This is also where we begin investigating lattices of logics and varieties, rather than particular examples.

It is well known that classical propositional logic can be interpreted in intuitionistic propositional logic. In particular Glivenko's theorem states that a formula is provable in the former iff its double negation is provable in the latter. We extend Glivenko's theorem and show that for every involutive substructural logic there exists a minimum substructural logic that contains the first via a double negation interpretation. Our presentation is algebraic and is formulated in the context of residuated lattices. In the last part (...) of the paper, we also discuss some extended forms of the Kolmogorov translation and we compare it to the Glivenko translation. (shrink)

Equivalences and translations between consequence relations abound in logic. The notion of equivalence can be defined syntactically, in terms of translations of formulas, and order-theoretically, in terms of the associated lattices of theories. W. Blok and D. Pigozzi proved in [4] that the two definitions coincide in the case of an algebraizable sentential deductive system. A refined treatment of this equivalence was provided by W. Blok and B. Jónsson in [3]. Other authors have extended this result to the cases of (...) k-deductive systems and of consequence relations on associative, commutative, multiple conclusion sequents. Our main result subsumes all existing results in the literature and reveals their common character. The proofs are of order-theoretic and categorical nature. (shrink)

We develop a general algebraic and proof-theoretic study of substructural logics that may lack associativity, along with other structural rules. Our study extends existing work on substructural logics over the full Lambek Calculus [34], Galatos and Ono [18], Galatos et al. [17]). We present a Gentzen-style sequent system that lacks the structural rules of contraction, weakening, exchange and associativity, and can be considered a non-associative formulation of . Moreover, we introduce an equivalent Hilbert-style system and show that the logic associated (...) with and is algebraizable, with the variety of residuated lattice-ordered groupoids with unit serving as its equivalent algebraic semantics. Overcoming technical complications arising from the lack of associativity, we introduce a generalized version of a logical matrix and apply the method of quasicompletions to obtain an algebra and a quasiembedding from the matrix to the algebra. By applying the general result to specific cases, we obtain important logical and algebraic properties, including the cut elimination of and various extensions, the strong separation of , and the finite generation of the variety of residuated lattice-ordered groupoids with unit. (shrink)

Background: Withdrawal of treatment is a common practice in intensive care units when treatment is considered futile. Compassion is an important aspect of care; however, it has not been explored much within the context of treatment withdrawal in intensive care units. Objectives: The aim was to examine how concepts of compassion are framed, utilised and communicated by intensive care nurses in the context of treatment withdrawal. Design: The study employed a qualitative approach conducting secondary analysis of an original data set. (...) In the primary study, 13 nurses were recruited from three intensive care units within a large hospital in United Kingdom. Deductive framework analysis was used to analyse the data in relation to compassionate care. Ethical considerations: The primary study was approved by the local Research Ethics Committee and the hospital’s Research and Development services. Findings: Compassionate care was mostly directed to the patient’s family and was demonstrated through care and emotional support to the family. It was predominantly expressed through attempts to maintain the patient’s dignity by controlling symptoms, maintaining patient cleanliness and removing technical apparatus. Conclusion: This study’s findings provide insight about compassionate care during treatment withdrawal which could help to understand and develop further clinicians’ roles. Prioritising the family over the patient raised concerns among nurses, who motivated by compassion, may feel justified in taking measures that are in the interests of the family rather than the patient. Further work is needed to explore the ethics of this. (shrink)

The purpose of this paper is first to show that if X is any locally compact but not compact perfect Polish space and stands for the one-point compactification of X, while E X is the equivalence relation which is defined on the Polish group C(X,R +*) by where f, g are in C(X,R +*), then E X is induced by a turbulent Polish group action. Second we show that given any if we identify the n-dimensional unit sphere S n with (...) the one-point compactification of R n via the stereographic projection, while E n , r is the equivalence relation which is defined on the Polish group C r (R n ,R +*) by where f, g are in C r (R n ,R +*), then E n , r is also induced by a turbulent Polish group action. (shrink)

Given a positive universal formula in the language of residuated lattices, we construct a recursive basis of equations for a variety, such that a subdirectly irreducible residuated lattice is in the variety exactly when it satisfies the positive universal formula. We use this correspondence to prove, among other things, that the join of two finitely based varieties of commutative residuated lattices is also finitely based. This implies that the intersection of two finitely axiomatized substructural logics over FL + is also (...) finitely axiomatized. Finally, we give examples of cases where the join of two varieties is their Cartesian product. (shrink)

Substructural logics have received a lot of attention in recent years from the communities of both logic and algebra. We discuss the algebraization of substructural logics over the full Lambek calculus and their connections to residuated lattices, and establish a weak form of the deduction theorem that is known as parametrized local deduction theorem. Finally, we study certain interpolation properties and explain how they imply the amalgamation property for certain varieties of residuated lattices.

All variants of pragmatism share the flaw that their concepts of practice rely on the idea of the local value of actions with respect only to locally defined aims and not on the criterion of a universal goodness. This paper claims that such a criterion can be found with the aid of an ontologically founded theory of the Good, which regards forms not as solely noematic universals, but as real, though abstract, entities. The idea of goodness is derived from the (...) thesis of the immediate knowledge of the Good. Further consequences of this form of theoretical foundation of goodness are the abandonment of the dogma of the immediate reference of language to the world as well as of the dogma of the primacy of acting over thinking. (shrink)

The Bioethics Committee of the Church of Greece, headed by Metropolitan Nikolaos of Mesogaia and Lavreotiki, was founded a few years ago, bringing together clergy, medical and legal experts. In his address, Metropolitan Nikolaos explains the concerns and the conditions that led to its foundation and he outlines some of the activities and the principles of the Committee. This address, a somewhat broad-brush presentation of the state of bioethics in Greece, is offered as a proposal as to how (...) the Church can work within a secular state, and how it can attempt to raise awareness about the spiritual aspect of bioethical concerns. Overall, the Bioethics Committee tries to promote public dialogue about sensitive bioethical issues, and to make sure that the spiritual perspective is duly informed by medical science. In addition, the Committee tries to make sure that the spiritual perspective is considered by the state in the face of existing or emerging law that touches on bioethical concerns. (shrink)

The article explores the right to cultural identity, advocating for the freedom to choose and change cultural affiliations as a critical human right. It emphasizes that while cultural identity is essential for human dignity and personal freedom, this right remains underexplored in contemporary human rights discourse, leading to the essentialization of culture and neglecting its dynamic nature. By promoting the recognition of the freedom to redefine cultural identity, the article positions this freedom as crucial to human rights. It supports this (...) view with key international legal instruments and examines relevant case law from the European Court of Human Rights and other bodies. Significant cases, including Ciubotaru v. Moldova and Sandra Lovelace v. Canada, illustrate the challenges individuals face in changing their identities and highlight existing legal protections. Ultimately, recognizing this right ensures respect for personal growth and evolving worldviews under human rights law. (shrink)

Freedom of will is crucial to the concept of guilt in criminal law, yet its existence is a major philosophical issue. Since the 17th century, classical mechanics and relativistic physics suggested determinism, negating free will. However, quantum mechanics, emerging in the 1930 s, challenged this deterministic view, leaving the reality of free will debatable. Natural sciences cannot conclusively resolve the free will question, allowing jurisprudence to adopt a normative-functional approach: people must be seen as free to uphold the rule of (...) law. Denying free will would eliminate the concept of guilt in criminal law, a riskier outcome than assuming free will, even if fictional. (shrink)

All known structural extensions of the substructural logic $\textbf{FL}_{\textbf{e}}$, the Full Lambek calculus with exchange/commutativity (corresponding to subvarieties of commutative residuated lattices axiomatized by $\{\vee, \cdot, 1\}$ -equations), have decidable theoremhood; in particular all the ones defined by knotted axioms enjoy strong decidability properties (such as the finite embeddability property). We provide infinitely many such extensions that have undecidable theoremhood, by encoding machines with undecidable halting problem. An even bigger class of extensions is shown to have undecidable deducibility problem (the (...) corresponding varieties of residuated lattices have undecidable word problem); actually with very few exceptions, such as the knotted axioms and the other prespinal axioms, we prove that undecidability is ubiquitous. Known undecidability results for non-commutative extensions use an encoding that fails in the presence of commutativity, so and-branching counter machines are employed. Even these machines provide encodings that fail to capture proper extensions of commutativity, therefore we introduce a new variant that works on an exponential scale. The correctness of the encoding is established by employing the theory of residuated frames. (shrink)

We provide an alternative proof of the decidability of the equational theory of lattices. The proof presented here is quite short and elementary.

Although Artificial Intelligence (AI) and Machine Learning (ML) are attracting a lot of scientific and engineering attention nowadays, nothing up to now has been achieved to reach the level of building machines that possess human-like intelligence. Though, the engineering community continuously claims that several engineering problems are solved using AI or ML. Here, it is argued that engineers are not being able to build intelligent machines, implying that the systems claimed to have AI/ML belong to different engineering domains. The base (...) of the syllogism is the existence of four main obstacles on which extensive elucidation is performed. In addition, attempt to clear out the developed confusion, mis-use and ab-use of the phrases “Artificial Intelligence” and “Machine Learning” by scientists and engineers is carried out. Furthermore, mathematical, and philosophical approaches are also mentioned that strengthen the argument against AI implementability as part of the whole syllogism. Finally, an alternative approach (not being unique) is suggested and discussed for performing research on AI and ML by the engineers. It is based on complexity theory and non-linear adaptive systems and provides the benefit of eliminating the before mentioned pragmatic and philosophical obstacles that engineers are facing and ignoring, without creating confusion on this scientific endeavor. (shrink)

Within a geometric and algebraic framework, the structures which are related to the spin-statistics connection are discussed. A comparison with the Berry-Robbins approach is made. The underlying geometric structure constitutes an additional support for this approach. In our work, a geometric approach to quantum indistinguishability is introduced which allows the treatment of singlevaluedness of wave functions in a global, model independent way.

If X is a locally compact Polish space, then LSC denotes the compact Polish space of lower semi-continuous real-valued functions on X equipped with the topology of epi-convergence.Our purpose in this article is to prove the following: if –∞ < α < β < ∞ and –∞ < a < b < ∞, while r ∈ ℕ \ {0}, then the set CV of all f ∈ LSC for which there is u ∈ Cr such that for any v ∈ (...) Cr we have that ∫αβf , v ′)dx ≥ ∫αβf , v )dx is not Borel. (shrink)

Με αφορμή ένα γεωμετρικό αμφορέα του επιγάστριου αττικο-κυκλαδικού τύπου από το νεκροταφείο της Ελεύθερνας, διακοσμημένο με τρεις μετόπες ομόκεντρων κύκλων στη ζώνη της κοιλιάς, συζητούνται τα προβλήματα τυπολογίας και προέλευσης αυτής της χαρακτηριστικής κατηγορίας αγγείων στο φως της νεότερης έρευνας. Η τεχνοτροπία όλων των αμφορέων αυτού του τύπου, δηλ. με δύο ή με τρεις μετόπες ομοκέντρων κύκλων, ακολουθεί πάντοτε τον αττικό ρυθμό, αλλά ενώ η αττική καταγωγή των διμετο- πικών αμφορέων του τύπου ουδέποτε αμφισβητήθηκε, η καταγωγή των τριμετοπικών αμφορέων, που (...) παρουσιάζουν ευρύτατη διάδοση στο Αιγαίο και κυρίως στις Κυκλάδες, άλλοτε αποδίδεται στην Αττική και άλλοτε στις Κυκλάδες. Στη μελέτη αυτή επιχειρείται ένας επαναπροσδιορισμός της τυπολογίας του επιγάστριου ομοκεντρικού αμφορέα και συζητείται διεξοδικά η κατηγορία των αμφορέων με τρεις μετόπες. (shrink)

Two constructions for adding an involution operator to residuated ordered monoids are investigated. One preserves integrality and the mingle axiom x 2x but fails to preserve the contraction property xx 2. The other has the opposite preservation properties. Both constructions preserve commutativity as well as existent nonempty meets and joins and self-dual order properties. Used in conjunction with either construction, a result of R.T. Brady can be seen to show that the equational theory of commutative distributive residuated lattices (without involution) (...) is decidable, settling a question implicitly posed by P. Jipsen and C. Tsinakis. The corresponding logical result is the (theorem-) decidability of the negation-free axioms and rules of the logic RW, formulated with fusion and the Ackermann constant t. This completes a result of S. Giambrone whose proof relied on the absence of t. (shrink)

Seize plaques de marbres du gymnase d’Andros ont été trouvées à l’état fragmentaire. De longueur et largeur irrégulières, mais de hauteur constante, elles vont s’épaississant régulièrement d’un long côté à l’autre. Des graffiti sont inscrits sur le côté vertical et étroit de ces plaques, plus rarement aussi sur un autre côté. Ces plaques rappellent les bancs des gymnases et des palestres où, dans ce que l’on appelle l’ephebeion, les jeunes gens assistaient aux leçons. Les graffiti trouvés dans cette pièce reflètent (...) la vanité des éphèbes. Le soin de la gravure sur deux des plaques (n° 1-2) permet de déduire que les graffiti ont d’abord été inscrits sous les auspices d’un magistrat tel que le gymnasiarque. Le nom de l’éphèbe parvenu à l’âge adulte pouvait être effacé et remplacé par un autre nom, mais la gravure ne révèle plus le même soin. L’étude de la prosopographie d’Andros et du contexte historique permet de dater les plaques du début du IIe s. av. J.-C. leur utilisation s’étendant au cours des IIe et Ier s. av. J.-C. (shrink)