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)

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].

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)

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)

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 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)

Includes essays on parapsychology and life after death by J. B. Rhine, David Hume, George Price, Plato, Rene Descartes, Thomas Hobbes, and John Locke, among others.

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.

Many researchers believe the tremendous industrial development over the past two centuries is unsustainable because it has led to unintended ecological deterioration. Despite the ever-growing attention sustainable supply-chain management has received, most SSCM research and models look at the consequences, rather than the antecedents or motives of such responsible practices. The few studies that explore corporate motives have remained largely qualitative, and large-scale empirical analyses are scarce. Drawing on multiple theories and combining supply-chain and business ethics literature, we purport that (...) instrumental, relational, and moral motives are behind a firm’s engagement in SSCM practices. Specifically, we examine the links between corporate motives, SSCM practices, and firm performance. Using a sample of 259 supply-chain firms in Germany, we empirically test five hypothesized relationships. Our results reveal that relational and moral motives are key drivers, and that firms exhibiting high levels of moral obligations tend to outperform those primarily driven by amoral considerations. Findings of this study contribute to multiple literatures espousing sustainability management and can help policy makers, stakeholder groups, and scholars develop more robust strategies for encouraging firms to practice SSCM. (shrink)

The Anatomy of Neoplatonism was the crowning achievement of A. C. Lloyd, the distinguished scholar of late ancient philosophy. He offers a rich and authoritative study of this school of thought, which was highly influential not only on subsequent philosophy but also on Christian theology. His discussion ranges over metaphysics, epistemology, logic, and language, and reveals the fundamental structure of Neoplatonist thought; the book is essential reading for all who work in this area. Lloyd shows that while Neoplatonism is not (...) a modern philosophy, it is indeed philosophy in the modern sense. (shrink)

This study investigated individual differences in the belief bias effect, which is the tendency to accept conclusions because they are believable rather than because they are logically valid. It was observed that the extent of an individual's belief bias effect was unrelated to a number of measures of reasoning competence. Instead, as predicted by mental models theory, it was related to a person's ability to generate alternative representations of premises: the more alternatives a person generated, the less likely they were (...) to show a belief bias effect. In contrast to belief effects, which were predicted by the number of alternatives generated, scores on logical reasoning tasks were predicted by an individual's understanding of the concept of logical necessity. These two mediating variables, ability to generate alternatives and understanding of logical necessity, were themselves predicted by similar variables, such as cognitive motivation and abstract reasoning ability. Our findings strongly support the mental models interpretation of the belief bias effect, and suggest that individual differences in this effect are associated with a specific ability to generate alternatives, rather than with a general logical competence. (shrink)

The current legal status and medical ethics of routine or religious penile circumcision of minors is a matter of ongoing controversy in many countries. We focus on the United Kingdom as an illustrative example, giving a detailed analysis of the most recent British Medical Association guidance from 2019. We argue that the guidance paints a confused and conflicting portrait of the law and ethics of the procedure in the UK context, reflecting deeper, unresolved moral and legal tensions surrounding child genital (...) cutting practices more generally. Of particular note is a lack of clarity around how to apply the “best interests” standard—ordinarily associated with time-sensitive proxy decision making regarding therapeutic options for a medically unwell patient—to a parental request for a medically unnecessary surgery to be carried out on the genitalia of a healthy child. Challenges arise in measuring and assigning weights to intended sociocultural or religious/spiritual benefits, and even to health-related prophylactic benefits, and in balancing these against potential physical, functional, and psychosexual risks or harms. Also of concern are apparently inconsistent safeguarding standards being applied to children based on their birth sex categorization or gender of rearing. We identify and discuss recent trends in British and international medical ethics and law, finding gradual movement toward a more unified standard for evaluating the permissibility of surgically modifying healthy children's genitals before they can meaningfully participate in the decision. (shrink)

Material Engagement Theory is currently driving a conceptual change in the archaeology of mind. Drawing upon the dictates of enactivism and active externalism, it specifically calls for a radical reconceptualization of mind and material culture. Unpersuaded by the common assumption that cognition is brain-bound, Malafouris argues in favour of a process ontology that situates thinking in action. In granting ontological primacy to material engagement, MET seeks to illuminate the emergence of human ways of thinking through the practical effects of the (...) material world. Considering that this is a characteristic example of a pragmatic take on cognition, this contemporary theoretical platform appears to share a lot with pragmatism. As of late, scholars working at the intersection of philosophy, semiotics, and cognitive science have made important steps towards the rapprochement between pragmatism and externalism. Looking to contribute to this growing corpus of work, the present paper focuses on MET’s relation to the pragmatism of Peirce, Dewey, and Mead. Having elsewhere recognized the overlap and complementarity between Malafouris’ and Peirce’s theories in particular, I developed a pragmatic and enactive theory of cognitive semiotics that is suitably geared to trace the nature, emergence, and evolution of material signs. Therefore, besides noting some obvious historical connections, I hereby aim to establish the theoretical backdrop upon which this composite theory is supposed to function, while also exploring new potential avenues. Given that this cognitive semiotic framework can be seen as a pragmatic extension of Malafouris’ enactivist approach to archaeology, the current paper delves into MET’s theoretical underpinnings, seeking to complement its working hypotheses and concepts with philosophical notions and ideas advanced long ago. This synthesis ultimately concludes with a call for the reconceptualization of ‘representation’ as a heuristic concept. (shrink)

In the 1960s and 1970s, scuba technology, underwater cameras, and documentarians revealed a long-hidden underwater world to the public. At this time oceanographic science was growing exponentially. Historians of the marine sciences have focused their studies of the period on institutional and military partnerships, and on the scientist-administrators who shaped oceanographic research institutions (such as the Scripps Institution of Oceanography, the Woods Hole Oceanographic Institution, and the British National Institute of Oceanography). Underwater habitat development during the 1960s and 1970s, however, (...) deserves greater attention than it has yet received, as it highlights a peculiar confluence of military, scientific, and popular interest, characteristic of the period, in the colonization of the seafloor. Existing accounts have focused on American habitats, notably Sealab and Tektite, since these were the largest and best funded. But this approach overemphasizes a Cold War narrative in which the sole protagonists of the habitat programs were the United States and the Soviet Union. At least 65 habitats were built between 1962 and 1991. Some were state-sponsored, with significant programs run by French, German, Japanese, and Canadian teams. This essay takes as a case study the Canadian Sublimnos habitat as well as the underwater exploration programs it helped launch in Newfoundland (Lora-I) and in the Arctic (Sub-Igloo). The Canadian case demonstrates that technological expertise and public enthusiasm for underwater exploration should not be solely understood with reference to Cold War interests of the two superpowers. Rather, the international range of habitat programs of the 1960s and 1970s reveals an expanding interest in the vertical underwater dimension that was fueled by numerous national scientific aims and territorial claims. (shrink)

As attention, in philosophy, is mainly discussed in the philosophy of mind, its ethical aspects have remained relatively unexplored. One notable exception is Iris Murdoch. Another philosopher, Maur...

In this brief but powerful book, acclaimed political philosopher C.B. Macpherson sets out in bold relief the essence of liberal democracy, both as it is currently conceived and as it might be reimagined. The Wynford edition includes a new Introduction by Frank Cunningham.

Studies of AGL have frequently used training and test stimuli that might provide multiple cues for learning, raising the question what subjects have actually learned. Using a selected subset of studies on humans and non‐human animals, Trotter et al. demonstrate how a meta‐analysis can be used to identify relevant experimental variables, providing a first step in asssessing the relative contribution of design features of grammars as well as of species‐specific effects on AGL.

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)