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

The paper considers some of the main trends of the recent development of mathematical fuzzy logic as an important tool in the toolbox of approximate reasoning techniques. Particularly the focus is on fuzzy logics as systems of formal logic constituted by a formalized language, by a semantics, and by a calculus for the derivation of formulas. Besides this logical approach also a more algebraic approach is discussed. And the paper ends with some hints toward applications which are based upon these theoretical considerations. 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

INTRODUCTION Advanced sensorimotor devices, like mobile robots, are often referred to as autonomous systems. The expression is used to intentionally remark on the difference between these systems and those of traditional industrial automation. Although no rigorous definition is easily available, a general informal consensus seems to exist on the features that denote autonomy: a system is considered the more autonomous the more reliably it can survive and perform tasks in the real world, without the need for human intervention. 171 ____________________________ *e-mail: claudio.sossai@isib.cnr.it J. of Mult.-Valued Logic & Soft Computing., Vol. 9, pp. 171-194 2003 Old City Publishing, Inc. Reprints available directly from the publisher Published by license under the OCP Science imprint, Photocopying permitted by license only a member of the Old City Publishing Group In the literature different ideas and techniques have been proposed and investigated to achieve these results, nonethe

An approach to fixed point theory based on the notion of fuzzy order is proposed. Such an approach extends both the fixed point theory in ordered sets and the fixed point theory in metric spaces. This since the fuzzy orders are strictly related with the quasi-metrics as defined by A. K. Seda in [10]. The aim is to give new tools for logic programming and for approximate reasoning.

Fuzzy logic is now one of the leading and most successful methodologies for the treatment of the vagueness phenomenon. It is a well-established sound formal system with numerous applications. In recent years, several significant books have been published where fuzzy logic is investigated deeply and so, its

The paper discusses some open problems in the field of mathematical fuzzy logic which may have a decisive influence for the future development of fuzzy logic within the next decade. 1

Abstract. Let L be a complete residuated lattice. Then we show that any L-preorder can be represented both by an implication-based graded inclusion as defined [1] and by a similarity-based graded inclusion as defined in [2]. Also, in accordance with a duality between [0,1]-orders and quasi-metrics, we obtain two corresponding representation theorems for quasi-metrics.

Abstract. The paper concerns fuzzy logic programming. As an example, we show that is not restrictive to confine ourselves to fuzzy Herbrand interpretations in giving a semantics for fuzzy programs. Also, we show that the resulting apparatus gives a unifying theoretical framework for fuzzy control.

This is an extended abstract of the published paper “Approximate Similarity and Poincaré Paradox to be published in Notre Dame Journal of Formal Logic, 49, 2008”. The startin point is an observation by De Cock and Kerre, in which, in considering Poincaré paradox, one observes that the intuitive notion of “approximate similarity ” cannot be adequately represented by the fuzzy equivalence relations. In this note we argue that the deduction apparatus of fuzzy logic gives adequate tools to face the question with. Indeed a first order theory is proposed whose fuzzy models are plausible candidates for the notion of approximate similarity. A connection between these structures and the point-free metric spaces is also established.

The paper concerns some paradoxes arising from the vagueness. For example, the Heap paradox, the Bold Man paradox and Poincaré’s paradox of the indiscernibility are considered. The solutions proposed by fuzzy logic in the framework of approximate reasoning theory are exhibited. Riassunto. Il lavoro considera alcuni paradossi che nascono da nozioni vaghe. Ad esempio si esaminano il paradosso del mucchio di grano, il paradosso dell’uomo calvo ed il paradosso sugli indiscernibili di Poincaré. Si espongono le soluzioni proposte dalla logica fuzzy nell’ambito della teoria dei ragionamenti approssimati. 1