A = 1} = 1 T (A). Theorem 3. For any uncertain second-order formulas we have and ( δ F )δ(a) and ( X S )X (ξ) T (( δ F )δ(a) ξ(a)) = 1 T (( X S )X (ξ) A(ξ)) = 1. Proof. It follows from the definition of truth value and the basic property of uncertain measure that T (( δ F )δ(a) ξ(a)) = M{( δ F )δ(a) ξ(a) = 1} = M{ ( δ F )δ(a) ξ(a) = 1} (1 min{δ(a)}) ξ(a) = 1 max {1 δ(a)} ξ(a) = 1. If max{1 δ(a)} = 0, then we have 1 ξ(a) = 0 from ξ F. Thus we have M { } max{1 δ(a)} ξ(a) = 1 = M{Γ} = 1. That is, T (( δ F )δ(a) ξ(a)) = 1. The other part may be proved by a similar way. The theorem is proved. Proof. It follows from the definition of truth value and the basic property of uncertain measure that T (ξ(a) ( δ F )δ(a)) = M{ξ(a) ( δ F )δ(a) = 1} = M{ ξ(a) ( δ F )δ(a) = 1} (1 ξ(a)) max {δ(a)} = 1. If max{δ(a)} = 0, then we have ξ(a) = 0 from ξ F, which leads 1 ξ(a) = 1. Thus we have M { } (1 ξ(a)) max {δ(a)} = 1 = M{Γ} = 1. That is, T (( δ F )δ(a) ξ(a)) = 1. The other part may be proved by a similar way. The theorem is proved. 6 Conclusion In this paper, uncertain second-order logic is introduced as an extension of uncertain propositional logic and uncertain predicate logic. The concepts of uncertain second-order logic proposition, uncertain second-order formula and truth value are proposed. Acknowledgments This work was supported by National Natural Science Foundation of China Grant No References [1] Chen, X., Ralescu, D., A note on truth value in uncertain logic, Proceedings of the Eighth International Conference on Information and Management Sciences, Kunming, China, July 20-28, 2009, [2] Chen, X., Liu, B., Existence and uniqueness theorem for uncertain differential equations, Fuzzy Optimization and Decision Making, vol.9, no.1, 69-81, 2010 [3] Gao, X., Gao, Y., Ralescu, D., On Liu s inference rule for uncertain systems, International Journal of Uncertainty, Fuzziness and Knowledge- Based Systems, vol.18, no.1, 1-11, [4] Gao, X., Some properties of continuous uncertain measure, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, vol.17, no.3, , 2009.

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