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 research on systems of logic programming with modules has followed two mainstreams, programming-in-the-large, where compositional operators are provided for combining separate and independent modules, and programming-in-the-small, which aims at enhancing logic programming with new logical connectives. In this paper, we present

A combinatorial argument for two finite structures to agree on all sentences with bounded quantifier rank in first-order logic with any set of unary generalized quantifiers, is given. It is known that connectivity of finite structures is neither in monadic \Sigma 1 1 nor in L !! (Q u ), where Q u is the set of all unary generalized quantifiers. Using this combinatorial argument and a combination of second-order EhrenfeuchtFra iss'e games developed by Ajtai and Fagin, we prove that connectivity of finite structures is not in monadic \Sigma 1 1 with any set of unary quantifiers, even if sentences are allowed to contain built-in relations of moderate degree. The combinatorial argument is also used to show that no class (if it is not in some sense trivial) of finite graphs defined by forbidden minors, is in L !! (Q u ). Especially, the class of planar graphs is not in L !! (Q u ). 1. Introduction The expressive power of first-order logic L !! is rather limited. This is beca...

The study of generalized quantifiers over the past 15 years has enriched enormously our understanding of natural language determiners (Dets). It has yielded answers to questions raised independently within generative grammar and it has provided us with new semantic generalizations, ones that were basically unformulable without the conceptual and technica apparatus of generalized quantifier theory. Here we overview results of both these types. historical note It was Montague (1969) who first interpreted natural language NPs as generalized quantifiers (though this term was not used by him). But it was only in the early 1980's with the publication of B&C (Barwise and Cooper, 1981) that the study of natural language Dets took on a life of its own. Also from this period are early versions of K&S (Keenan and Stavi, 1986) and Higginbotham and May (1981). The former fed into subsequent formal studies such as van Benthem (1984, 1986) and Westerstähl (1985). The latter focussed on specific linguistic applications of binary quantifiers, a topic initiated in Altham and Tennant (1974), drawing on the mathematical work of Mostowski (1957), and pursued later in a more general linguistic setting in van Benthem (1989) and Keenan (1987b, 1992). Another precursor to the mathematical study o generalized quantifiers is Lindstr_m (1969) who provides the type notation used to classif

. We introduce a new framework for classifying logics on finite structures and studying their expressive power. This framework is based on the concept of almost everywhere equivalence of logics, that is to say, two logics having the same expressive power on a class of asymptotic measure 1. More precisely, if L, L 0 are two logics and # is an asymptotic measure on finite structures, then L j a.e. L 0 (#) means that there is a class C of finite structures with #(C ) = 1 and such that L and L 0 define the same queries on C. We carry out a systematic investigation of j a.e. with respect to the uniform measure and analyze the j a.e. -equivalence classes of several logics that have been studied extensively in finite model theory. Moreover, we explore connections with descriptive complexity theory and examine the status of certain classical results of model theory in the context of this new framework. x1. Introduction and summary of results. Finite model theory can be succinct...

In this paper some of the basics of classification theory for abstract elementary classes are discussed. Instead of working with types which are sets of formulas (in the first-order case) we deal instead with Galois types which are essentially orbits of automorphism groups acting on the structure. Some of the most basic results in classification theory for non elementary classes are presented. The motivating point of view is Shelah's categoricity conjecture for L# 1 ,# . While only very basic theorems are proved, an effort is made to present number of different technologies: Flavors of weak diamond, models of weak set theories, and commutative diagrams. We focus in issues involving existence of Galois types, extensions of types and Galois-stability.

. We provide tools for a concise axiomatization of a broad class of quantifiers in many-valued logic, so-called distribution quantifiers. Although sound and complete axiomatizations for such quantifiers exist, their size renders them virtually useless for practical purposes. We show that for quantifiers based on finite distributive lattices compact axiomatizations can be obtained schematically. This is achieved by providing a link between skolemized signed formulas and filters/ideals in Boolean set lattices. Then lattice theoretic tools such as Birkhoff's representation theorem for finite distributive lattices are used to derive tableau-style axiomatizations of distribution quantifiers. Introduction The aim of this paper 1 is to provide concise axiomatizations of certain quantifiers in many-valued logic which were introduced by Mostowski (1961) and baptized distribution quantifiers by Carnielli (1987). The task of axiomatizing such quantifiers has been solved satisfactorily in theor...

We show how sequent calculi for some generalized quantifiers can be obtained by generalizing the Herbrand approach to ordinary first order proof theory. Typical of the Herbrand approach, as compared to plain sequent calculus, is increased control over relations of dependence between variables. In the case of generalized quantifiers, explicit attention to relations of dependence becomes indispensible for setting up proof systems. It is shown that this can be done by turning variables into structured objects, governed by various types of structural rules. These structured variables are interpreted semantically by means of a dependence relation. This relation is an analogue of the accessibility relation in modal logic. We then isolate a class of axioms for generalized quantifiers which correspond to first-order conditions on the dependence relation.

The concept of a generalized quantifier of a given similarity type was defined in [Lin66]. Our main result says that on finite structures different similarity types give rise to different classes of generalized quantifiers. More exactly, for every similarity type t there is a generalized quantifier of type t which is not definable in the extension of first order logic by all generalized quantifiers of type smaller than t. This was proved for unary similarity types by Per Lindstrom [Wes] with a counting argument. We extend his method to arbitrary similarity types. 1 Introduction According to Lindstrom [Lin66], generalized quantifiers are simply classes of structures of a fixed similarity type such that the class is closed under isomorphisms. We identify similarity types with finite sequences of positive integers. A structure A of (similarity) type t = (t 1 ; : : : ; t u ) consists of a finite Key words: generalized quantifier, finite model theory, abstract model theory, y Par...

We investigate the expressive power of various extensions of first-order, inductive, and infinitary logic with counting quantifiers. We consider in particular a LOGSPACE extension of first-order logic, and a PTIME extension of fixpoint logic with counters. Counting is a fundamental tool of algorithms. It is essential in the case of unordered structures. Our aim is to understand the expressive power gained with a limited counting ability. We consider two problems: (i) unnested counters, and (ii) counters with no free variables. We prove a hierarchy result based on the arity of the counters under the first restriction. The proof is based on a game technique that is introduced in the paper. We also establish results on the asymptotic probabilities of sentences with counters under the second restriction. In particular, we show that first-order logic with equality of the cardinalities of relations has a 0/1 law. 1 Introduction Counting is a fundamental operation of numerous algorithms. Cou...