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)

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).

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.

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)

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, 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)

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.

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 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.

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.

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.

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)

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.

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.

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)

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.

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)

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. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim).

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)

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)

Logics designed to deal with vague statements typically allow algebraic semantics such that propositions are interpreted by elements of residuated lattices. The structure of these algebras is in general still unknown, and in the cases that a detailed description is available, to understand its significance for logics can be difficult. So the question seems interesting under which circumstances residuated lattices arise from simpler algebras in some natural way. A possible construction is described in this paper.Namely, we consider pairs consisting of (...) a Brouwerian algebra and an equivalence relation. The latter is assumed to be in a certain sense compatible with the partial order, with the formation of differences, and with the formation of suprema of pseudoorthogonal elements; we then call it an s-equivalence relation. We consider operations which, under a suitable additional assumption, naturally arise on the quotient set. The result is that the quotient set bears the structure of a residuated lattice. Further postulates lead to dual BL-algebras. In the case that we begin with Boolean algebras instead, we arrive at dual MV-algebras. (shrink)

Quantum B-algebras, the partially ordered implicational algebras arising as subreducts of quantales, are introduced axiomatically. It is shown that they provide a unified semantic for non-commutative algebraic logic. Specifically, they cover the vast majority of implicational algebras like BCK-algebras, residuated lattices, partially ordered groups, BL- and MV-algebras, effect algebras, and their non-commutative extensions. The opposite of the category of quantum B-algebras is shown to be equivalent to the category of logical quantales, in the way that every quantum B-algebra admits a (...) natural embedding into a logical quantale, the enveloping quantale. Partially defined products of algebras related to effect algebras are handled efficiently in this way. The unit group of the enveloping quantale of a quantum B-algebra X is shown to be always contained in X, which gives a functorial subgroup X× of X. Similar subfunctors are obtained for the non-commutative extensions of BCK-algebras and effect algebras. The results of Galatos, Jónsson, and Tsinakis on the splitting of generalized BL-algebras into a semidirect product of a partially ordered group operating on an integral residuated poset are extended to a characterization of twisted semidirect products of a po-group by a quantum B-algebra. (shrink)

A continuoxis t- norm is a continuous map * from [0, 1]² into [0,1] such that is a commutative totally ordered monoid. Since the natural ordering on [0,1] is a complete lattice ordering, each continuous t-norm induces naturally a residuation → and becomes a commutative naturally ordered residuated monoid, also called a hoop. The variety of basic hoops is precisely the variety generated by all algebras, where * is a continuous t-norm. In this paper we investigate the structure of the (...) variety of basic hoops and some of its subvarieties. In particular we provide a complete description of the finite subdirectly irreducible basic hoops, and we show that the variety of basic hoops is generated as a quasivariety by its finite algebras. We extend these results to Hájek's BL-algebras, and we give an alternative proof of the fact that the variety of BL-algebras is generated by all algebras arising from continuous t-norms on [ 0,1] and their residua. The last part of the paper is devoted to the investigation of the subreducts of BL- algebras, of Gödel algebras and of product algebras. (shrink)

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)

A continuoxis t- norm is a continuous map * from [0, 1]² into [0,1] such that ([ 0,1], *, 1) is a commutative totally ordered monoid. Since the natural ordering on [0,1] is a complete lattice ordering, each continuous t-norm induces naturally a residuation → and ([ 0,1], *, →, 1) becomes a commutative naturally ordered residuated monoid, also called a hoop. The variety of basic hoops is precisely the variety generated by all algebras ([ 0,1], *, →, 1), where (...) * is a continuous t-norm. In this paper we investigate the structure of the variety of basic hoops and some of its subvarieties. In particular we provide a complete description of the finite subdirectly irreducible basic hoops, and we show that the variety of basic hoops is generated as a quasivariety by its finite algebras. We extend these results to Hájek's BL-algebras, and we give an alternative proof of the fact that the variety of BL-algebras is generated by all algebras arising from continuous t-norms on [ 0,1] and their residua. The last part of the paper is devoted to the investigation of the subreducts of BL- algebras, of Gödel algebras and of product algebras. (shrink)

Pseudo-BCK-algebras are a non-commutative generalization of well-known BCK-algebras. The paper describes a situation when a linearly ordered pseudo-BCK-algebra is an ordinal sum of linearly ordered cone algebras. In addition, we present two identities giving such a possibility of the decomposition and axiomatize the residuation subreducts of representable pseudo-hoops and pseudo-BL-algebras.

An effect algebra is a partial algebraic structure, originally formulated as an algebraic base for unsharp quantum measurements. In this article we present an approach to the study of lattice effect algebras (LEAs) that emphasizes their structure as algebraic models for the semantics of (possibly) non-standard symbolic logics. This is accomplished by focusing on the interplay among conjunction, implication, and negation connectives on LEAs, where the conjunction and implication connectives are related by a residuation law. Special cases of LEAs are (...) MV-algebras and orthomodular lattices. The main result of the paper is a characterization of LEAs in terms of so-called Sasaki algebras. Also, we compare and contrast LEAs, Hájek's BL-algebras, and the basic algebras of Chajda, Halaš, and Kühr. (shrink)

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)

of monadic or relational predicate calculus (Fa, Gb, Rab, Hcd, etc.). • The Boolean Algebra BL set-up by such a language will be such that: – BL will have 2 n states (corresponding to the state descriptions of L) – BL will contain 2 2n propositions, in total. ∗ This is because each proposition p in BL is equivalent to a disjunction of state descriptions. Thus, each subset of the set of..

Using the theory of BL-algebras, it is shown that a propositional formula ϕ is derivable in Łukasiewicz infinite valued Logic if and only if its double negation ˜˜ϕ is derivable in Hájek Basic Fuzzy logic. If SBL is the extension of Basic Logic by the axiom (φ & (φ→˜φ)) → ψ, then ϕ is derivable in in classical logic if and only if ˜˜ ϕ is derivable in SBL. Axiomatic extensions of Basic Logic are in correspondence with subvarieties of the (...) variety of BL-algebras. It is shown that the MV-algebra of regular elements of a free algebra in a subvariety of BL-algebras is free in the corresponding subvariety of MV-algebras, with the same number of free generators. Similar results are obtained for the generalized BL-algebras of dense elements of free BL-algebras. (shrink)

We present the logic BLChang, an axiomatic extension of BL (see [23]) whose corresponding algebras form the smallest variety containing all the ordinal sums of perfect MV-chains. We will analyze this logic and the corresponding algebraic semantics in the propositional and in the first-order case. As we will see, moreover, the variety of BLChang-algebras will be strictly connected to the one generated by Chang’s MV-algebra (that is, the variety generated by all the perfect MV-algebras): we will also give some new (...) results concerning these last structures and their logic. (shrink)

Many results in fuzzy logic depend on the mathematical structure the truth value set obeys. In this textbook the algebraic foundations of many-valued and fuzzy reasoning are introduced. The book is self-contained, thus no previous knowledge in algebra or in logic is required. It contains 134 exercises with complete answers, and can therefore be used as teaching material at universities for both undergraduated and post-graduated courses. Chapter 1 starts from such basic concepts as order, lattice, equivalence and residuated lattice. It (...) contains a full section on BL-algebras. Chapter 2 concerns MV-algebra and its basic properties. Chapter 3 applies these mathematical results on Lukasiewicz-Pavelka style fuzzy logic, which is studied in details; besides semantics, syntax and completeness of this logic, a lot of examples are given. Chapter 4 shows the connection between fuzzy relations, approximate reasoning and fuzzy IF-THEN rules to residuated lattices. (shrink)

Fuzzy logics are in most cases based on an ad-hoc decision about the interpretation of the conjunction. If they are useful or not can typically be found out only by testing them with example data. Why we should use a specific fuzzy logic can in general not be made plausible. Since the difficulties arise from the use of additional, unmotivated structure with which the set of truth values is endowed, the only way to base fuzzy logics on firm ground is (...) the development of alternative semantics to all of whose components we can associate a meaning. In this paper, we present one possible approach to justify ex post Łukasiewicz Logic as well as Basic Logic. The notion of ambiguity is central. Our framework consists of a Boolean or a Heyting algebra, respectively, endowed with an equivalence relation expressing ambiguity. The quotient set bears naturally the structure of an MV- or a BL-algebra, respectively, and thus can be used to interpret propositions of the mentioned logics. (shrink)

In general, there is only one fuzzy logic in which the standard interpretation of the strong conjunction is a strict triangular norm, namely, the product logic. We study several equations which are satisfied by some strict t-norms and their dual t-conorms. Adding an involutive negation, these equations allow us to generate countably many logics based on strict t-norms which are different from the product logic.

This paper deals with Kripke-style semantics for many-valued logics. We introduce various types of Kripke semantics, and we connect them with algebraic semantics. As for modal logics, we relate the axioms of logics extending MTL to properties of the Kripke frames in which they are valid. We show that in the propositional case most logics are complete but not strongly complete with respect to the corresponding class of complete Kripke frames, whereas in the predicate case there are important many-valued logics (...) like BL, Ł and Π, which are not even complete with respect to the class of all predicate Kripke frames in which they are valid. Thus although very natural, Kripke semantics seems to be slightly less powerful than algebraic semantics. (shrink)

We investigate the expressivity of many-valued modal logics based on an algebraic structure with a complete linearly ordered lattice reduct. Necessary and sufficient algebraic conditions for admitting a suitable Hennessy–Milner property are established for classes of image-finite and modally saturated models. Full characterizations are obtained for many-valued modal logics based on complete BL-chains that are finite or have the real unit interval [0, 1] as a lattice reduct, including Łukasiewicz, Gödel, and product modal logics.

The monoidal t-norm based logic MTL is obtained from Hájek''s Basic Fuzzy logic BL by dropping the divisibility condition for the strong (or monoidal) conjunction. Recently, Jenei and Montgana have shown MTL to be standard complete, i.e. complete with respect to the class of residuated lattices in the real unit interval [0,1] defined by left-continuous t-norms and their residua. Its corresponding algebraic semantics is given by pre-linear residuated lattices. In this paper we address the issue of standard and rational completeness (...) (rational completeness meaning completeness with respect to a class of algebras in the rational unit interval [0,1]) of some important axiomatic extensions of MTL corresponding to well-known parallel extensions of BL. Moreover, we investigate varieties of MTL algebras whose linearly ordered countable algebras embed into algebras whose lattice reduct is the real and/or the rational interval [0,1]. These embedding properties are used to investigate finite strong standard and/or rational completeness of the corresponding logics. (shrink)

The monoidal t-norm based logic MTL is obtained from Hájek's Basic Fuzzy logic BL by dropping the divisibility condition for the strong (or monoidal) conjunction. Recently, Jenei and Montgana have shown MTL to be standard complete, i.e. complete with respect to the class of residuated lattices in the real unit interval [0,1] defined by left-continuous t-norms and their residua. Its corresponding algebraic semantics is given by pre-linear residuated lattices. In this paper we address the issue of standard and rational completeness (...) (rational completeness meaning completeness with respect to a class of algebras in the rational unit interval [0,1]) of some important axiomatic extensions of MTL corresponding to well-known parallel extensions of BL. Moreover, we investigate varieties of MTL algebras whose linearly ordered countable algebras embed into algebras whose lattice reduct is the real and/or the rational interval [0,1]. These embedding properties are used to investigate finite strong standard and/or rational completeness of the corresponding logics. (shrink)

We sharpen Hájek’s Completeness Theorem for theories extending predicate product logic, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Pi\forall}$$\end{document}. By relating provability in this system to embedding properties of ordered abelian groups we construct a universal BL-chain L in the sense that a sentence is provable from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Pi\forall}$$\end{document} if and only if it is an L-tautology. As well we characterize the class of lexicographic sums that have this universality property.

Classification of certain group-like FL $_e$ -chains is given: We define absorbent-continuity of FL $_e$ -algebras, along with the notion of subreal chains, and classify absorbent-continuous, group-like FL $_e$ -algebras over subreal chains: The algebra is determined by its negative cone, and the negative cone can only be chosen from a certain subclass of BL-chains, namely, one with components which are either cancellative (that is, those components are negative cones of totally ordered Abelian groups) or two-element MV-algebras, and with no (...) two consecutive cancellative components. It is shown that the classification theorem does not hold if we drop the absorbent-continuity condition. Our result is the first classification theorem in the literature on FL $_e$ -algebras that does not assume the condition of being naturally ordered (which, under certain conditions, corresponds to continuity of the monoid operation). In our classification theorem, continuity is replaced by the much weaker absorbent-continuity. (shrink)

This note contains a correct proof of the fact that the set of all first-order formulas which are valid in all predicate Kripke frames for Hájek's many-valued logic BL is not arithmetical. The result was claimed in [5], but the proof given there was incorrect.

This book is not a list or an overview of various theories of ethics. Nor is it a didactic manual for specific teaching units on moral education aimed at some group based on age or a particular theme (although some educational frameworks will be proposed). As the title suggests, the book intends to seek the starting points or foundations without which no moral education would be possible. The goal is to formulate and tackle the key questions that precede all moral (...) education. What makes ""good vs. evil"" language possible and meaningful? Can virtue be taught and learned? What makes our actions good? What is the condition of human nature? Are we naturally good, or evil? What constitutes an educator's right to morally influence anyone else (not just a child)? What is the goal of moral education? What does a morally educated person look like? And how can we ensure the coveted moral result? Or--in the words of Jan Amos Comenius, the ""teacher of nations""--how to educate a person to not only know what is good, but also to want what is good, and to do what is good ""even when no one is looking?"" ""I wish I would have had Jan Habl's concise, passionate, and eminently readable book available as a text for my ethics courses during my thirty-six years of teaching at colleges and universities in Canada. Habl not only raises the alarm about a moral crisis in Western societies today, but also provides a careful treatment of some oft-neglected presuppositions of moral education. The concluding chapter introduces us to Jan Amos Comenius, a seventeenth century Czech philosopher, theologian, and pedagogue, from whom we have much to learn about a better approach to moral education."" --Elmer Thiessen, Emmanuel Bible College ""With a sound and broad grasp of ethical (moral) theorists, Jan Habl incisively critiques the fallacies of popular theories, rendering them ineffectual if not dangerous. Then, drawing on Czech educator/philosopher Comenious, he persuasively advances a theistic proposal for nurturing moral persons and communities. Immensely timely, never more needed, a rich, compelling read for our contemporary societies if we wish for a civil and peaceful existence for ourselves and future generations. Highly recommended."" --Duane H. Elmer, Trinity International University ""You have in your hands a text by someone I take to be one of the very serious young moral philosophers of our time. The profundity of his analysis of the need for moral education reminds me of the style of analysis of one of the great moral philosophers of the final decades of the twentieth century, Alasdair MacIntyre."" --Thomas K. Johnson, World Evangelical Alliance Jan Habl is a professor of pedagogy at universities in Hradec Kralove and Usti nad Labem (Czech Republic). He has taught systematic theology and ethics at the Evangelical Theological Seminary in Prague. He has authored a number of books and studies in the areas of philosophy of education, ethics, and pedagogy, including On Being Human(e) (2016); Teaching and Learning Through Story, Comenius' Labyrinth and the Educational Potential of Narrative Allegory (2014). (shrink)