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

Residuated fuzzy logic calculi are related to continuous t-norms, which are used as truth functions for conjunction, and their residua as truth functions for implication. In these logics, a negation is also definable from the implication and the truth constant 0, namely :' is ' ! 0. However, this negation behaves quite differently depending on the t-norm. For a nilpotent t-norm (a t-norm which is isomorphic to / Lukasiewicz t-norm), it turns out that : is an involutive negation. However, for t-norms without non-trivial zero divisors, : is Godel negation. In this paper we investigate the residuated fuzzy logics arising from continuous t-norms without nontrivial zero divisors and extended with an involutive negation. 1 1 Introduction Residuated fuzzy (many-valued) logic calculi are related to continuous t-norms which are used as truth functions for the conjunction connective, and their residua as truth functions for the implication. Main examples are / Lukasiewicz (/L), Godel...

Herbrand’s Theorem £¥ ¤ ¦ for, i.e., Gödel logic enriched by the projection § operator is proved. As a consequence we obtain a “chain normal form” and a translation of £ ¤ ¦ prenex into (order) clause logic, referring to the classical theory of dense total orders with endpoints. A chaining calculus provides a basis for efficient theorem proving.

Hypersequent calculi arise by generalizing standard sequent calculi to refer to whole contexts of sequents instead of single sequents. We present a number of results using hypersequents to obtain a Gentzen-style characterization for the family of Gödel logics. We first describe analytic calculi for propositional finite and infinite-valued Gödel logics. We then show that the framework of hypersequents allows one to move straightforwardly from the propositional level to first-order as well as propositional quantification. A certain type of modalities, enhancing the expressive power of Gödel logic, is also considered. 1

Entailment in propositional Gödel logics can be defined in a natural way. While all infinite sets of truth values yield the same sets of tautologies, the entailment relations differ. It is shown that there is a rich structure of infinite-valued Gödel logics, only one of which is compact. It is also shown that the compact infinite-valued Gödel logic is the only one which interpolates, and the only one with an r.e. entailment relation. 1

First-order Gödel logics are a family of infinite-valued logics where the sets of truth values V are closed subsets of [0,1] containing both 0 and 1. Different such sets V in general determine different Gödel logics GV (sets of those formulas which evaluate to 1 in every interpretation into V). It is shown that GV is axiomatizable iff V is finite, V is uncountable with 0 isolated in V, or every neighborhood of 0 in V is uncountable. Complete axiomatizations for each of these cases are given. The r.e. prenex, negation-free, and existential fragments of all first-order Gödel logics are also characterized.

A t-tautology (triangular tautology) is a propositional formula which is a tautology in all fuzzy logics defined by continuous triangular norms. In this paper we show that the problem of recognizing t-tautologies is coNP complete, and thus decidable. 1 Introduction Triangular Logics. A triangular logic is a propositional fuzzy logics whose truth functions are defined by continuous triangular norms (t-norms) [8, 9]; a formal definition is given in section 2. Triangular logics have attracted a lot of research in recent years, since on the one hand they retain an appealing theory akin to the theory of Boolean algebras in classical logic, while on the other hand they subsume major fuzzy formalisms such as / Lukasiewicz logic / L, Product logic \Pi, and Godel logic G. In turn, every continuous t-norm can be represented as the ordinal sum of the / Lukasiewicz, Product and Godel t-norms [7]. The tautologies of / L, \Pi, and G are coNP complete, and thus not harder than the classical prop...

We construct a faithful interpretation of / Lukasiewicz's logic in the product logic (both propositional and predicate). Using known facts it follows that the product predicate logic is not recursively axiomatizable. We prove a completeness theorem for the product logic extended by a unary connective 4 of Baaz [1]. We show that Godel's logic is a sublogic of this extended product logic. We also prove NP-completeness of the set of propositional formulas satisfiable in product logic (resp. in Godel's logic). 1 Introduction We shall be concerned with many-valued logics in this paper; in particular, in / Lukasiewicz's logic / L, Godel's logic G and product logic P. Our aim is to obtain information about complexity of these logics in terms of recursive theory (in the case of predicate logic) or in terms of computational complexity theory (in the case of propositional logic). Scarpellini [13] and Mundici [9] provide such information for / Lukasiewicz's logic. Hence we shall concentrate on ...

We present a general framework that allows to construct systematically analytic calculi for a large family of (propositional) many-valued logics --- called projective logics --- characterized by a special format of their semantics. All finite-valued logics as well as infinite-valued Godel logic are projective. As a case-study, sequent of relations calculi for Godel logics are derived. A comparison with some other analytic calculi is provided.

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