Consistency degrees of theories in some systems of propositional fuzzy logic

doi:10.1016/j.fss.2004.11.005

Fuzzy Sets and Systems

Volume 152, Issue 2, 1 June 2005, Pages 321-331

https://doi.org/10.1016/j.fss.2004.11.005 Get rights and content

Abstract

Based on the analysis of relationships among the properties of consistent, inconsistent and fully divergent theories, and by means of the concept of divergence degrees, the present paper proposes a new membership function for reflecting the consistency degrees of general theories in four different types of fuzzy logic systems, i.e., Łukasiewicz fuzzy logic system Luk, Gödel fuzzy logic system Göd, Product fuzzy logic system Pro and the $R_{0}$ -fuzzy logic system $L^{*}$ .

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The additive transitivity is adopted in this study to measure consistency degrees of FPRs. The consistency measure methods based on other concepts such as divergence degrees [38,47,48] deserve further study. Ting Huang: Investigation, Writing – original draft.
This study investigates linguistic information-based granular computing for supporting multi-attribute group decision-making where relative importance of experts/attributes, and outcomes of pairwise comparisons over alternatives are recorded in the form of distributed linguistic preference relations. First, we propose two linguistic information granulation models based on the measures of consistency and consensus to realize the operability of two types of linguistic information, respectively. The granulation of the weight linguistic information is closely linked with that of the alternative preference linguistic information in a way that two sets of control factors are introduced. Second, an estimation method for expert/attribute weights is developed based on the optimal control factors and solutions to the aforesaid linguistic information granulation models. The proposed linguistic information granulation models are in fact constrained non-linear optimization problems which may be neither effectively nor efficiently solved by general optimization methods. Third, we therefore propose a novel solution approach named tournament selection operator-guided particle swarm optimization (TSOG-PSO). Compared with the standard PSO and its variants, the TSOG-PSO enjoys more feasibility and lower complexity. A new energy vehicle evaluation problem is analyzed using the proposals to demonstrate their applicability. Some discussions and comparative studies are conducted to illustrate the effectiveness of the proposals.
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The generalized truth degree of quantitative logic in the logic system L<inf>n</inf><sup>*</sup> (n-valued N M-logic system)
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Many-valued logic was far from the mainstreams of mathematical research until 1965, in which time, the “fuzzy boom” started with Zadeh’s paper “Fuzzy Sets” [7]. Since then, there appeared a number of papers on fuzzy logic and its applications, part of which are references [8–32]. [19,22,25]
The concept of the truth degree of a formula is the crucial tool and the building block in quantitative logic, from which the concept of logic metric in quantitative logic is derived. Logic metric takes an important role in quantitative logic, related to which are other concepts in quantitative logic such as divergence, consistency, etc. In the present paper, having combined the theory of generalized tautologies with the theory of truth degrees in quantitative logic, we have proposed the theory of $Σ_{Γ}$ -truth degrees of formulas related to theory $Γ$ in the logic system $L_{n}^{*}$ ( $n$ -valued $N M$ -logic system), and discussed some of its properties. With the help of the properties of $Σ_{Γ}$ -truth degrees: $τ_{Γ} (A) + τ_{Γ} (A \to B) \leq 1 + τ_{Γ} (B)$ , we have obtained the $Γ$ -logic metric on the set $F (S)$ of formulas in the propositional logic system $L_{n}^{*}$ ( $n$ -valued $N M$ -logic system). By the work of this paper we can generalize the theory of quantitative logic in all-round way and establish an approximate reasoning’s framework related to theory $Γ$ in the logic system $L_{n}^{*}$ ( $n$ -valued $N M$ -logic system).
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In this paper, we introduce the concept of the degree of the truth in the framework of so called many-valued propositional logic systems with the intention of measuring to what extent a given formula is true. Such a concept can indeed induce, in a very natural way, the degree of the similarity as well as to induce a pseudo-metric among formulas as the ‘graded’ version of the notion of the logical equivalence. The basic properties of such induced logic metric space hence are investigated in this paper. We further introduce different concepts of the degree of the divergence and the degree of the consistency in order to grade the extent of the consistency of a logic theory. To the end, all the basic logic notions are graded. Lastly, we propose in this paper three patterns of ‘approximate reasoning’ so far as the many-valued propositional logic systems are concerned. Hence, we can claim that we have laid the foundation for the “quantitative logic”. Since this theory has touched upon several key concepts of similarity, comparison, measurements, etc. in information theory, we are convinced that the theory may have profound impact in contemporary information technology. In addition, quantitative logic can be regarded as a highly representative example of mathematics of uncertainty capable of handling vagueness.
Consistency degrees of theories and methods of graded reasoning in n-valued R<inf>0</inf>-logic (NM-logic)
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In the present paper the idea of Wang [G.J. Wang, Theory of truth degrees of formulas in Łukasiewicz n-valued propositional logic and a limit theorem, Sci. China Inform. Sci. E 35(6) (2005) 561–569 (in Chinese)] is firstly extended to the n-valued R₀-logic $L_{n}^{*}$ and the concept of truth degrees of formulas in $L_{n}^{*}$ is proposed. A limit theorem saying that the truth function τ_n induced by truth degrees converges to the integrated truth function τ when n converges to infinity is obtained. This theorem builds a bridge between discrete valued R₀-logic and continuous valued R₀-logic. Secondly, based on deduction theorem, completeness theorem and the concept of truth degrees of formulas in $L_{n}^{*}$ , the concept of consistency degrees of theories is given. It is proved that a theory Γ over $L_{n}^{*}$ is a useless theory(i.e., the deductions of Γ are all tautologies) iff the consistency degree consist_n(Γ) of Γ is equal to 1, Γ is consistent iff $\frac{1}{2} ⩽ {consist}_{n} (Γ) ⩽ 1$ , and Γ is inconsistent iff consist_n(Γ) = 0. Lastly, the concept of consistency degrees of theories is generalized and a method of graded reasoning in $L_{n}^{*}$ is obtained.
Generalized consistency degrees of theories w.r.t. formulas in several standard complete logic systems
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The present paper carries out a deep analysis on the inconsistency of theories by means of deduction theorems, completeness theorems and satisfiability degrees of formulas, and introduces the concept of the degree of entailment of a contradiction from a theory in classical two-valued logic system, Łukasiewicz fuzzy logic system, Gödel fuzzy logic system, product fuzzy logic system and the $R_{0}$ -fuzzy logic system. It is proved that in classical two-valued logic system, Łukasiewicz fuzzy logic system and the $R_{0}$ -fuzzy logic system, respectively, the concept of consistency degrees of theories defined by the divergence degrees of theories in [X.N. Zhou, G.J. Wang, Consistency degrees of theories in some systems of propositional fuzzy logic, Fuzzy Sets and Systems 152 (2005) 321–331; H.J. Zhou, G.J. Wang, A new theory index based on deduction theorems in several logic systems, Fuzzy Sets and Systems 157(2006) 427–443] is reasonable and can accurately measure the consistency degrees of theories. The concept of the degree of entailment of a contradiction from a theory is generalized by replacing the contradiction with a general formula and then the generalized consistency degrees of theories w.r.t. general formulas in the above-mentioned logic systems is established.

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^☆: Supported by the NSF of China.

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Consistency degrees of theories in some systems of propositional fuzzy logic☆

Abstract

Fuzzy Sets and Systems

Fuzzy Sets and Systems

Algebraic Foundations of Many-Valued Reasoning

Metamathematics of Fuzzy Logic