The aim of this book is to explore the possibilities and limits of Comenius's greatest and most important work, General Consultation Concerning Restoration of Human Affairs. The humanity of people is not quite right, but not quite lost--that is the foundational anthropological assumption of the Consultation. How does Comenius understand humanity? What are human affairs? What's wrong with them? And the most important question: Can they be somehow corrected, improved, or restored?

The continuing viability of the free‐radical theory of ageing has been questioned following apparently incompatible recent results. We show by modelling positional effects of the generation and primary targets of reactive oxygen species that many of the apparently negative results are likely to be misleading. We conclude that there is instead a need to look more closely at the mechanisms by which free radicals contribute to age‐related dysfunction in living systems. There also needs to be deeper understanding of the dynamics (...) of accumulation and removal of the various kinds of molecular damage, in particular mtDNA mutations. Finally, the expectation that free‐radical damage on its own might cause ageing needs to be relinquished in favour of the recognition that the free‐radical theory is just one of the multiple mechanisms driving the ageing process. (shrink)

Bei der Herausbildung des modernen sozialtheoretischen Denkens haben zwei Theorieströmungen eine zentrale Rolle gespielt: der amerikanische Pragmatismus und die deutsche Tradition des Historismus. Eine Zusammenführung der beiden damals unverbundenen Strömungen, so die These des Bandes, wäre historisch höchst fruchtbar gewesen. Die Autoren zeigen, wie eine vergleichende Bezugnahme noch heute revitalisierend auf die aktuelle Sozialtheorie und die empirischen Forschungsprogramme des Sozialen wirkt.

Recently Flaminio and Montagna (Proceedings of the 5th EUSFLAT Conference, II: 201–206. Ostrava, 2007), (Inter. J. Approx. Reason. 50:138–152, 2009) introduced the notion of a state MV-algebra as an MV-algebra with internal state. We have two kinds: state MV-algebras and state-morphism MV-algebras. These notions were also extended for state BL-algebras in (Soft Comput. doi:10.1007/s00500-010-0571-5). In this paper, we completely describe subdirectly irreducible state-morphism BL-algebras and this generalizes an analogous result for state-morphism MV-algebras presented in (Ann. Pure Appl. Logic 161:161–173, 2009).

Generalizations of Boolean elements of a BL-algebra L are studied. By utilizing the MV-center MV(L) of L, it is reproved that an element x L is Boolean iff x x * = 1. L is called semi-Boolean if for all x L, x * is Boolean. An MV-algebra L is semi-Boolean iff L is a Boolean algebra. A BL-algebra L is semi-Boolean iff L is an SBL-algebra. A BL-algebra L is called hyper-Archimedean if for all x L, xn is Boolean (...) for some finite n 1. It is proved that hyper-Archimedean BL-algebras are MV-algebras. The study has application in mathematical fuzzy logics whose Lindenbaum algebras are MV-algebras or BL-algebras. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim). (shrink)

Bl. Ladislaus of Gielniów is commonly regarded as the first major literary figure in Poland to write in Polish, as well as Latin. He is also the most important writer among the friars of the early Franciscan observant reform movement in Poland, which grew vigorously there after the visit of St. Giovanni of Capestrano in 1453. There, they took on the name of "Bernardines" to distinguish them from the Conventual Franciscans, after the cult of St. Bernardino of Siena, which Capestrano (...) promoted so strongly on all his travels.1 The Bernardine contribution to the development of the Polish literary language was significant, and the Bernardine contribution to the development of popular... (shrink)

BL-algebras rise as Lindenbaum algebras from many valued logic introduced by Hájek [2]. In this paper Boolean ds and implicative ds of BL-algebras are defined and studied. The following is proved to be equivalent: (i) a ds D is implicative, (ii) D is Boolean, (iii) L/D is a Boolean algebra. Moreover, a BL-algebra L contains a proper Boolean ds iff L is bipartite. Local BL-algebras, too, are characterized. These results generalize some theorems presented in [4], [5], [6] for MV-algebras which (...) are BL-algebras fulfiling an additional double negation law x = x **. (shrink)

In this paper we define sheaf spaces of BL-algebras (or BL-sheaf spaces), we study completely regular and compact BL-sheaf spaces and compact representations of BL-algebras and, finally, we prove that the category of non-trivial BL-algebras is equivalent with the category of compact local BL-sheaf spaces.

In this paper we study and equationally characterize the subvarieties of BL, the variety of BL-algebras, which are generated by families of single-component BL-chains, i.e. MV-chains, Product-chain or Gödel-chains. Moreover, it is proved that they form a segment of the lattice of subvarieties of BL which is bounded by the Boolean variety and the variety generated by all single-component chains, called ŁΠG.

Une question souvent posée par les spécialistes de la culture byzantine est celle qui concerne les relations entre littérature et réalité à Byzance - relation réputée «obscure» (comme si la dite relation était «claire» ou «évidente» pour les autres littératures). Le mélange entre théologie et paganisme, intellectualisme et passion, haute culture et amoralisme, réalisme et abstraction, continue à poser des problèmes, jadis escamotés sous le nom de «byzantinisme». Dans ce cadre général, on doit inclure une question plus concrète sur la (...) philosophie et son exercice à Byzance et, plus précisément, celle qui concerne la «persona» du philosophe, ses attributs et sa vocation. En tant que cas exemplaire, et en même temps extraordinaire, nous avons choisi le destin de Jean Italos. (shrink)

In this paper, the notion of the radical of a filter in BL-algebras is defined and several characterizations of the radical of a filter are given. Also we prove that A/F is an MV-algebra if and only if Ds ⊆ F. After that we define the notion of semi maximal filter in BL-algebras and we state and prove some theorems which determine the relationship between this notion and the other types of filters of a BL-algebra. Moreover, we prove that A/F (...) is a semi simple BL-algebra if and only if F is a semi maximal filter of A. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. (shrink)

In this paper we investigate splitting algebras in varieties of logics, with special consideration for varieties of BL-algebras and similar structures. In the case of the variety of all BL-algebras a complete characterization of the splitting algebras is obtained.

In this paper we introduce a poset of subvarieties of BL-algebras, whose completion is the entire lattice of subvarietes; we exhibit also a description of this poset in terms of finite sequences of functions on the natural numbers.

We define the set of double complemented elements in BL-algebras and state and prove some theorems which determines properties of these sets. We introduce the notion of an almost top element and study the properties of these elements.

In this paper we introduce n -fold implicative basis logic and the related algebras called n -fold implicative BL-algebras. Also we define n -fold implicative filters and we prove some relations between these filters and construct quotient algebras via these filters.

In the field of many-valued logics, Hájek’s Basic Logic BL was introduced in Hájek (Metamathematics of fuzzy logic, trends in logic. Kluwer Academic Publishers, Berlin, 1998). In this paper we will study four families of n-contractive (i.e. that satisfy the axiom \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\phi^n\rightarrow\phi^{n+1}}$$\end{document}, for some \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${n\in\mathbb{N}^+}$$\end{document}) axiomatic extensions of BL and their corresponding varieties: BLn, SBLn, BLn and SBLn. Concerning BLn we have that (...) every BLn-chain is isomorphic to an ordinal sum of MV-chains of at most n + 1 elements, whilst every BLn-chain is isomorphic to an ordinal sum of MVn-chains (for SBLn and SBLn a similar property holds, with the difference that the first component must be the two elements boolean algebra); all these varieties are locally finite. Moving to the content of the paper, after a preliminary section, we will study generic and k-generic algebras, completeness and computational complexity results, amalgamation and interpolation properties. Finally, we will analyze the first-order versions of these logics, from the point of view of completeness and arithmetical complexity. (shrink)

De acordo com BL. Hijmans, diante da pergunta sobre o caráter filosófico do pensamento de Epicteto é possível afirmar que ele não se constitua, de fato, como uma filosofia. Para Hijmans, o estoicismo epictetiano é construído a partir de um conjunto de crenças inquestionáveis que têm como base a relação de submissão a um princípio divino responsável por criar e comandar o cosmos. Por esse motivo, Hijmans afirma que faltam dois elementos ao pensamento de Epicteto para caracterizá-lo como filosofia, são (...) eles: o sentimento de maravilhamento ou perplexidade típico de uma reflexão filosófica e a capacidade de autocrítica. Nesse sentido, nosso artigo irá propor duas objeções aos argumentos de Hijmans com o intuito de demonstrar que o estoicismo de Epicteto detém as características necessárias para ser considerado, de fato, filosofia. (shrink)

Different constructions of BL-chains are compared. We establish when the ordinal sum and the poset product of the same family of BL-chains coincide. We also compare the poset product of MV-chains and product chains with saturated BL-chains.

In [10] it is claimed that the set of predicate tautologies of all complete BL-chains and the set of all standard tautologies coincide. As noticed in [11], this claim is wrong. In this paper we show that a complete BL-chain B satisfies all standard BL-tautologies iff for any transfinite sequence of elements of B, the condition ∧i ∈ I = 2 holds in B.

The main aim of the paper is to solve a problem posed in Di Nola et al. (Multiple Val. Logic 8:715–750, 2002) whether every pseudo BL-algebra with two negations is good, i.e. whether the two negations commute. This property is intimately connected with possessing a state, which in turn is essential in quantum logical applications. We approach the solution by describing the structure of pseudo BL-algebras and pseudo hoops as important families of quantum structures. We show when a pseudo hoop (...) can be embedded into the negative cone of the reals. We give an equational base characterizing representable pseudo hoops. We also describe some subvarieties: normal-valued, and varieties where each maximal filter is normal. We produce some noncommutative covers and extend the area where each algebra is good. Finally, we show that there are uncountably many subvarieties of pseudo BL-algebras having members that are not good. (shrink)

This paper is a report of the 700th anniversary of the death of Bl. Jan Duns Scotus. International Jubilee Symposium in Lublin.

BL-algebras are the Lindenbaum algebras for Hájek's Basic Logic, just as Boolean algebras correspond to the classical propositional calculus. The finite totally ordered BL-algebras are ordinal sums of MV-chains. We develop a natural duality, in the sense of Davey and Werner, for each subvariety generated by a finite BL-chain, and we use it to describe the injective and the weak injective members of these classes.

Abstract.Given a class C of t-norm BL-algebras, one may wonder which is the complexity of the set Taut(C∀) of predicate formulas which are valid in any algebra in C. We first characterize the classes C for which Taut(C∀) is recursively axiomatizable, and we show that this is the case iff C only consists of the Gödel algebra on [0,1]. We then prove that in all cases except from a finite number Taut(C∀) is not even arithmetical. Finally we consider predicate monadic (...) logics TautM(C∀) of classes C of t-norm BL-algebras, and we prove that (possibly with finitely many exceptions) they are undecidable. (shrink)

This work presents a model-theoretic approach to the study of first-order theories of classes of BL-chains. Among other facts, we present several classes of BL-algebras, generating the whole variety of BL-algebras, whose first-order theory has quantifier elimination. Model-completeness and decision problems are also investigated. Then we investigate classes of BL-algebras having (or not having) the amalgamation property or the joint embedding property and we relate the above properties to the existence of ultrahomogeneous models.

ABSTRACTIn this paper, we consider non-associative generalisations of Hájek's logics BL and psBL. As it was shown by Cignoli, Esteva, Godo, and Torrens, the former is the logic of continuous t-norms and their residua. Botur introduced logic naBL which is the logic of non-associative continuous t-norms and their residua. Thus, naBL can be viewed as a non-associative generalisation of BL. However, Botur has not presented axiomatization of naBL. We fill this gap by constructing an adequate Hilbert-style calculus for naBL. Although, (...) as was shown by Flondor, Georgescu, and Iorgulescu, there are no non-commutative continuous t-norms, Hájek's psBL can be viewed as BL's non-commutative generalisation. We present the logic psnaBL of psnaBL-algebras which can be viewed as naBL's non-commutative generalisation as well as psBL's non-associative generalisation and BL's both non-commutative and non-associative generalisation. (shrink)

For each integer n≥0, we provide an explicit functional characterization of the free n-generated BL-algebra, together with an explicit construction of the corresponding normal forms.

In this paper, the notions of tense operators and tense filters in \-algebras are introduced and several characterizations of them are obtained. Also, the relation among tense \-algebras, tense \-algebras and tense Boolean algebras are investigated. Moreover, it is shown that the set of all tense filters of a \-algebra is complete sublattice of \\) of all filters of \-algebra \. Also, maximal tense filters and simple tense \-algebras and the relation between them are studied. Finally, the notions of tense (...) congruence relations in tense \-algebras and strict tense \-algebras are introduced and an one-to-one correspondence between tense filters and tense congruences relations induced by tense filters are provided. (shrink)

In this paper, the notions of tense operators and tense filters in \-algebras are introduced and several characterizations of them are obtained. Also, the relation among tense \-algebras, tense \-algebras and tense Boolean algebras are investigated. Moreover, it is shown that the set of all tense filters of a \-algebra is complete sublattice of \\) of all filters of \-algebra \. Also, maximal tense filters and simple tense \-algebras and the relation between them are studied. Finally, the notions of tense (...) congruence relations in tense \-algebras and strict tense \-algebras are introduced and an one-to-one correspondence between tense filters and tense congruences relations induced by tense filters are provided. (shrink)

In this paper, we consider a collection of filters of a BL-algebra A. We use the concept of congruence relation with respect to filters to construct a uniformity which induces a topology on A. We study the properties of this topology regarding different filters.

Consider any subvariety of BL-algebras generated by a single BL-chain which is the ordinal sum of the standard MV-algebra on [0, 1] and a basic hoop H. We present a geometrical characterization of elements in the finitely generated free algebra of each of these subvarieties. In this characterization there is a clear insight of the role of the regular and dense elements of the generating chain. As an application, we analyze maximal and prime filters in the free algebra.

In this paper we show that the subvarieties of BL, the variety of BL-algebras, generated by single BL-chains on [0, 1], determined by continous t-norms, are finitely axiomatizable. An algorithm to check the subsethood relation between these subvarieties is provided, as well as another procedure to effectively find the equations of each subvariety. From a logical point of view, the latter corresponds to find the axiomatization of every residuated many-valued calculus defined by a continuous t-norm and its residuum. Actually, the (...) paper proves the results for a more general class than t-norm BL-chains, the so-called regular BL-chains. (shrink)