The aim of this paper is to give a description of the free algebras in some varieties of Glivenko MTL-algebras having the Boolean retraction property. This description is given (generalizing the results of [9]) in terms of weak Boolean products over Cantor spaces. We prove that in some cases the stalks can be obtained in a constructive way from free kernel DL-algebras, which are the maximal radical of directly indecomposable Glivenko MTL-algebras satisfying the equation in the title. We include examples (...) to show how we can apply the results to describe free algebras in some well known varieties of involutive MTL-algebras and of pseudocomplemented MTL-algebras. (shrink)

The classical Glivenko theorem asserts that a propositional formula admits a classical proof if and only if its double negation admits an intuitionistic proof. By a natural expansion of the BCK-logic with negation we understand an algebraizable logic whose language is an expansion of the language of BCK-logic with negation by a family of connectives implicitly defined by equations and compatible with BCK-congruences. Many of the logics in the current literature are natural expansions of BCK-logic with negation. The validity of (...) the analogous of Glivenko theorem in these logics is equivalent to the validity of a simple one-variable formula in the language of BCK-logic with negation. (shrink)

The aim of this paper is to give a description of the free algebras in some varieties of Glivenko MTL-algebras having the Boolean retraction property. This description is given (generalizing the results of [9]) in terms of weak Boolean products over Cantor spaces. We prove that in some cases the stalks can be obtained in a constructive way from free kernel DL-algebras, which are the maximal radical of directly indecomposable Glivenko MTL-algebras satisfying the equation in the title. We include examples (...) to show how we can apply the results to describe free algebras in some well known varieties of involutive MTL-algebras and of pseudocomplemented MTL-algebras. (shrink)

In this paper we prove that the equational class generated by bounded BCK-algebras is the variety generated by the class of finite simple bounded BCK-algebras. To obtain these results we prove that every simple algebra in the equational class generated by bounded BCK-algebras is also a relatively simple bounded BCK-algebra. Moreover, we show that every simple bounded BCK-algebra can be embedded into a simple integral commutative bounded residuated lattice. We extend our main results to some richer subreducts of the class (...) of integral commutative bounded residuated lattices and to the involutive case. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim). (shrink)

AbstractŁukasiewicz implication algebras are {→,1}-subreducts of Wajsberg algebras (MV-algebras). They are the algebraic counterpart of Super-Łukasiewicz Implicational logics investigated in Komori, Nogoya Math J 72:127–133, 1978. The aim of this paper is to study the direct decomposability of free Łukasiewicz implication algebras. We show that freely generated algebras are directly indecomposable. We also study the direct decomposability in free algebras of all its proper subvarieties and show that infinitely freely generated algebras are indecomposable, while finitely free generated algebras can be (...) only decomposed into a direct product of two factors, one of which is the two-element implication algebra. (shrink)

Łukasiewicz implication algebras are {→,1}-subreducts of Wajsberg algebras (MV-algebras). They are the algebraic counterpart of Super-Łukasiewicz Implicational logics investigated in Komori, Nogoya Math J 72:127–133, 1978. The aim of this paper is to study the direct decomposability of free Łukasiewicz implication algebras. We show that freely generated algebras are directly indecomposable. We also study the direct decomposability in free algebras of all its proper subvarieties and show that infinitely freely generated algebras are indecomposable, while finitely free generated algebras can be (...) only decomposed into a direct product of two factors, one of which is the two-element implication algebra. (shrink)

We show that the class of all isomorphic images of Boolean Products of members of SR [1] is the class of all archimedean W-algebras. We obtain this result from the characterization of W-algebras which are isomorphic images of Boolean Products of CW-algebras.

In any variety of bounded integral residuated lattice-ordered commutative monoids the class of its semisimple members is closed under isomorphic images, subalgebras and products, but it is not closed under homomorphic images, and so it is not a variety. In this paper we study varieties of bounded residuated lattices whose semisimple members form a variety, and we give an equational presentation for them. We also study locally representable varieties whose semisimple members form a variety. Finally, we analyze the relationship with (...) the property “to have radical term”, especially for k-radical varieties, and for the hierarchy of varieties k>0 defined in Cignoli and Torrens. (shrink)

In a classical paper [15] V. Glivenko showed that a proposition is classically demonstrable if and only if its double negation is intuitionistically demonstrable. This result has an algebraic formulation: the double negation is a homomorphism from each Heyting algebra onto the Boolean algebra of its regular elements. Versions of both the logical and algebraic formulations of Glivenko’s theorem, adapted to other systems of logics and to algebras not necessarily related to logic can be found in the literature (see [2, (...) 9, 8, 14] and [13, 7, 14]). The aim of this paper is to offer a general frame for studying both logical and algebraic generalizations of Glivenko’s theorem. We give abstract formulations for quasivarieties of algebras and for equivalential and algebraizable deductive systems and both formulations are compared when the quasivariety and the deductive system are related. We also analyse Glivenko’s theorem for compatible expansions of both cases. (shrink)

In this paper we characterize the MV-algebras containing as subalgebras Post algebras of finitely many orders. For this we study cyclic elements in MV-algebras which are the generators of the fundamental chain of the Post algebras.

In this paper we see that the answer of this question is affirmative. We prove this for Dco-algebras and as special case we obtain the result for Positive Implication algebras. First we give, without proof, the properties of Dco-algebras and S-algebras and their connection with Positive Implication algebras and Implication algebras. These results can be found in [T] and [IT].

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)

Let ${\mathbb{BRL}}$ denote the variety of commutative integral bounded residuated lattices (bounded residuated lattices for short). A Boolean retraction term for a subvariety ${\mathbb{V}}$ of ${\mathbb{BRL}}$ is a unary term t in the language of bounded residuated lattices such that for every ${{\bf A} \in \mathbb{V}, t^{A}}$ , the interpretation of the term on A, defines a retraction from A onto its Boolean skeleton B(A). It is shown that Boolean retraction terms are equationally definable, in the sense that there is (...) a variety ${\mathbb{V}_{t} \subsetneq \mathbb{BRL}}$ such that a variety ${\mathbb{V} \subsetneq \mathbb{BRL}}$ admits the unary term t as a Boolean retraction term if and only if ${\mathbb{V} \subseteq \mathbb{V}_{t}}$ . Moreover, the equation s(x) = t(x) holds in ${\mathbb{V}_{s} \cap \mathbb{V}_{t}}$ . The radical of ${{\bf A} \in \mathbb{BRL}}$ , with the structure of an unbounded residuated lattice with the operations inherited from A expanded with a unary operation corresponding to double negation and a a binary operation defined in terms of the monoid product and the negation, is called the radical algebra of A. To each involutive variety ${\mathbb{V} \subseteq \mathbb{V}_{t}}$ is associated a variety ${\mathbb{V}^{r}}$ formed by the isomorphic copies of the radical algebras of the directly indecomposable algebras in ${\mathbb{V}}$ . Each free algebra in such ${\mathbb{V}}$ is representable as a weak Boolean product of directly indecomposable algebras over the Stone space of the free Boolean algebra with the same number of free generators, and the radical algebra of each directly indecomposable factor is a free algebra in the associated variety ${\mathbb{V}^{r}}$ , also with the same number of free generators.A hierarchy of subvarieties of ${\mathbb{BRL}}$ admitting Boolean retraction terms is exhibited. (shrink)

We give a presentation of Post algebras of ordern+1 (n1) asn+1 bounded Wajsberg algebras with an additional constant, and we show that a Wajsberg algebra admits a P-algebra reduct if and only if it isn+1 bounded.

In this paper we prove that the equational class generated by bounded BCK-algebras is the variety generated by the class of finite simple bounded BCK-algebras. To obtain these results we prove that every simple algebra in the equational class generated by bounded BCK-algebras is also a relatively simple bounded BCK-algebra. Moreover, we show that every simple bounded BCK-algebra can be embedded into a simple integral commutative bounded residuated lattice. We extend our main results to some richer subreducts of the class (...) of integral commutative bounded residuated lattices and to the involutive case. (shrink)

In this paper we characterize, classify and axiomatize all axiomatic extensions of the IMT3 logic. This logic is the axiomatic extension of the involutive monoidal t-norm logic given by ¬φ3 ∨ φ. For our purpose we study the lattice of all subvarieties of the class IMT3, which is the variety of IMTL-algebras given by the equation ¬(x 3) ∨ x ≈ ⊤, and it is the algebraic counterpart of IMT3 logic. Since every subvariety of IMT3 is generated by their totally (...) ordered members, we study the structure of all IMT3-chains in order to determine the lattice of all subvarieties of IMT3. Given a family of IMT3-chains the number of elements of the largest odd finite subalgebra in the family and the number of elements of the largest even finite subalgebra in the family turns out to be a complete classifier of the variety generated. We obtain a canonical set of generators and a finite equational axiomatization for each subvariety and, for each corresponding logic, a finite set of characteristic matrices and a finite set of axioms. (shrink)

Lattice BCK logic is the expansion of the well known Meredith implicational logic BCK expanded with lattice conjunction and disjunction. Although its natural axiomatization has three rules named modus ponens, ∨‐rule and ∧‐rule, we show that we can give an equivalent presentation with just modus ponens and ∧‐rule, however it is impossible to obtain an equivalent presentation with modus ponens as unique rule. In this paper we study and characterize all axiomatic extensions of lattice BCK logic with modus ponens as (...) unique rule. We obtain an infinite chain of proper axiomatic extensions with this property. Moreover, we prove that there is no weakest axiomatic extension of Lattice BCK‐logic admitting modus ponens as unique rule. (shrink)

In this paper we show that the quasivariety generated by an infinite simple MV-algebra only depends on the rationals which it contains. We extend this property to arbitrary families of simple MV-algebras.

Łukasiewicz’s infinite-valued logic is commonly defined as the set of formulas that take the value 1 under all evaluations in the Łukasiewicz algebra on the unit real interval. In the literature a deductive system axiomatized in a Hilbert style was associated to it, and was later shown to be semantically defined from Łukasiewicz algebra by using a “truth-preserving” scheme. This deductive system is algebraizable, non-selfextensional and does not satisfy the deduction theorem. In addition, there exists no Gentzen calculus fully adequate (...) for it. Another presentation of the same deductive system can be obtained from a substructural Gentzen calculus. In this paper we use the framework of abstract algebraic logic to study a different deductive system which uses the aforementioned algebra under a scheme of “preservation of degrees of truth”. We characterize the resulting deductive system in a natural way by using the lattice filters of Wajsberg algebras, and also by using a structural Gentzen calculus, which is shown to be fully adequate for it. This logic is an interesting example for the general theory: it is selfextensional, non-protoalgebraic, and satisfies a “graded” deduction theorem. Moreover, the Gentzen system is algebraizable. The first deductive system mentioned turns out to be the extension of the second by the rule of Modus Ponens. (shrink)

Causation appears to present us with an interpretative difficulty. While arguably a redundant relation given fundamental physics, it is nevertheless apparently pragmatically indispensable. This chapter revisits certain arguments made previously by the author for these claims with the benefit of hindsight, starting with the role of causal models in the human sciences, and attempting to explain why it is not possible to straightforwardly ground such models in fundamental physics. This suggests that further constraints, going beyond physics, are needed to legitimate (...) such models. These supplementary constraints could be reified, but that would seem to conflict with the completeness of physics. A response is to emphasize the practical role of causal talk, and the author suggests that a fictionalist approach might be worth exploring. After clarifying fictionalism as a general approach, they carry out in some detail the project of clarifying what a fictionalist attitude to causation would involve. (shrink)

Early work on the frequency theory of probability made extensive use of the notion of randomness, conceived of as a property possessed by disorderly collections of outcomes. Growing out of this work, a rich mathematical literature on algorithmic randomness and Kolmogorov complexity developed through the twentieth century, but largely lost contact with the philosophical literature on physical probability. The present chapter begins with a clarification of the notions of randomness and probability, conceiving of the former as a property of a (...) sequence of outcomes, and the latter as a property of the process generating those outcomes. A discussion follows of the nature and limits of the relationship between the two notions, with largely negative verdicts on the prospects for any reduction of one to the other, although the existence of an apparently random sequence of outcomes is good evidence for the involvement of a genuinely chancy process. (shrink)

Atheists are frequently demonized as arrogant intellectuals, antagonistic to religion, devoid of moral sentiments, advocates of an "anything goes" lifestyle. Now, in this revealing volume, nineteen leading philosophers open a window on the inner life of atheism, shattering these common stereotypes as they reveal how they came to turn away from religious belief. These highly engaging personal essays capture the marvelous diversity to be found among atheists, providing a portrait that will surprise most readers. Many of the authors, for example, (...) express great affection for particular religious traditions, even as they explain why they cannot, in good conscience, embrace them. None of the contributors dismiss religious belief as stupid or primitive, and several even express regret that they cannot, or can no longer, believe. Perhaps more important, in these reflective pieces, they offer fresh insight into some of the oldest and most difficult problems facing the human mind and spirit. For instance, if God is dead, is everything permitted? Philosophers without Gods demonstrates convincingly, with arguments that date back to Plato, that morality is independent of the existence of God. Indeed, every writer in this volume adamantly affirms the objectivity of right and wrong. Moreover, they contend that secular life can provide rewards as great and as rich as religious life. A naturalistic understanding of the human condition presents a set of challenges--to pursue our goals without illusions, to act morally without hope of reward--challenges that can impart a lasting value to finite and fragile human lives. Collectively, these essays highlight the richness of atheistic belief--not only as a valid alternative to religion, but as a profoundly fulfilling and moral way of life. "This Atheists R Us compilation differs markedly in tone from Hitchens and Dawkins. Excellent fare for Christian small groups whose members are genuinely interested in the arguments raised by atheists." --Christianity Today "Readable, personal, and provocative.... Contrary to the popular image, atheism isn't all rebellious trumpets and defiant drums.... Here we have all the varieties of unreligious experience, a full symphony of unbelief." --Free Inquiry "Compelling and sophisticated arguments that religious people ought to confront." --Tikkun. (shrink)

Michel Onfray est l'un des philosophes français les plus connus - mais que sait-on de lui au juste? Apprécié ou controversé, il déclenche des réactions viscérales, trop souvent approximatives. Si certains l'aiment et le défendent sans réserve, d'autres le détestent sans le lire. De fait, on a souvent réduit son travail à deux ou trois livres médiatisés alors qu'il a écrit plus de soixante ouvrages, fort divers, où il est question de philosophie, de littérature, de poésie, de musique, de peinture, (...) de photographie. Cet essai biographique se propose donc d'explorer l'originalité de sa démarche. Il retrace, à partir des textes de Michel Onfray, les rencontres décisives qui ont propulsé ou infléchi sa pensée. Et dégage les caractéristiques d'une écriture singulière, tissée de morale, de vécu et de métaphysique incandescente. (shrink)

_Philosophy of Probability: Contemporary Readings_ is the first anthology to collect essential readings in this important area of philosophy. Featuring the work of leading philosophers in the field such as Carnap, Hájek, Jeffrey, Joyce, Lewis, Loewer, Popper, Ramsey, van Fraassen, von Mises, and many others, the book looks in depth at the following key topics: subjective probability and credence probability updating: conditionalization and reflection Bayesian confirmation theory classical, logical, and evidential probability frequentism physical probability: propensities and objective chances. The book (...) features a useful primer on the mathematics of probability, and each section includes an introduction by the editor, as well as a guide to further reading. A broad-ranging and highly accessible exploration of the subject, _Philosophy of Probability_ is ideal for any student of formal epistemology, philosophy of science, metaphysics, or philosophy of mathematics. (shrink)

This chapter discusses fundamental problems and prospects for ethical naturalism. Section 1 explains what is meant by “ethical naturalism” and surveys different versions of the view. Section 2 discusses the central philosophical challenge to ethical naturalism, viz., the “Normativity Objection.” Section 3 offers a battery of responses to it on behalf of the ethical naturalist. Section 4 explores a promising and novel approach to ethical naturalism, viz., a moral nativist theory that that combines a Chomskian approach to moral competence with (...) a structuralist metaphysics. (shrink)

Le statut de Pierre Bayle dans l'histoire des idées philosophiques et religieuses de l'époque classique a changé du tout au tout. D'observateur, il est devenu, dans l'historiographie actuelle, un acteur majeur dans les débats philosophiques, religieux et politiques de son époque, dont les écrits nourriront la réflexion des philosophes clandestins qui fondent les Lumières radicales. Ce volume d'études contribue à cette nouvelle interprétation du statut et de la portée du rationalisme de Bayle.

Pascal founds his interpretation of the Augustinian doctrine of the corruption of human nature on a philosophy of faith inherited from Montaigne: « we are Christians in just the same way as we are Périgordians or Germans » (Essais, II, 12) : this conception of « human faith » is analysed, in turn, by means of concepts drawn from Descartes (passion) and Gassendi (imagination). He thus leads us to a very modern conception of « human faith » – without grace (...) – as being the product of education and habit, a « sentiment » like so many others. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Response 2: “Pessimism of the Intellect, Optimism of the Will”Antonis BalasopoulosLet me begin with a few words on my title, which was chosen as reflecting the nature of the orientation of my work in the field of utopian studies and therefore also of my orientation toward the theme of this roundtable. As Francesca Antonini puts it in a recent essay, the phrase, which became associated with the work of (...) Antonio Gramsci though it originates in Romain Rolland, describes “the (seemingly contradictory) coexistence of a realistic description of the status quo, on the one hand, and a genuine commitment to the possibility of transforming reality, on the other.”1Anyone who has read Thomas More’s Utopia knows that Rolland’s and Gramsci’s formula very much describes that book’s bifurcation between a voice of relentless critique of what exists in Europe—and particularly in England—in More’s time (in Book 1) and a voice preoccupied with the description of an alternative, and arguably better, world (in Book 2). This bifurcation, it is equally well known, is part of the genetic material More’s book bequeathed to its successors, so that one calls a “utopia” a (mostly, though not always) literary text, where the “realistic description of the status quo” coexists with a vision arising out of a “commitment to the possibility of transforming reality”—admittedly in different admixtures and proportions of [End Page 544] the critical or negative and the anticipatory, projective or prefigurative elements. Though Ernst Bloch’s distinction between “cold” and warm” streams within Marxism is not identical to this first binary, it is certainly related to it, so that the “cold” stream is one that privileges the rational and analytical element (usually, I would propose, in relation to the part of the Marxist tradition we know as “critique of political economy”), while the “warm stream” is one that rather foregrounds that which relates to affect, to desire, hope, or fidelity to the transformative project (these are frequently correlated to those elements of Marxism that correspond to its also being a political and philosophically imbued creed).2 What follows here, then, explores the implications of the dialectical interplay between these two sets of oppositions—and hopefully justifies such insistence on dialectic itself.3To the extent that I can see, the basic themes and areas of concentration in my colleagues’ essays involve decolonizing utopian studies as a project; addressing, in that connection, the heterogeneity of utopianisms that emerge out of a number of marginalized subjectivities; the provenance of the questions of ecology, sustainability, animal life for the present and future of utopian studies (and to some degree, for utopianism’s past); and finally, the question of utopian forms of social activism and therefore also the search for spaces beyond the walls of academic insularity. These are important and largely interconnected questions, but I am afraid I will only have the space to touch on them briefly and sketchily here.In her cogent and multifaceted intervention, Caroline Edwards begins her reflections on the task of decolonizing utopian studies by noting the high investment of fin-de-siècle utopian fiction in notions of industrial progress, noting that such notions have involved “lasting damage” for “Black writers, artists, and thinkers.” Within this framework, she evokes Joseph R. Winters’s argument that “modern notions of progress and freedom are inherently flawed,” since they are founded on the oppressive traditions of capitalism and colonialism. In turn, this leads to an argument against the teleological futurism of most nineteenth-century utopias and in support of the deconstruction of “the idea of historical progress unfolding in Hegelian dialectical stages.” What is to replace this is a hauntological (after Derrida) conception of the relation between the present and the past, for which “the present is always a site of potential breaks and ruptures that can signify both dislocation and pain as well as novelty and possibility.” In a similar vein, Heather Alberro questions the teleological futurism of an earlier tradition of utopia, evoking Laurence Davis’s observation on the importance of [End Page 545] “neglected or suppressed possibilities for qualitatively better forms of life” within the “Now.” The desideratum is, again... (shrink)

La question du nihilisme a une longue histoire. Elle émerge au sein de l'idéalisme allemand, se déploie ensuite dans l'intelligentsia et la littérature russes, puis trouve chez Nietzsche et Heidegger ses explorations les plus influentes. Si le mot semble moins courant depuis un demi-siècle -- comme d'autres termes aussi chargés, pensons à celui d'aliénation -- les enjeux qu'il soulève demeurent toutefois au cœur de la pensée contemporaine. Les risques environnementaux nourrissent d'ailleurs plus que jamais l'inquiétante intuition que de puissantes dynamiques (...) de dévalorisation du réel sont à l'œuvre dans les sociétés modernes. Ces douze essais, qui font largement entendre la voix d'une nouvelle génération, n'ont pourtant pas pour but de défendre une thèse commune. Ils explorent plutôt les multiples aspects du nihilisme et se confrontent à ses grandes interprétations. Sont ainsi proposées autant de pistes pour rejoindre les débats sur le capitalisme, la technique, l'éducation, le patriarcat, le roman ou la démocratie. Pluralité revendiquée, qui illustre néanmoins un même ancrage dans les défis du monde actuel. (shrink)

Despite the readily available discussion on counseling supervision models for over a quarter of a century, there is little attention in the literature with respect to how developmental supervision models align with existential philosophy. One model, The Integrated Developmental Model (IDM), is a robust and well-accepted model of supervision with embedded undertones of existentialism requiring scholarly discussion. The primary goal of this article is to emphasize the parallels between the IDM and Sartre’s philosophical principles of existentialism thereby creating a meaning (...) making framework for supervisors to enhance developmental growth of their supervisees. (shrink)

In order to understand what we are talking about when we talk about joint attention, I will scrutinize how the mainstream view that builds on representational and intentionalist theories of mind is constituted. My aim is to show that much of the theory of joint attention is quite narrowly constructed and comes with tacit disciplinary biases that exclude much of what is existentially important in our practices of sharing our perceptions and guiding others to attend to the world in novel (...) ways. By using the frameworks of classical phenomenology and enactivist theories of mind, I aim to show how we, in our interactions, share our perceptions even with beings that are distinctly different from us. And, that this _difference_ in our personal worldview, our cognitive capacities, and our sensory modalities should be seen as a constitutive aspect of our practices of shared attention. I will articulate attention as a revelatory practice that aids us in discovering a shared world through the plurality, rather than similarity, of our perceptual actions. Attention is, in this sense, not solely driven by subjective intentions; it is also formed and co-constituted by my being addressed by the other. (shrink)

This chapter contains section titles: Abilities Concepts Conclusion.

Epistemological internalists have a problem about perceptual knowledge: how can perceptual experience both provide faithful information about the external world and justification for empirical belief? This is Sellars’s famous problem about “the given.” Chapter 12 argues, first, that this problem is not just for internalists—a version of it arises for naturalistic externalists. But, second, it argues that the problem can be solved within naturalistic bounds, by appealing to a category of causal relations called “intelligible causation.”.