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 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)

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)

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)

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 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].

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)

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)

In this long-awaited book, Antony Duff offers a new perspective on the structures of criminal law and criminal liability. His starting point is a distinction between responsibility (understood as answerability) and liability, and a conception of responsibility as relational and practice-based. This focus on responsibility, as a matter of being answerable to those who have the standing to call one to account, throws new light on a range of questions in criminal law theory: on the question of criminalisation, which can (...) now be cast as the question of what we should have to answer for, and to whom, under the threat of criminal conviction and punishment; on questions about the criminal trial, as a process through which defendants are called to answer, and about the conditions (bars to trial) given which a trial would be illegitimate; on questions about the structure of offences, the distinction between offences and defences, and the phenomena of strict liability and strict responsibility; and on questions about the structures of criminal defences. The net result is not a theory of criminal law; but it is an account of the structure of criminal law as an institution through which a liberal polity defines a realm of public wrongdoing, and calls those who perpetrate (or are accused of perpetrating) such wrongs to account. (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)

The question "What can justify criminal punishment ?" becomes especially insistent at times, like our own, of penal crisis, when serious doubts are raised not only about the justice or efficacy of particular modes of punishment, but about the very legitimacy of the whole penal system. Recent theorizing about punishment offers a variety of answers to that question-answers that try to make plausible sense of the idea that punishment is justified as being deserved for past crimes; answers that try to (...) identify some beneficial consequences in terms of which punishment might be justified; as well as abolitionist answers telling us that we should seek to abolish, rather than to justify, criminal punishment. This book begins with a critical survey of recent trends in penal theory, but goes on to develop an original account (based on Duff's earlier Trials and Punishments) of criminal punishment as a mode of moral communication, aimed at inducing repentance, reform, and reconciliation through reparation-an account that undercuts the traditional controversies between consequentialist and retributivist penal theories, and that shows how abolitionist concerns can properly be met by a system of communicative punishments. In developing this account, Duff articulates the "liberal communitarian" conception of political society (and of the role of the criminal law) on which it depends; he discusses the meaning and role of different modes of punishment, showing how they can constitute appropriate modes of moral communication between political community and its citizens; and he identifies the essential preconditions for the justice of punishment as thus conceived-preconditions whose non-satisfaction makes our own system of criminal punishment morally problematic. Punishment, Communication, and Community offers no easy answers, but provides a rich and ambitious ideal of what criminal punishment could be-an ideal of what criminal punishment cold be-and ideal that challenges existing penal theories as well as our existing penal theories as well as our existing penal practices. (shrink)

In a series of influential articles, Rae Langton has argued that Austinian speech-act theory can illuminate the way in which pornography contributes to the subordination of women. I will argue that Langton’s application of Austin is incorrect. In earlier work, I have argued against Langton’s view on the grounds that being subordinated is not the sort of condition that can be brought about through an illocutionary act. In this paper, however, I will set aside that objection and focus instead on (...) Langton’s claims about the type of illocutionary act pornography might be. I will argue, first, that Langton cannot say what she wants to say about the content of pornography and still have it that pornography is a verdictive or exercitive speech act, in Austin’s sense. Verdictive and exercitive speech acts do not, and cannot describe the states of affairs they bring about. Second, I will argue that pornography does not in any case satisfy the conditions necessary for a speech act to be verdictive or exercitive. Specifically, there do not exist accepted background conventions that would give the pornographer authority to render verdicts or to take executive action. This objection to Langton’s view has been made before; I’ll show that her response to the objection betrays a confusion of authority with power. (shrink)

The present article commences analyzing the origins and influences of the religious discourse on the configuration of the modern constitutional discourse and the contributions of the jus-positivism in the consolidation of this sacred-civil language. The second issue is the definition of the U.S. Constitution as a mixed and not as a democratic constitution, with regard to the influences of Plato, Aristotle, Cicero and Polybius to the Drafters of the first modern constitutional text; stability and equilibrium took preference over democracy in (...) a wide sense. I also analyze how the Drafter’s decision has conditioned the modern constitutional system up to the present. (shrink)

In ”Embodiment and Epistemology,” Louise Antony considers a kind of ”Cartesian epistemology” according to which, so far as knowing goes, knowers could be completely disembodied, that is, pure Cartesian egos. Antony examines a number of recent challenges to Cartesian epistemology, particularly challenges from feminist epistemology. She contends that we might have good reason to think that theorizing about knowledge can be influenced by features of our embodiment, even if we lack reason to suppose that knowing itself varies relative to such (...) features. She also argues that a masculinist bias can result in mishandling of cognitive differences in cases where they actually exist and she examines a number of the ways in which the maleness of philosophy has, according to feminists, distorted epistemology. Even if a Cartesian approach offers one indispensable part of a comprehensive epistemology, according to Antony, we still need an epistemology that answers questions raised by our everyday, embodied lives. (shrink)

A complete history of Islamic political thought from early Islam to the present Now in its 2nd edition, this textbook describes and interprets all schools of Islamic political thought, their origins, inter-connections and meaning. It examines the Qur'an, the early Caliphate, classical Islamic philosophy and the political culture of the Ottoman and other empires. It covers major thinkers such as Averroes and Ibn Taymiyya as well as a number of lesser authors, and Ibn Khaldun is presented as one of the (...) most original political theorists ever. It draws on a wide range of sources including writings on religion, law, philosophy and statecraft expressed in treatises, handbooks and political rhetoric. The new edition analyses the connections between religion and politics, covering the most recent developments in Islamic political thought and the most recent historical scholarship. It ends with a critical survey of reformism and Islamism from the late-19th century up to the present day. (shrink)

Given Locke’s views on primary and secondary qualities, it seems he is committed to there being real underlying properties in objects, the arrangement and disposition of which underlies and produces the observed properties of that object. It might be natural to think that these primary qualities provide a general system for classifying objects into classes: that we could delineate the real kinds of objects in nature by looking at what their real primary qualities were. A list of the particular qualities (...) of some object would define that object perfectly precisely; by looking at which properties were shared or divergent between this object and others, we group the object with others in the same kind. These real kinds in nature can play several important theoretical roles. They can support real scientific understanding, including inductive inferences (iof things are of the same real kind, then observed properties of one support the inference to unobserved properties of the other); they can form the basis for our judgements about the powers and capacities of particular objects, based on the what other things of that type can do; and these real kinds can provide the meanings for general terms: the thought being that the meanings of class names are empty without real classes to which they correspond. (shrink)

Does the unconscious exist? Cultural critic Antony Easthope answers with a witty, lucid, informed "yes" and draws out its implications for the way we live, how we enjoy art, and how we think about people in society and history. Drawing on the writings of Freud and Lacan, he argues that the study of the unconscious is a way of analyzing meanings across culture as an effect of desire. Easthope tests for unconscious significance in an amazing variety of examples, including jokes, (...) Tampax advertisements, Hamlet , Hitchcock's Psycho , the life and death of Princess Di, Terminator 2 , Bob Geldof's autobiography, and the film Titanic. (shrink)

Rather than entailing that a particular outcome will occur, many scientific theories only entail that an outcome will occur with a certain probability. Because scientific evidence inevitably falls short of conclusive proof, when choosing between different theories it is standard to make reference to how probable the various options are in light of the evidence. A full understanding of probability in science needs to address both the role of probabilities in theories, or chances, as well as the role of probabilistic (...) judgment in theory choice. In this chapter, the author introduces and distinguishes the two sorts of probability from one another and attempt to offer a satisfactory characterization of how the different uses for probability in science are to be understood. A closing section turns to the question of how views about the chance of some outcome should guide our confidence in that outcome. (shrink)

Con el fin de desentrañar el papel que las más actuales corrientes filosóficas desempeñan en la educación, los autores de este libro parten de un crítico análisis de los acontecimientos de mayo del 68.Se intuyen dos vías de intervención de la filosofía sobre el pensamiento y la práctica educativa: por un lado, la tendencia anti constructivista a los discursos posmodernos, sistémicos y tecnológicos ; por otro, la persistencia del humanismo asentado ahora en el intento de prolongar la modernidad" y en (...) nuevas perspectivas tales como el pacifismo y el ecologismo. (shrink)

Studie over het theologische en filosofische gedachtengoed van de franciscaanse scholasticus (ca. 1265-1308).

This study proposes that Neoplatonism, while not a modern philosophy, is philosophy in the modern sense. Lloyd analyzes the key structures that underlie the dogmas of the Neoplatonic world picture, including the concept of emanation, the return of the soul to the One, the place of mystical knowledge, epistemology, and Porphyry's theory of predication, and shows that they rest on original but intelligible concepts and arguments.