The paper considers the fundamental notions of many- valued logic together with some of the main trends of the recent development of infinite valued systems, often called mathematical fuzzy logics. Besides this logical approach also a more algebraic approach is discussed. And the paper ends with some hints toward applications which are based upon actual theoretical considerations about infinite valued logics. Key words: mathematical fuzzy logic, algebraic semantics, continuous t-norms, left-continuous t-norms, Pavelka-style fuzzy logic, fuzzy set theory, non-monotonic fuzzy reasoning 1 Basic ideas 1.1 From classical to many-valued logic Logical systems in general are based on some formalized language which includes a notion of well formed formula, and then are determined either semantically or syntactically. That a logical system is semantically determined means that one has a notion of interpretation or model 1 in the sense that w.r.t. each such interpretation every well formed formula has some (truth) value or represents a function into

This thesis develops the idea of probabilistic arithmetic. The aim is to replace arithmetic operations on numbers with arithmetic operations on random variables. Specifically, we are interested in numerical methods of calculating convolutions of probability distributions. The long-term goal is to be able to handle random problems (such as the determination of the distribution of the roots of random algebraic equations) using algorithms which have been developed for the deterministic case. To this end, in this thesis we survey a number of previously proposed methods for calculating convolutions and representing probability distributions and examine their defects. We develop some new results for some of these methods (the Laguerre transform and the histogram method), but ultimately find them unsuitable. We find that the details on how the ordinary convolution equations are calculated are

There are many examples in the literature that suggest that indistinguishability is intransitive, despite the fact that the indistinguishability relation is typically taken to be an equivalence relation (and thus transitive). It is shown that if the uncertainty perception and the question of when an agent reports that two things are indistinguishable are both carefully modeled, the problems disappear, and indistinguishability can indeed be taken to be an equivalence relation. Moreover, this model also suggests a logic of vagueness that seems to solve many of the problems related to vagueness discussed in the philosophical literature. In particular, it is shown here how the logic can handle the sorites paradox. 1

a rule-based expert system. This expert system checks results to ensure their consistency, identifies whether the results come from arterial or venous vessels, and then produces an interpretation of their meaning. This `crisp' expert system was validated, verified and commercially released, and has since been installed at twenty two hospitals all around the United Kingdom. The assessment of umbilical acid-base status is characterised by uncertainty in both the basic data and the knowledge required for its interpretation. Fuzzy logic provides a technique for representing both these forms of uncertainty in a single framework. A `preliminary' fuzzy-logic based expert system to interpret error-free results was developed, based on the knowledge embedded in the crisp expert system. Its performance was compared against clinicians in a validation test, but initially its performance was found to be poor in comparison with the clinicians and inferior to the crisp expert system. An automatic tuni

This paper explores the stability of a class of feedback fuzzy systems. The class consists of generalized additive fuzzy systems that compute a system output as a convex sum of linear operators. Continuous versions of these systems are globally asymptotically stable if all rule matrices are stable (negative definite). So local rule stability leads to global system stability. This relationship between local and global system stability does not hold for the better known discrete versions of feedback fuzzy systems. A corollary shows that it does hold for the discrete versions in the special but practical case of diagonal rule matrices. The paper first reviews additive fuzzy systems and then extends them to the class of generalized additive fuzzy systems. The Appendix derives the basic ratio structure of additive fuzzy systems and shows how supervised learning can tune their parameters.

Soft computing (SC) is an association of computing methodologies that includes as its principal members fuzzy logic, neuro-computing, evolutionary computing and probabilistic computing. We present a collection of methods and tools that can be used to perform diagnostics, estimation, and control. These tools are a great match for real-world applications that are characterized by imprecise, uncertain data, and incomplete domain knowledge. We outline the advantages of applying SC techniques and in particular the synergy derived from the use of hybrid SC systems. We illustrate some combinations of hybrid SC systems, such as fuzzy logic controllers (FLCs) tuned by neural networks (NNs) and evolutionary computing (EC), NNs tuned by EC or FLCs, and EC controlled by FLCs. We discuss three successful real-world examples of SC applications to industrial equipment diagnostics, freight train control, and residential property valuation. 1. Introduction 1.1 Motivation There is a w...

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Introduction The study of human judgment under uncertainty has a history that is almost contemporaneous with that of probability theories. This is not a coincidence. From the outset, the idea of using probability to describe cognitive states or aspects of subjective judgment has provoked debate, theory construction, and empirical research. It is no exaggeration to say that probability theories have exerted a strong prescriptive influence on the study of judgment and decision making (see Gigerenzer 1994 [21] and Smithson 1989 [41] for overviews). In the modern era, proponents of the Subjective Expected Utility (SEU) framework advocated a version of Bayesianism as the benchmark for rational judgment and decision making, and this viewpoint dominated studies of human judgment and decision making during the 50's and 60's. By the late 70's and early 80's, some scholars had begun to question whether we should regard deviations from probability theories as "irrational" (cf. Cohen 1981

This thesis describes the architecture of learning systems which can explain their decisions through a rule-based knowledge representation. Two problems in learning are addressed: pattern classification and function approximation. In Part I, a pattern classifier for discrete-valued problems is presented. The system utilizes an information-theoretic algorithm for constructing informative rules from example data. These rules are then used to construct a computational network to perform parallel inference and posterior probability estimation. The network can be extended incrementally; that is, new data can be incorporated without repeating the training on previous data. It is shown that this technique performs comparably with other techniques including the backpropagation network while having unique advantages in incremental learning capability, training efficiency, and knowledge representation. Examples are shown of rulebased classification and explanation. In Part II, we present a metho...