On contradiction in fuzzy logic

 We clarify which space of functions in [0, 1] ^E would be reasonable in fuzzy logic in order to avoid self-contradiction.     

On fuzzy equality and approximation in fuzzy logic

The paper is a contribution to the theory of fuzzy logic in narrow sense with evaluated syntax (FLn). We show that the concepts of fuzzy equality and the provability degree enable to generalize the concept of fuzzy approximation. In the second part of the paper we return to the Mamdani-Assilian formula, which is formed on the basis of the so called totally bounded fuzzy equality and using which we can approximate any function with the prescribed accuracy.This paper has been supported by Grant A1187901/99 of the GA AV R and the project VS96037 of MMT of the Czech Republic.     

On the implication operator in fuzzy logic

We investigate the different possible ways of translating an implication proposition in approximate reasoning. We study a general class of these operators.     

On the power of fuzzy logic

In this article we address the issues brought up by Elkan in his article, “The paradoxical success of fuzzy logic,” [IEEE Expert, 3–8 (1994)]. Elkan's work has caused concern since it purportedly reveals a Fuzzy Logic weakness regarding its theoretical foundations. A further investigation of Elkan's theorem (“Theorem 1”) revealed that its conclusion is not correct. After indicating the points where we disagree with Elkan, we reformulate Theorem 1, calling this new version “Theorem 2.” Theorems 1 and 2 have the same hypotheses but different conclusions. According to Theorem 2 there is a region of points that do hold the equivalence in the hypotheses of Theorem 1. In other words, one does not need to change the definition of logical equivalence in Theorem 1 in order to prove that Fuzzy Logic does not collapse to a two-valued logic. In a further analysis of Theorem 2 we show that Elkan's work does not affect the power of Fuzzy Logic to model vagueness. © 1996 John Wiley & Sons, Inc.     

Problem-oriented theorem-proving method in fuzzy logic (po-method)

Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 58–68, September–October, 1995.     

On the strict logic foundation of fuzzy reasoning

This paper focuses on the logic foundation of fuzzy reasoning. At first, a new complete first-order fuzzy predicate calculus system K^* corresponding to the formal system L^* is built. Based on the many-sort system K_ms^* corresponding to K^*, the triple I methods of FMP and FMT for fuzzy reasoning and their consistency are formalized, thus fuzzy reasoning is put completely and rigorously into the logic framework of fuzzy logic.The author is indebted to anonymous referee for his useful comments which have helped to improve the paper.     

On copulas, quasicopulas and fuzzy logic

We investigate (quasi)copulas as possible truth functions of fuzzy conjunction which is not necessarily associative and present some axiom systems for such fuzzy logics. In particular, we study an expansion of Łukasiewicz (infinite valued propositional) logic by a new connective interpreted as an arbitrary quasicopula (and also by a new connective interpreted as the residuum of the copula). Main results concern standard completeness.     

On linguistic approximation in the frame of fuzzy logic deduction

 This paper presents a new linguistic approximation algorithm and its implementation in the frame of fuzzy logic deduction. The algorithm presented is designed for fuzzy logic deduction mechanism implemented in Linguistic Fuzzy Logic Controller (LFLC).     

Jumps and logic in the law

The main stream of legal theory tends to incorporate unwritten principles into the law. Weighing of principles plays a great role in legal argumentation, inter alia in statutory interpretation. A weighing and balancing of principles and other prima facie reasons is a jump. The inference is not conclusive.To deal with defeasibility and weighing, a jurist needs both the belief-revision logic and the nonmonotonic logic. The systems of nonmonotonic logic included in the present volume provide logical tools enabling one to speak precisely about various kinds of rules about rules, dealing with such things as applicability of rules, what is assumed by rules, priority between rules and the burden of proof. Nonmonotonic logic is an example of an extension of the domain of logic. But the more far-reaching the extension is, the greater problems it meets. It seems impossible to make logical reconstruction of the totality of legal argumentation.The lawyers' search for reasons has no obvious end point. Ideally, the search for reasons may end when one arrives at a coherent totality of knowledge. In other words, coherence is the termination condition of reasoning. Both scientific knowledge and knowledge of legal and moral norms progresses by trial and error, and that one must resort to a certain convention to define what error means. The main difference is, however, that conventions of science are much more precise than those of legal scholarship.Consequently, determination of error in legal science is often holistic and circular. The reasons determining that a legal theory is erroneous are not more certain than the contested theory itself. A strict and formal logical analysis cannot give us the full grasp of legal rationality. A weaker logical theory, allowing for nonmonotonic steps, comes closer, at the expense of an inevitable loss of computational efficiency. Coherentist epistemology grasps even more of this rationality, at the expense of a loss of preciseness.     

Knowledge representation in fuzzy logic

The author presents a summary of the basic concepts and techniques underlying the application of fuzzy logic to knowledge representation. He then describes a number of examples relating to its use as a computational system for dealing with uncertainty and imprecision in the context of knowledge, meaning, and inference. It is noted that one of the basic aims of fuzzy logic is to provide a computational framework for knowledge representation and inference in an environment of uncertainty and imprecision. In such environments, fuzzy logic is effective when the solutions need not be precise and/or it is acceptable for a conclusion to have a dispositional rather than categorical validity. The importance of fuzzy logic derives from the fact that there are many real-world applications which fit these conditions, especially in the realm of knowledge-based systems for decision-making and control     

A self-organizing fuzzy logic controller for dynamic systems usinga fuzzy auto-regressive moving average (FARMA) model

The paper proposes a complete design method for an online self-organizing fuzzy logic controller without using any plant model. By mimicking the human learning process, the control algorithm finds control rules of a system for which little knowledge has been known. In a conventional fuzzy logic control, knowledge on the system supplied by an expert is required in developing control rules, however, the proposed new fuzzy logic controller needs no expert in making control rules, Instead, rules are generated using the history of input-output pairs, and new inference and defuzzification methods are developed. The generated rules are stored in the fuzzy rule space and updated online by a self-organizing procedure. The validity of the proposed fuzzy logic control method has been demonstrated numerically in controlling an inverted pendulum     

The chaining syllogism in fuzzy logic

We analyze the validity of the chaining syllogism in fuzzy systems, i.e., whether two fuzzy rules IF F, THEN G, and IF G, THEN H imply the rule IF F, THEN H. Conditions are given under which this basic deduction scheme holds. "If A is predicated of all B, and B of all C, A must necessarily be predicated of all C." ;-The chaining syllogism according to Aristotle's Prior Analytics.     

Some properties of fuzzy reasoning in propositional fuzzy logic systems

In order to analyze the logical foundation of fuzzy reasoning, this paper first introduces the concept of generalized roots of theories in ?ukasiewicz propositional fuzzy logic ?uk, Gödel propositional fuzzy logic Göd, Product propositional fuzzy logic Π, and nilpotent minimum logic NM (the R₀-propositional fuzzy logic L^∗). Next, it is proved that all consequences of a theory Γ, named D(Γ), are completely determined by its generalized root whenever Γ has a generalized root. Moreover, it is proved that every finite theory Γ has a generalized root, which can be expressed by a specific formula. Finally, we demonstrate the existence of a non-fuzzy version of Fuzzy Modus Ponens (FMP) in ?uk, Göd, Π and NM (L^∗), and we provide its numerical version as a new algorithm for solving FMP.     

Comments on “FuzzyShell: a large-scale expert system shellusing fuzzy logic for uncertainty reasoning” [and reply]

The above paper by Pan et al. (ibid. vol.6 (1998)) presents a rather complicated mechanism allowing fuzzy evidence to be aggregated when fuzzy inferences are made about the same fuzzy variable by different rules. The purpose of this comment is to present a simpler way to address this FuzzyCLIPS problem. In reply, the authors point out that the commenter's main conclusions about their work are ill founded. They believe that his erroneous conclusions were caused by his misunderstanding the difference between rule firing and evidence aggregation in their work     

On the monotonicity of (LDL) logic programs with set

LDL is one of the recently proposed logical query languages, which incorporate set, for data and knowledge base systems. Since LDL programs can simulate negation, they are not monotonic in general. On the other hand, there are monotonic LDL programs. This paper addresses the natural question of “When are the generally nonmonotonic LDL programs monotonic?” and investigates related topics such as useful applications for monotonicity. We discuss four kinds of monotonicity, and examine two of them in depth. The first of the two, called “ω-monotonicity”, is shown to be undecidable even when limited to single-stratum programs. The second, called “uniform monotonicity”, is shown to implyω-monotonicity. We characterize the uniform monotonicity of a program (i) by a relationship between its Bancilhon-Khoshafian semantics and its LDL semantics, and (ii) with a useful property called subset completion independence. Characterization (ii) implies that uniformly monotonie programs can be evaluated more efficiently by discarding dominated facts. Finally, we provide some necessary and/or sufficient, syntactic conditions for uniform monotonicity. The conditions pinpoint (a) enumerated set terms, (b) negations of membership and inclusion, and (c) sharing of set terms as the main source for nonuniform monotonicity.     

On the robustness of Type-1 and Interval Type-2 fuzzy logic systems in modeling

Research on the robustness of fuzzy logic systems (FLSs), an imperative factor in the design process, is very limited in the literature. Specifically, when a system is subjected to small deviations of the sampling points (operating points), it is of great interest to find the maximum tolerance of the system, which we refer to as the system’s robustness. In this paper, we present a methodology for the robustness analysis of interval type-2 FLSs (IT2 FLSs) that also holds for T1 FLSs, hence, making it more general. A procedure for the design of robust IT2 FLSs with a guaranteed performance better than or equal to their T1 counterparts is then proposed. Several examples are performed to demonstrate the effectiveness of the proposed methodologies. It was concluded that both T1 and IT2 FLSs can be designed to achieve robust behavior in various applications, and preference one or the other, in general, is application-dependant. IT2 FLSs, having a more flexible structure than T1 FLSs, exhibited relatively small approximation errors in the several examples investigated. The methodologies presented in this paper lay the foundation for the design of FLSs with robust properties that will be very useful in many practical modeling and control applications.     

Fuzzy logic and the calculi of fuzzy rules, fuzzy graphs, and fuzzy probabilities

The past few years have witnessed a rapid growth in the number and variety of applications of fuzzy logic, ranging from consumer products and industrial process control to medical instrumentation, information systems, and decision analysis. The foundations of fuzzy logic have become firmer and its impact within the basic sciences—and especially in mathematical and physical sciences—has become more visible and more substantive. And yet, there are still many misconceptions about the aims of fuzzy logic and misjudgments of its strengths and limitations.One of the common misconceptions is rooted in semantics: as a label, fuzzy logic, FL, has two different meanings. More specifically, in a narrow sense, fuzzy logic, FLn, is a logical system which aims at a formalization of approximate reasoning. In this sense, fuzzy logic is an extension of multivalued logic but its agenda is quite different from that of conventional multivalued systems.In a wide sense, fuzzy logic, FLw, is coextensive with fuzzy set theory, FST. Flw is far broader than FLn and contains Fln as one of its branches. Today, the term fuzzy logic is used predominantly in its wide sense. Thus, effectively, FL = FLw = FST.Another important point is that any field X can be fuzzified by replacing crisp sets in X by fuzzy sets. For example, through fuzzification, arithmetic can be generalized to fuzzy arithmetic, topology to fuzzy topology, control theory to fuzzy control theory, etc.In this perspective, the calculi of fuzzy rules (CFR), fuzzy graphs (CFG), and fuzzy probabilities (CFP) may be viewed as generalizations of the calculi of rules, graphs, and probabilities. The importance of CFR, CFG, and CFP derives from the fact that they play a central role in most of the applications of FL. In particular, the calculus of fuzzy graphs, which is a subset of the calculus of fuzzy rules, accounts for most of the applications of fuzzy logic in control, systems analysis, and related fields.Central to the calculus of fuzzy rules is a language referred to as the Fuzzy Dependency and Command Language (FDCL). The syntax of FDCL is concerned with the form of rules, while the semantics of FDCL is concerned with their meaning. An important issue in CFR is that of the induction of rules from observations.In CFG, a fuzzy graph is defined as the disjunction of Cartesian products of fuzzy sets. In effect, a fuzzy graph may be viewed as a compressed representation of a functional or relational dependence. Operations on fuzzy graphs play an important role in CFG.In the calculus of fuzzy probabilities, probabilities are assumed to be represented as fuzzy rather than crisp numbers. In a related way, probability distributions are represented as fuzzy graphs. A major aim of CFP is to provide a framework for linguistic decision analysis—a type of qualitative analysis in which fuzzy numbers and fuzzy graphs are employed to represent both probabilities and utilities.In an essential way, the methodologies of fuzzy rules, fuzzy graphs, and fuzzy probabilities reflect the fact that

1. (a) imprecision and uncertainty are pervasive; and

2. (b) precision and certainty carry a cost.

In the final analysis, the principal aim of these methodologies is to exploit the tolerance for imprecision and uncertainty to achieve tractability, robustness, and low solution cost.     

On choosing models for linguistic connector words for Mamdani fuzzy logic systems

We examine ten antecedent connector models in the framework of a singleton or nonsingleton fuzzy logic system (FLS), to establish which models can be used. In this work, a usable connector model must lead to a separable firing degree that is a closed-form and piecewise-differentiable function of the membership function parameters and also the parameter characterizing that connector model. Our analysis shows that: for a singleton FLS where the Mamdani-product or Mamdani-minimum implication method is used, all ten antecedent connector models are usable; for a nonsingleton FLS where the Mamdani-product implication method is used, only one antecedent connector model is usable; and for a nonsingleton FLS where the Mamdani-minimum implication method is used, none of the ten antecedent connector models is usable. We also show, by examples, that the parameter of the antecedent connector model provides additional freedom in adjusting a FLS, so that the FLS has the potential to achieve better performance than a FLS that uses the traditional product or minimum t-norm for the antecedent connections.     

Modifiers based on some t-norms in fuzzy logic

Modifiers generated by n-placed functions are considered. The subject matter of modifiers are fuzzy sets, i.e. membership functions defined on the unit interval I = [0, 1]. Two sets of modifiers are considered as example cases. One of them is a set of modifiers generated by t-norms and t-conorms. Here different dual pairs of norms create modifiers of different grade of strength. These norms are examples of two-placed functions. Another case is to generate a series of modifiers using only one DeMorgan class of norms. Norms are generalized to be n-placed functions. The place number n takes effect to the strength of a modifier. Two different DeMorgan classes are taken into the consideration. The first steps to the direction of many-valued modifier logics are taken.