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This volume celebrates the work of Petr Hájek on mathematical fuzzy logic and presents how his efforts have influenced prominent logicians who are continuing his work. The book opens with a discussion on Hájek's contribution to mathematical fuzzy logic and with a scientific biography of him, progresses to include two articles with a foundation flavour, that demonstrate some important aspects of Hájek's production, namely, a paper on the development of fuzzy sets and another paper on some fuzzy versions of set theory and arithmetic.

Articles in the volume also focus on the treatment of vagueness, building connections between Hájek's favorite fuzzy logic and linguistic models of vagueness. Other articles introduce alternative notions of consequence relation, namely, the preservation of truth degrees, which is discussed in a general context, and the differential semantics. For the latter, a surprisingly strong standard completeness theorem is proved. Another contribution also looks at two principles valid in classical logic and characterize the three main t-norm logics in terms of these principles.

Other articles, with an algebraic flavour, offer a summary of the applications of lattice ordered-groups to many-valued logic and to quantum logic, as well as an investigation of prelinearity in varieties of pointed lattice ordered algebras that satisfy a weak form of distributivity and have a very weak implication.

The last part of the volume contains an article on possibilistic modal logics defined over MTL chains, a topic that Hájek discussed in his celebrated work, Metamathematics of Fuzzy Logic, and another one where the authors, besides offering unexpected premises such as proposing to call Hájek's basic fuzzy logic HL, instead of BL, propose a very weak system, called SL as a candidate for the role of the really basic fuzzy logic. The paper also provides a generalization of the prelinearity axiom, which was investigated by Hájek in the context of fuzzy logic.

Inhaltsverzeichnis

Frontmatter

Introduction

Frontmatter

Chapter 1. Introduction

Abstract
Since Petr Hájek, the scientist we are going to celebrate, is the main contributor to Mathematical Fuzzy Logic, we will first spend a few words about this subject
Francesc Esteva, Lluís Godo, Siegfried Gottwald, Franco Montagna

Chapter 2. Petr Hájek: A Scientific Biography

Abstract
Petr Hájek is a renowned Czech logician, whose record in mathematical logic spans half a century. His results leave a permanent imprint in all of his research areas, which can be delimited roughly as set theory, arithmetic, fuzzy logic and reasoning under uncertainty, and information retrieval; some of his results have enjoyed successful applications.
Zuzana Haniková

Foundational Aspects of Mathematical Fuzzy Logic

Frontmatter

Chapter 3. The Logic of Fuzzy Set Theory: A Historical Approach

Abstract
Actually it is a well accepted fact that fuzzy set theory, as a mathematical theory of a sort of generalized sets, has a natural relationship to particular kinds of non-classical logics of comparable truth degrees, called mathematical fuzzy logics. The chapter discusses aspects of the historical development of the relationship of fuzzy sets with formal logics suitable for a natural presentation of fuzzy set theory. It also sketches the core mathematical ideas behind mathematical fuzzy logics, explains the key rôle played by Petr Hájek in this development, presents some of the basic results for these logics, and offers some indications for ongoing developments in the field.
Siegfried Gottwald

Chapter 4. Set Theory and Arithmetic in Fuzzy Logic

Abstract
This chapter offers a review of Petr Hájek’s contributions to first-order axiomatic theories in fuzzy logic (in particular, ZF-style fuzzy set theories, arithmetic with a fuzzy truth predicate, and fuzzy set theory with unrestricted comprehension schema). Generalizations of Hájek’s results in these areas to MTL as the background logic are presented and discussed.
Libor Běhounek, Zuzana Haniková

Chapter 5. Bridges Between Contextual Linguistic Models of Vagueness and T-Norm Based Fuzzy Logic

Abstract
Linguistic models of vagueness usually record contexts of possible precisifications. A link between such models and fuzzy logic is established by extracting fuzzy sets from context based word meanings and by analyzing standard logical connectives in this setting. In this manner the three fundamental t-norms (Łukasiewicz, minimum, and product) highlighted in Petr Hájek’s approach to fuzzy logic reappear as bounds for degrees extracted from contextual models. In a further step, two semantic scenarios for fuzzy logic—similarity based reasoning and Giles’s dialogue game—are adapted to obtain t-norm based truth functions from the point of view of context update semantics.
Christian G. Fermüller, Christoph Roschger

Semantics and Consequence Relation in Many-Valued Logic

Frontmatter

Chapter 6. Consequence and Degrees of Truth in Many-Valued Logic

Abstract
I argue that the definition of a logic by preservation of all degrees of truth is a better rendering of Bolzano’s idea of consequence as truth-preserving when “truth comes in degrees”, as is often said in many-valued contexts, than the usual scheme that preserves only one truth value. I review some results recently obtained in the investigation of this proposal by applying techniques of abstract algebraic logic in the framework of Łukasiewicz logics and in the broader framework of substructural logics, that is, logics defined by varieties of (commutative and integral) residuated lattices. I also review some scattered, early results, which have appeared since the 1970’s, and make some proposals for further research. 2010 Mathematics Subject Classification: 03B50, 03G27, 03B47, 03B22
Josep Maria Font

Chapter 7. The Differential Semantics of Łukasiewicz Syntactic Consequence

Abstract
The classical condition “c is a semantic consequence of a set P of premises” in infinite-valued propositional Łukasiewicz logic Ł\(_{\infty }\) is refined using enriched valuations that take into account the effect on the formula c of the stability of the truth-value of each formula p in P under small perturbations (or, measurement errors) of the models of P. The differential properties of the functions represented by c and by all p in P naturally lead to a new notion of semantic consequence that turns out to coincide with syntactic consequence.
Daniele Mundici

Chapter 8. Two Principles in Many-Valued Logic

Abstract
Classically, two propositions are logically equivalent precisely when they are true under the same logical valuations. Also, two logical valuations are distinct if, and only if, there is a formula that is true according to one valuation, and false according to the other. By a real-valued logic we mean a many-valued logic in the sense of Petr Hájek that is complete with respect to a subalgebra of truth values of a BL-algebra given by a continuous triangular norm on [0, 1]. Abstracting the two foregoing properties from classical logic leads us to two principles that a real-valued logic may or may not satisfy. We prove that the two principles are sufficient to characterise Łukasiewicz and Gödel logic, to within extensions. We also prove that, under the additional assumption that the set of truth values be closed in the Euclidean topology of [0, 1], the two principles also afford a characterisation of Product logic.
Stefano Aguzzoli, Vincenzo Marra

Algebra for Many-Valued Logic

Frontmatter

Chapter 9. How Do $$\ell $$ ℓ -Groups and Po-Groups Appear in Algebraic and Quantum Structures?

Abstract
In this survey we give an account of the relationships between \(\ell \)-groups and some important algebraic structures like MV-algebras, BL-algebras, and their non-commutative versions given by pseudo MV-algebras and pseudo BL-algebras. In a similar way we show how partially ordered groups are connected with quantum structures like orthomodular lattices, effect algebras and pseudo effect algebras. For the latter classes, an important role is played by various types of the Riesz Decomposition Property.
Anatolij Dvurečenskij

Chapter 10. Semi-linear Varieties of Lattice-Ordered Algebras

Abstract
We consider varieties of pointed lattice-ordered algebras satisfying a restricted distributivity condition and admitting a very weak implication. Examples of these varieties are ubiquitous in algebraic logic: integral or distributive residuated lattices; their \(\left\{ \cdot \right\} \)-free subreducts; their expansions (hence, in particular, Boolean algebras with operators and modal algebras); and varieties arising from quantum logic. Given any such variety \(\fancyscript{V}\), we provide an explicit equational basis (relative to \(\fancyscript{V}\)) for the semi-linear subvariety \(\fancyscript{W}\) of \(\fancyscript{V}\). In particular, we show that if \(\fancyscript{V}\) is finitely based, then so is \(\fancyscript{W}\). To attain this goal, we make extensive use of tools from the theory of quasi-subtractive varieties.
Antonio Ledda, Francesco Paoli, Constantine Tsinakis

More Recent Trends

Frontmatter

Chapter 11. On Possibilistic Modal Logics Defined Over MTL-Chains

Abstract
In this chapter we revisit a 1994 chapter by Hájek et al. where a modal logic over a finitely-valued Łukasiewicz logic is defined to capture possibilistic reasoning. In this chapter we go further in two aspects: first, we generalize the approach in the sense of considering modal logics over an arbitrary finite MTL-chain, and second, we consider a different possibilistic semantics for the necessity and possibility modal operators. The main result is a completeness proof that exploits similar techniques to the ones involved in Hájek et al.’s previous work.
Félix Bou, Francesc Esteva, Lluís Godo

Chapter 12. The Quest for the Basic Fuzzy Logic

Abstract
The quest for basic fuzzy logic was initiated by Petr Hájek when he proposed his basic fuzzy logic BL, complete with respect to the semantics given by all continuous t-norms. Later weaker systems, such as MTL, UL or \(\mathrm{psMTL}^r\), complete with respect to broader (but still meaningful for fuzzy logics) semantics, have been introduced and disputed the throne of the basic fuzzy logic. We contribute to the quest with our own proposal of a basic fuzzy logic. Indeed, we put forth a very weak logic called \(\mathrm {SL}^\ell \), introduced and studied in earlier works of the authors, and propose it as a base of a new framework which allows to work in a uniform way with both propositional and first-order fuzzy logics.
Petr Cintula, Rostislav Horčík, Carles Noguera

Backmatter

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