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

We present two embeddings of infinite-valued ̷Lukasiewicz logic ̷L into Meyer and Slaney’s abelian logic A, the logic of lattice-ordered abelian groups. We give new analytic proof systems for A and use the embeddings to derive corresponding systems for ̷L. These include: hypersequent calculi for A and ̷L and terminating versions of these calculi; labelled single sequent calculi for A and ̷L of complexity co-NP; unlabelled single sequent calculi for A and ̷L. 1

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

Abstract. Monoidal t-norm logic MTL and related fuzzy logics are extended with various modalities distinguished by the axiom �(A ∨ B) → (�A ∨ �B). Such modalities include Linear logic-like exponentials, the globalization (or Delta) operator, and truth stressers like “very true”. Extensions of MTL with modalities are presented here via axiomatizations, hypersequent calculi, and algebraic semantics, and related to standard algebras based on t-norms. Embeddings of logics, decidability, and the finite embedding property are also investigated. 1

This paper proposes a possibility of developing an axiomatic set theory, as first-order theory within the framework of fuzzy logic in the style of [13]. In classical ZFC, we use an analogy of the construction of a Booleanvalued universe---over a particular algebra of truth values---to show the non-triviality of our theory. We present a list of problems and research tasks. 1

In this paper we investigate the addition of arbitrary independent involutive negations to t-norm based logics. We deal with several extensions of MTL and establish general completeness results. Indeed, we will show that, given any t-norm based logic satisfying some basic properties, its extension by means of an involutive negation preserves algebraic and (finite) strong standard completeness. We will deal with both propositional and predicate logics. 1

Artificial intelligence is a relatively new scientific and technological field which studies the nature of intelligence by using computers to produce intelligent behaviour. Initially, the main goal was a purely scientific one, understanding human intelligence, and this remains the aim of cognitive scientists. Unfortunately, such an ambitious and fascinating goal is not only far from being achieved but has yet to be satisfactorily approached. Fortunately, however, artificial intelligence also has an engineering goal: building systems that are useful to people even if the intelligence of such systems has no relation whatsoever with human intelligence, and therefore being able to build them does not necessarily provide any insight into the nature of human intelligence. This engineering goal has become the predominant one among artificial intelligence researchers and has produced impressive results, ranging from knowledge-based systems to autonomous robots, that have been applied to many different domains. Furthermore, artificial intelligence products and services today represent an annual market of tens of billions of dollars worldwide.

There is a number of completely integrable gravity theories in two dimensions. We study the metric-affine approach on a 2-dimensional spacetime and display a new integrable model. Its properties are described and compared with the known results of Poincaré gauge gravity. PACS: 04.50.+h, 04.20.Fy, 04.20.Jb, 04.60.Kz, 02.30.Ik I.

In fuzzy logic in wider sense, t-norms got a prominent r^ole in recent times. In many-valued logic, the Lukasiewicz systems, the Godel sytems, and also the product logic are t-norm based. The present paper discusses the more general problem of the adequate axiomatizability for such t-norm based logical systems in general, surveying results of the last years.

Hajek's BL logic is the fuzzy logic capturing the tautologies of continuous t-norms and their residua. In this paper we investigate