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Quantitative Method Based on Cotangent Similarity Degree in Three‐Valued Łukasiewicz Logic

2021; Institution of Engineering and Technology; Volume: 30; Issue: 1 Linguagem: Inglês

10.1049/cje.2020.11.011

2075-5597

Peng Yu,

Multi-Criteria Decision Making

Chinese Journal of ElectronicsVolume 30, Issue 1 p. 134-144 Original Research PaperFree Access Quantitative Method Based on Cotangent Similarity Degree in Three-Valued Łukasiewicz Logic Yu Peng, Corresponding Author Yu Peng yupeng@sust.edu.cn School of Arts and Sciences, Shaanxi University of Science and Technology, Xi'an, 710021 ChinaSearch for more papers by this author Yu Peng, Corresponding Author Yu Peng yupeng@sust.edu.cn School of Arts and Sciences, Shaanxi University of Science and Technology, Xi'an, 710021 ChinaSearch for more papers by this author First published: 01 January 2021 https://doi.org/10.1049/cje.2020.11.011 This work is supported by the National Natural Science Foundation of China (No.61976130, No.61871260), Scientific Research Program Funded by Shaanxi Provincial Education Department (No.18JK0099), and Natural Science Foundation of Shaanxi Province (No.2020JQ-698). AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract The main purpose of this paper is to establish a type of quantitative model by using the contangent similarity function in the three-valued Łukasiewicz propositional logic system Ł3. We introduce the concepts of the cotangent similarity degree, cotangent pseudo-distance and cotangent truth degree of the propositions, together with their basic properties in Ł3. We investigate the relationship between the cotangent truth degree and contangent pseudo-distance, and prove the continuity of the logical connectives and in the logical metric space. We propose a graded reduction method and three types of graded reasoning frameworks on the propositions set F(S), and provide several examples and basic properties of it. I. Introduction Given two propositions and , we can observe that the former is reliable and the latter is unreliable because the former is always true and the latter is always false, regardless of the truth valuation of p is. In general, in any one of the propositional logic systems, we can view tautologies as reliable propositions and contradictions as unreliable propositions. Then, the following question arises: what are the reliabilities of the propositions and This leads to the question of the manner in which to distinguish the reliability of a formula. In fact, the method of distinguishing the reliability of a formula was first proposed by Roser and Turquette in 1952[1]. This issue subsequently attracted attention from numerous scholars, and various solutions emerged[2-7]. Representative research results included the truth assignment method proposed by Pavelka[2]; the probabilistic logic approach proposed by Adam[3]; the state theory in MV algebra proposed by Mundici[4]; the fuzzy probabilistic logic method proposed by Flaminio [5]; the possibilistic logic approach proposed by Dubios[6]; and the credibilistic logic method proposed by Lee[7]. Quantitative logic, which is a new method for describing the reliability of a formula, was proposed by Wang[8-13]. The core of this logic lies in the formula truth degree, which not only can distinguish the reliability of the formula, but can also be used to describe the compatibility of the theory. From the existing quantitative logic literature[14-17], it can be found that quantitative logic is applicable to common logic systems, such as the classical logic system L, fuzzy logic system uk and L*(NM logic), as well as for defining the similarity degree and distance between φ and φ by calculating the truth degree of . However, it cannot be applied to the more extensive MTL(Monoidal t-norm based Logic)logic system, because for a certain left continuous t-norm, the triangular inequalities making up the distance no longer hold. For example, the operator 0 can be defined as follows: We can prove that ⊗ is a left continuous t-norm, but not a strong regular one. We can not establish the distance between formulae by using this operator and its implication operator. In order to overcome these deficiencies, by limiting the implication operator to a strong regular implication operator, a type of special truth degree theory in the MTL logic system was established[18-20], which was referred to as the SMTL (Strong monoidal t-norm based Logic) logic system. However, the strong MTL logic system remained unable to eliminate this flaw. In order to establish and apply graded reasoning over a wider range, in this paper, we use the cotangent similarity function as a tool to define the cotangent similarity degree and cotangent distance between formulae, thereby proposing a new quantitative model distinguished from quantitative logic, and attempt to introduce the fuzzy similarity degree into the research on multi-valued logic. Moreover, we introduce two specific applications of this quantitative model, graded reduction, and three types of graded reasoning frameworks on the formula set F(S). Furthermore, the cotangent truth degree defined in this paper does not meet the axiomatic definition of the probability truth degree in Ref.[10], and is an effective extension of quantitative logic. II. Cotangent Similarity Measures Between Logical Formulas In this section, we outline several basic concepts that will be used in the following. Definition 1[21, 22] Let U be an ordinary set, F(U) denote the set of all fuzzy sets on , , and : be a mapping. If S satisfies: S1) ; S2) if and only if ; S3) ; and S4) if , , then S is known as the similarity measure between fuzzy sets A and B. Let U be an nonempty finite set, and . Then, we can prove that is a similarity measure on fuzzy sets satisfying definition 1, which we refer to as a cotangent similarity function[23]. Next, we use the above cotangent similarity function to establish a type of quantitative model on formula set F(S) in the three-valued Łukasiewicz propositional logic system Ł3. Suppose that and F(S) is the free algebra of type generated by S, where is a unary operator, and ∨ and → are two binary operators on F(S). Then, the elements of S are atomic propositions (atomic formulae), while the elements of F(S) are propositions (formulae). Assume that , and then define on I3 a unary operator and two binary operators ∨ and → as follows: where . Then, I3 is an algebra of type , known as the valuation domain of logical system Ł3. A homomorphism : of type from F(S) into I3 is known as a valuation of F(S). Moreover, is known as a valuation of φ, and the set of all valuations of F(S) will be denoted by , is referred to as a tautology if for every . Furthermore, φ is referred to as a contradiction if for every . Let and , and φ are said to be logically equivalent if for every , denoted by . Assume that is a finite atomic propositional set, and let be a formula containing m atomic formulae . represents the formulae generated by Sm, while Ωm represents the set of all valuations of F(Sm). Let , then, we obtain a vector in . Conversely, for every there exists only one such that . Hence, there is a one-to-one mapping between v and , and can be expressed as . In the following, for the convenience of description, we denote Ωm by Um. Definition 2 Let , , and define as then the formula φ can be viewed as the three-valued fuzzy set on domain Um. Definition 3 Let , and define then, is known as the cotangent similarity degree between formulae φ and φ. Theorem 1 Suppose that , , and , then , and Remark 1 The proof of Theorem 1 is similar to the proof of theorem 1 in Ref.[24] and theorem 1 in Ref.[25]. Example 1 In Ł3, suppose that . Let us calculate the cotangent similarity degree between formulae and . According to Definitions 2 and 3, we have , , . Then Proposition1 Let ; then i) ; ii) if and only if , and and are both Boolean formulae, particularly where is a contradiction and T is a tautology; iii) if and only if iv) v) ; vi) vii) viii) ix) x) if then xi) if , then xii) . Proof i) This is demonstrated from Definition 3. ii) Therefore, if and only if , and and are Boolean formulae. In particular, and T are Boolean formulae, and , . iii) If , for every , we have . Otherwise, there exists , such that . Furthermore, we have , , which contradicts . Therefore, when , , . In contrast, when , we have for every , iv) v) We can prove that when , . Hence vi) This can be derived from the inequality and the proof of v). vii) Similar to the proof of iv). viii) This can be proven by the inequality and monotonicity of the function on the [0, 1] interval. ix) This can be proven by the inequality and monotonicity of function on the [0, 1] interval. x) . As we have . Furthermore, we obtain . Moreover, the function is a strict monotonically decreasing function on the [0,1] interval, and we have , . xi) This can be proven by the inequality () and monotonicity of function on the [0,1] interval. xii) We first prove when the following inequality holds: Case 1: Any two values in a, b and c are equal. When a = b, we have hence, inequality (1) holds. When and , similarly, inequality (1) is valid. Case 2: a, b and c are not equal to one another. Firstly, when or , it is easy to observe that inequality (1) holds. Secondly, when or , we have Hence, inequality (1) holds. Now, we prove vii). , we have Furthermore, III. Cotangent Truth Degree of Logical Formula In this section, we use the cotangent similarity degree to define the concept of the cotangent truth degree of a formula, which can be used to distinguish the reliability of different formulae. Definition 4 Let , and be a tautology, define Then, is known as the cotangent truth degree of formula φ. A tautology is a reliable formula. A formula compared to a reliable formula can naturally express the reliable or unreliable degree of this formula. Hence, the definition of the cotangent truth degree of a formula is reasonable. Proposition 2 Let , then Remark 2 In the classical propositional logic system L, if we define the truth degree of formula φ as Definition 4, according to Proposition 2, the cotangent truth degree of formula φ is equal to , which is precisely the correct truth degree of formula φ in quantitative logic[2]. Example 2 In Ł3, calculate the cotangent truth degrees of the following formulae: i) ii) . Solution i) In this case, . It can be verified that, when , , , , , , , , , , , , , , . When , , , , , , . According to Proposition . ii) It can be verified that, only when , . In order to make , the component of x must take a value in , and there exists at least one component that is equal to . It can be verified that the number of all vectors x satisfying the above condition is . According to Proposition 2, Proposotion3 Let , then i) ii) φ is tautology if and only if ; iii) φ is contradiction if and only if ; iv) If φ and φ are logical equivalence, then ; v) If , then ; vi) vii) viii) ix) x) xi) here, ; xii) Proof i), ii), iii), iv) can be proven by the Definition 4 and Proposition 2. v) As , we have and . Taking then . Assume that , then . vi) Firstly, Moreover, . Hence, we have vii) viii) and ix) are obviously the corollary of vii). x) As is a topology in 3, and according to v)in Proposition 3 we have . xi) This is the same as the proof of x). xii) Firstly, Secondly, Finally, Proposition 4 Let and , then IV. Contangent Distance Among Formula Set F(S) In this section, we introduce the concept of the cotangent distance on formula set F(S), and establish the logical metric space . Thereafter, we investigate the relationship between the cotangent distance ρCot and cotangent truth degree τCot, as well as the continuity problems of the logical connection in . Theorem 2 Let and , then, is the distance between formulae φ and φ. We refer to this as the cotangent distance, while is known as the logical metric space. Remark 3 i) In the above theorem, the similarity degree ξCot satisfies i), iii), and xii) in Proposition 1. Hence, ρCot is the distance between the formulae φ and φ. However, this distance is only the pseudo-distance among the formula set F(S), because two formulae with a zero distance may be equal or only have a logical equivalence. ii) It can be observed from Theorem 2 and Definition 3 that the method for defining the similarity degree and distance of formulae in this paper can be applied to other logic systems, particularly the MTL logic system, which weakens the condition that the t-norm operator must be a regular operator[14, 15], thereby improving its results and offering an advantage. Proposition 5 Let and be a tautology, then i) ii) if and only if iii) iv) v) vi) vii) viii) ix) If then x) If , then Now, we provide the relationship between the cotangent distance and cotangent truth degree . Proposition 6 Suppose that , then Let Γ be a theory of F(S), and D(Γ) denotes the all conclusitions of , , then we define the distance between and as . Proposition 7 Suppose that , i) if , then ; ii) if , then . Here, Proof i) Firstly, we prove that , if , As is a tautology in Ł3, by the MP inference rule, if is a tautology, then is a tautology. Furthermore, according to v) in Proposition 3, we obtain . In this case, Hence when is a tautology, Secondly, as is a tautology for every , we have , and , which demonstrates that is the minimum value in set . Hence, . ii) As is a tautology in Ł3, we have which is a monotonically decreasing sequence in the logical metric space , and . The limit of sequence exists, denoted by , that is . Because , there exists such that , therefore . Proposition 8 and are continuous operators in the logical metric space . Proof Assume that , and 1) According to Proposition 5, we have Hence is continuous with respect to ρcot. 2) , and from , we can observe that there exists From Proposition 5, we have Taking , then when Hence ∨ is continuous with respect to ρcot. 3) From vii) in Proposition 5, we have Hence → is continuous with respect to ρCot. V. Graded Reduction Theory Based on Cotangent Truth Degree In propositional logic, if the proposition set Γ is provided, and D(Γ) represents the set of all conclusions of Γ, the complexity of D(Γ) is closely related to Γ. Therefore, it is quite necessary to simplify proposition set Γ before discussing D(Γ). In the following, we provide a new type of method for studying this problem by using the cotangent truth degree, which can be viewed as a specific application of cotangent truth degree. Definiton 5 Assume that . If , we say that φ is a reducible element of Γ; otherwise, it is an unreducible element. If any formula contained in Γ is an unreducible element, we refer to Γ as an independent proposition set. Obviously, if Γ is an independent proposition set, then any subset of Γ is also an independent proposition set. Definiton 6 Let , if and is an independent proposition set, then Γ0 is known as a reduction of Γ. The above definition is the classical reduction theory of the proposition set. In the following, we provide another reduction method by using the formula cotangent truth degree. Theorem 3 Suppose that Γ is a finite proposition set, Then if and only if . Proof As , the necessity is obvious; thus, we only prove the sufficiency. Assume that . For every , by means of the generalized deductive theorem in 3, we have . According to Proposiiton 3, we can observe that and are tautologies in Ł3. From the HS inference rule, we determine that is a tautology, therefore Definiton 7 Suppose that , . If , then φ is known as a truth degree reducible element of Γ, otherwise φ is a truth degree unreducible element of Γ. Definiton 8 Suppose that . If and such that ,then is called a truth degree reduction of Γ. Example 3 In Ł3, let , then , but , and , hence is a truth degree reduction of . Theorem 4 Suppose that . Then Γ0 is a truth degree reduction of Γ if and only if and is an independent proposition set. Proof Necessity. Because Γ0 is a truth degree reduction of Γ; that is holds owing to . Meanwhile ; hence, . However, as Γ0 is a truth degree reduction of , such that , is an independent proposition set. Sufficiency. Because , we have for every . Furthermore, Γ0 is an independent proposition set, and we have . Let then, holds, but does not hold, and is a truth degree reduction of Γ Theorem 5 Suppose that , then the truth degree reduction of Γ always exists. Proof If , we have , and itself a truth degree reduction of If there exists such that , then studying . If , then is a truth degree reduction of . Theorem 6 Γ0 is a truth degree reduction of Γ if and only if i) ; ii) ; iii) . Proof The sufficiency is obvious. We only prove the necessity. Let . Assume that there exist such that , by ii) in Proposition 3, is a tautology in 3. This means that any element in Γ0 can be inferred by . Furthermore, for every can be inferred by . Since Γ0 is a truth degree reduction of Γ, by Theorem 4, . Therefore, for every , we have , this contradicts with such that . Proposition 9 Suppose that If and , then is a truth degree reduction of Γ. Definition 9 Suppose that are all truth degree reductions of Γ, and let Then, and are known as the core set, relative essential set, and unnecessary set of the truth degree reduction of Γ, respectively. Moreover, , , and are known as the core element, relative essential element, and unnecessary element of the truth degree reduction of Γ. Theorem 7 Suppose that are all truth degree reductions of Γ. Then i) if and only if ii) if and only if and ; iii) if and only if for an arbitrary truth degree reduction of , \vspace*{0.5mm} Proof i) Suppose that , then . If . According to Theorem 5, there exists such that Γi is a truth degree reduction of Γ, which conflicts with . Therefore, . However, suppose that , then cannot hold. For an arbitrary , is not the truth degree reduction of Γ. Therefore, φ belongs to the arbitrary truth degree reduction of . ii) Necessity. Suppose that , then and . Meanwhile, there exists at least Γi of Γ such that . Hence , . Sufficiency. Because , we\vspace*{0.5mm} have . However, , so\vspace*{0.5mm} there exists at least of such that , and we have . Hence . iii) Necessity. This follows immediately from the Definitions 6 and 7. Sufficiency. Suppose that . Then, there exists of , which is a truth degree reduction of , such that , hence which is a contradiction. In fact, the truth degree reduction method proposed in this paper is based on an accurate reduction of the proposition set, which limits the serviceable reduction range. The reason for this is that the reduction of Γ is not required to be accurate in actual applications. Naturally, the problem of the graded reduction of the proposition set is proposed, which is the starting point for the following two definitions. Definition 10 Suppose that . If , is known as an α-reducible formula; if there exists such that , is known as an -unreducible formula. Definition 11 Suppose that , . If , and such that , is known as an α-truth degree reduction of Γ. Example 4 In Ł3, let , let , . Then . But . Hence, when is an α-truth degree reduction of Γ. From Definitions 8 and 11, and Example 4, we can observe that the α-truth degree reduction method defined in this paper achieves the goal of graded reduction. It lays the foundation for graded reasoning on the formula sets. VI. Graded Reasoning Theory Based on Cotangent Truth Degree In order to describe the degree of a statement inferred by a premise set x, Chakraborty introduced the graded consequence method in Ref.[26,27]. However, when studying this approach in multi-valued propositional logic, we identified several deficiencies. The graded consequence method does not consider the existence of a generalized deduction theorem in the multi-valued proposition logic system, resulting in certain shortcomings. Let , in . For example, according to the graded consequence method, we determine that the degree of φ inferred by is , and can not be inferred by Γ (for further details, please refer to the literature [26,27]). However, according to the generalized deductive theorem in Ł3, we know that , and the above definition exhibits certain flaws. Fortunately, the method proposed in this paper can improve this deficiency. Definition 12 In Ł3, let , and define then is known as the degree of χ inferred by Γ. Example 5 Let ; then, . The degree of q → p inferred by is . Proposition 10 In Ł3, let be finite theories of F(S). Then i) if , then ; ii) if , then iii) if , then ; iv) if , then ; v) if , then ; and vi) if , then Γ is an inconsistency theory; otherwise, Γ is a consistency proposition set. Proof i) and ii) are proven from Definition 12 and the generalized deduction theorem of Ł3. iii) In Ł3, we have Furthermore, iv) Because is a axiom in . We have and . Therefore However, is an axiom in , and from vi) in Proposition 3, we can observe that . Hence, . v) This can be proven by and being tautologies in and v) in Proposition 3. vi) If from iii) in Proposition 3, we determine that is a tautology in Ł3. According to the generalized deduction theorem, we can observe that , and Γ is an inconsistency theory. If then is not the conclusion of Γ, Γ is a consistency theory. Finally, we propose three types of graded reasoning frameworks on formula set F(S) by using the cotangent truth and cotangent distance, and study the relationship between the grade reduction method and the three types of graded reasoning frameworks. Definition 13 Let , i) if , then is known as the I-type error of theory Γ, denoted by ; ii) if , then is known as the II-type error of theory , denoted by ; iii) if , then is known as the III-type error of theory , denoted by . Here, . Theorem 8 In , if and only if Proof Theorem 9 In Ł3, If , Proof Because is a tautology in Ł3,when , we determine that is a Tautology. By means of the MP inference rule, we determine that is a tautology in Ł3. Furthermore, according to v) in Proposition 3, we can observe that . , we have . Furthermore, . Hence . Owing to , we have The following is an important property that establishes the relationship between the α-truth degree reduction and three types of graded reasoning frameworks. Theorem 10 In Ł3, suppose that , is a α-truth degree reduction of Γ and , then . Proof When Γ0 is an α-truth degree reduction of , Meanwhile , , and we have , Therefore, Theorem 11 Let be an α-truth degree reduction of Γ and . Then . Proof Therefore, VII. Conclusions In this paper, we have introduced the concepts of the cotangent similarity degree, cotangent truth degree, and cotangent distance on the formula set F(S). The basic properties of these concepts were presented. Moreover, we proved the continuity of the different logical connections in the logical metric space (F(S), ρCot). Finally, based on the cotangent truth degree, we proposed a type of graded reduction method and three types of graded reasoning frameworks, and the link between these was investigated. We obtained several interesting conclusions and effectively expanded the application scope of quantitative logic. Biography YU Peng (corresponding author) was born in 1981. He received the Ph.D. degree in mathematics from shaanxi normal University in 2019. He is currently an associate professor of Shaanxi University of Science and Technology. His research interests include non-classic logic, fuzzy reasoning and rough set. 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