Recently the cosmological evolution of the universe has been considered where 3-dimensional spatial topology undergone drastic changes. The process can explain, among others, the observed smallness of the neutrino masses and the speed of inflation. However, the entire evolution is perfectly smooth from 4-dimensional point of view. Thus the raison d’être for such topology changes is the existence of certain non-standard 4-smoothness on R4 already at very early stages of the universe. We show that the existence of such smoothness can (...) be understood as a byproduct of the quantumness of the origins of the universe. Our analysis is based on certain formal aspects of the quantum mechanical lattice of projections of infinite dimensional Hilbert spaces where formalization reaches the level of models of axiomatic set theory. (shrink)

For a finite von Neumann algebra factor M, the projections form a modular ortholattice L(M). We show that the equational theory of L(M) coincides with that of some resp. all L(ℂ n × n ) and is decidable. In contrast, the uniform word problem for the variety generated by all L(ℂ n × n ) is shown to be undecidable.

We analyse the projected future evolution of the distribution of voting power and related quantities under the qualified majority decision rule for the Council of Ministers of the EU, prescribed by the forthcoming EU Reform Treaty. Our projections are based on the demographic changes forecast by eurostat [4] for the period stretching from the present to the middle of the 21st Century. We use a method similar to the one we used in [6], [7], [8] and [9].

We characterize all finitary consequence relations over S4.3, both syntactically, by exhibiting so-called passive rules that extend the given logic, and semantically, by providing suitable strongly adequate classes of algebras. This is achieved by applying an earlier result stating that a modal logic L extending S4 has projective unification if and only if L contains S4.3. In particular, we show that these consequence relations enjoy the strong finite model property, and are finitely based. In this way, we extend the known (...) results by Bull and Fine, from logics, to consequence relations. We also show that the lattice of consequence relations over S4.3 is countable and distributive and it forms a Heyting algebra. (shrink)

Volume 8 of this landmark edition follows Peirce from May 1890 through July 1892—a period of turmoil as his career unraveled at the U.S. Coast and Geodetic Survey. The loss of his principal source of income meant the beginning of permanent penury and a lifelong struggle to find gainful employment. His key achievement during these years is his celebrated Monist metaphysical project, which consists of five classic articles on evolutionary cosmology. Also included are reviews and essays from The Nation in (...) which Peirce critiques Paul Carus, William James, Auguste Comte, Cesare Lombroso, and Karl Pearson, and takes part in a famous dispute between Francis E. Abbot and Josiah Royce. Peirce's short philosophical essays, studies in non-Euclidean geometry and number theory, and his only known experiment in prose fiction complete his production during these years. Peirce's 1883-1909 contributions to the Century Dictionary form the content of volume 7 which is forthcoming. (shrink)

Praise for Volume 1: "... a first-rate edition, which supersedes all other portable Peirces.... all the Peirce most people will ever need." —Louis Menand, The New York Review of Books Volume 2 of this convenient two-volume chronological reader’s edition provides the first comprehensive anthology of the brilliant American thinker Charles Sanders Peirce’s mature philosophy. A central focus of Volume 2 is Peirce’s evolving theory of signs and its appplication to his pragmatism.

The Bentham Project is delighted to announce a call for papers for “Bentham and Australia: Convicts, Utility, and Empire”, a conference to be held at University College London on 11-12 April 2019 to mark the forthcoming publication of Writings on Australia, a volume of The Collected Works of Jeremy Bentham. The conference will explore themes such as the influence and impact of Bentham’s ideas on the theory and practice of punishment in convict Australia, on advocates and opponents of co...

Praise for Volume 1: "... a first-rate edition, which supersedes all other portable Peirces.... all the Peirce most people will ever need." —Louis Menand, The New York Review of Books Volume 2 of this convenient two-volume chronological reader’s edition provides the first comprehensive anthology of the brilliant American thinker Charles Sanders Peirce’s mature philosophy. A central focus of Volume 2 is Peirce’s evolving theory of signs and its appplication to his pragmatism.

In this paper subvarieties of pseudocomplemented distributive lattices are classified by their unification type. We determine the unification type of every particular unification problem in each subvariety of pseudocomplemented distributive lattices.

The research discussed in this paper centers around the convergence of extended reality (XR) platforms, computational design, digital fabrication, and critical urban study practices. Its aim is to cultivate interdisciplinary and multiscalar approaches within these domains. The research endeavor represents a collaborative effort between two primary disciplines: critical urban studies, which prioritize socio-environmental justice, and integrated digital design to production, which emphasize the realization of volumetric or voxel-based structural systems. Moreover, the exploration encompasses augmented reality to assess its utilization in (...) both the assembly process of the structures and the integration of phygital (physical and digital) data with the physical environment. Within the context of these research scopes, this paper introduces FabriCity-XR as an interactive phygital installation. In addition to presenting an overview of the integrated research driven and performative design to production methodologies, the project showcases the practical implementation of web-based augmented reality trails, eliminating the requirement for external applications for interaction. This approach allows users to seamlessly navigate and engage with phygital content overlaid on physical objects using their personal smart devices. The result is a captivating and immersive user experience that effectively merges the physical and digital realms. (shrink)

The projection postulate is the source of a long standing controversy in the interpretation of the axiomatic foundations of quantum mechanics. In a sense which is made precise in chapter II the projection postulate is a mathematical theorem easily derivable within the mathematical framework of the theory. This theorem receives a clear and straightforward interpretation if Luders' rule is given only statistical significance. Under the assumption that an interpretation of quantum mechanics has to provide an account of the process of (...) measurement as a change on the state of individual systems, however, additional interpretive assumptions are needed in order to arrive at a satisfactory interpretation of Luders' rule. ;I show in chapter III that there are serious problems with the usual interpretation by making explicit in a quantum logical framework the concept of individual state and minimal disturbance implicit in this view. In chapter IV I show that it is possible to formulate a natural interpretation of Luders' rule as a description of 'dispersion free' state transformations within a semantic framework that takes states of individual systems to be represented by Boolean ultrafilters , as opposed to the usual approach that takes individual states to be represented by lattice ultrafilters. This lay the ground for the clarification of several interpretive issues in the foundations of quantum mechanics. It is shown that Luders' rule can be justified as a description of individual state transformations without the need of imposing additional assumptions. The derivation of Luders' rule within the semantic framework of B-states shows an important connection between interpretation that take individual state to be 'relative' to experimental situations and the statistical structure of the theory. (shrink)

Part I of this paper is developed in the tradition of Stone-type dualities, where we present a new topological representation for general lattices (influenced by and abstracting over both Goldblatt's [17] and Urquhart's [46]), identifying them as the lattices of stable compact-opens of their dual Stone spaces (stability refering to a closure operator on subsets). The representation is functorial and is extended to a full duality.In part II, we consider lattice-ordered algebras (lattices with additional operators), extending the Jónsson and (...) Tarski representation results [30] for Boolean algebras with Operators. Our work can be seen as developing, and indeed completing, Dunn's project of gaggle theory [13, 14]. We consider general lattices (rather than Boolean algebras), with a broad class of operators, which we dubb normal, and which includes the Jónsson-Tarski additive operators. Representation of l-algebras is extended to full duality. (shrink)

I discuss the connection between the Hamiltonian and path integral approaches for fermionic fields. I show how the temporal Wilson projection operators appear naturally in a lattice action. I also carefully treat the insertion of a chemical potential term.

The extended semantic realism (ESR) model proposes a new theoretical perspective which embodies the mathematical formalism of standard (Hilbert space) quantum mechanics (QM) into a noncontextual framework, reinterpreting quantum probabilities as conditional instead of absolute. We provide in this review an overall view on the present status of our research on this topic. We attain in a new, shortened way a mathematical representation of the generalized observables introduced by the ESR model and a generalization of the projection postulate of elementary (...) QM. Basing on these results we prove that the Bell-Clauser-Horne-Shimony-Holt (BCHSH) inequality, a modified BCHSH inequality and quantum predictions hold together in the ESR model because they refer to different parts of the picture of the physical world supplied by the model. Then we show that a new mathematical representation of mixtures must be introduced in the ESR model which does not coincide with the standard representation in QM and avoids some deep problems that arise from the representation of mixtures provided by QM. Finally we get a nontrivial generalization of the Lüders postulate, which is justified in a special case by introducing a reasonable physical assumption on the evolution of the compound system made up of the measured system and the measuring apparatus. (shrink)

Following Birkhoff and von Neumann, quantum logic has traditionally been based on the lattice of closed linear subspaces of some Hilbert space, or, more generally, on the lattice of projections in a von Neumann algebra A. Unfortunately, the logical interpretation of these lattices is impaired by their nondistributivity and by various other problems. We show that a possible resolution of these difficulties, suggested by the ideas of Bohr, emerges if instead of single projections one considers elementary (...) propositions to be families of projections indexed by a partially ordered set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{C}(A)}$$\end{document} of appropriate commutative subalgebras of A. In fact, to achieve both maximal generality and ease of use within topos theory, we assume that A is a so-called Rickart C*-algebra and that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{C}(A)}$$\end{document} consists of all unital commutative Rickart C*-subalgebras of A. Such families of projections form a Heyting algebra in a natural way, so that the associated propositional logic is intuitionistic: distributivity is recovered at the expense of the law of the excluded middle. Subsequently, generalizing an earlier computation for n × n matrices, we prove that the Heyting algebra thus associated to A arises as a basis for the internal Gelfand spectrum (in the sense of Banaschewski–Mulvey) of the “Bohrification” \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\underline A}$$\end{document} of A, which is a commutative Rickart C*-algebra in the topos of functors from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{C}A}$$\end{document} to the category of sets. We explain the relationship of this construction to partial Boolean algebras and Bruns–Lakser completions. Finally, we establish a connection between probability measures on the lattice of projections on a Hilbert space H and probability valuations on the internal Gelfand spectrum of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\underline A}$$\end{document} for A = B(H). (shrink)

Following Birkhoff and von Neumann, quantum logic has traditionally been based on the lattice of closed linear subspaces of some Hubert space, or, more generally, on the lattice of projections in a von Neumann algebra A. Unfortunately, the logical interpretation of these lattices is impaired by their nondistributivity and by various other problems. We show that a possible resolution of these difficulties, suggested by the ideas of Bohr, emerges if instead of single projections one considers elementary (...) propositions to be families of projections indexed by a partially ordered set C(A) of appropriate commutative subalgebras of A. In fact, to achieve both maximal generality and ease of use within topos theory, we assume that A is a so-called Rickart C*-algebra and that C(A) consists of all unital commutative Rickart C*-subalgebras of A. Such families of projections form a Hey ting algebra in a natural way, so that the associated propositional logic is intuitionistic: distributivity is recovered at the expense of the law of the excluded middle. Subsequently, generalizing an earlier computation for n x n matrices, we prove that the Heyting algebra thus associated to A arises as a basis for the internal Gelfand spectrum (in the sense of Banaschewski-Mulvey) of the "Bohrification" A̱ of A, which is a commutative Rickart C*-algebra in the topos of functors from C(A) to the category of sets. We explain the relationship of this construction to partial Boolean algebras and Bruns-Lakser completions. Finally, we establish a connection between probability measures on the lattice of projections on a Hubert space H and probability valuations on the internal Gelfand spectrum of A̱ for A = B(H). (shrink)

A normal state on a von Neumann algebra defines a countably additive probability measure over its projection lattice. The von Neumann algebras familiar from ordinary QM are algebras of all the bounded operators on a Hilbert space H, aka Type I factor von Neumann algebras. Their normal states are density operator states, and can be pure or mixed. In QFT and the thermodynamic limit of QSM, von Neumann algebras of more exotic types abound. Type III von Neumann algebras, for (...) instance, have no pure normal states; the pure states they do have fail to be countably additive. I will catalog a number of temptations to accord physical significance to non-normal states, and then give some reasons to resist these temptations: pure though they may be, non-normal states on non-Type I factor von Neumann algebras can't do the interpretive work we've come to expect from pure states on Type I factors; our best accounts of state preparation don't work for the preparation of non-normal states; there is a sense in which non-normal states fail to instantiate the laws of quantum mechanics. (shrink)

This paper explores the status of the von Neumann-Luders state transition rule (the "projection postulate") within "real-logic" quantum logic. The entire discussion proceeds from a reading of the Luders rule according to which, although idealized in applying only to "minimally disturbing" measurements, it nevertheless makes empirical claims and is not a purely mathematical theorem. An argument (due to Friedman and Putnam) is examined to the effect that QL has an explanatory advantage over Copenhagen and other interpretations which relativize truth-value assignments (...) to experimental arrangements. Two versions of QL, the lattice-theoretic (LT) and partial-Boolean-algebra (PBA), are considered. It turns out that the projection postulate is intimately connected with the choice of conditional connective for QL. The effect of the projection postulate is obtained with the Sasaki conditional. However, this choice is found to require extra assumptions, on both the LT and PBA versions, which are either just as ad hoc as the projection postulate itself or indefensible from within the real-logic QL framework. (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)

The thirteenth-century logician Lambert of Auxerre was well known for his Summa Lamberti, or simply Logica, written in the mid-1250s, which became an authoritative textbook on logic in the Western tradition. Our knowledge of medieval logic comes in great part from Lambert's Logica and three other texts: William of Sherwood's Introductiones in logicam, Peter of Spain's Tractatus, and Roger Bacon's Summulae dialectics. Of the four, Lambert's work is the best example of question-summas that proceed principally by asking and answering questions (...) on the subject matter. Thomas S. Maloney's translation of Logica, the only complete translation of this work in any language, is a milestone in the study of medieval logic. More than simply a translation, Maloney's project is a critical, comprehensive study of Lambert's logic situated in the context of his contemporaries and predecessors. As such, it offers a wealth of annotation and commentary. The lengthy introduction and extensive notes to the text explain the origin, theoretical context, and intricacies of the text and its doctrines. Maloney also addresses the disputed issues of authorship, date, and place of publication of the Summa Lamberti and makes available to the English-only audience the French, German, and Italian secondary sources--all translated--that are needed to enter the discussion. "Thomas S. Maloney fully commands the primary and secondary sources necessary to elucidate Lambert's Logica. An expert on Roger Bacon's philosophy, he demonstrates a rare proficiency in medieval Latin and scholastic logic. His references to sources from the ancient (Aristotle and Boethius) and medieval worlds are apposite, perspicuous, and useful. The volume's presentation with an appropriate introduction and commentary in the endnotes will no doubt establish it as an indispensable resource for scholars in the twenty-first century." --Alan Perreiah, University of Kentucky. (shrink)

The variety DMM of De Morgan monoids has just four minimal subvarieties. The join-irreducible covers of these atoms in the subvariety lattice of DMM are investigated. One of the two atoms consisting of idempotent algebras has no such cover; the other has just one. The remaining two atoms lack nontrivial idempotent members. They are generated, respectively, by 4-element De Morgan monoids C4 and D4, where C4 is the only nontrivial 0-generated algebra onto which finitely subdirectly irreducible De Morgan monoids (...) may be mapped by noninjective homomorphisms. The homomorphic preimages of C4 within DMM constitute a proper quasivariety, which is shown to have a largest subvariety U. The covers of the variety within U are revealed here. There are just ten of them. In exactly six of these ten varieties, all nontrivial members have C4 as a retract. In the varietal join of those six classes, every subquasivariety is a variety—in fact, every finite subdirectly irreducible algebra is projective. Beyond U, all covers of [or of ] within DMM are discriminator varieties. Of these, we identify infinitely many that are finitely generated, and some that are not. We also prove that there are just 68 minimal quasivarieties of De Morgan monoids. (shrink)

By quoting extensively from unpublished letters written by John von Neumann to Garret Birkhoff during the preparatory phase (in 1935) of their ground-breaking 1936 paper that established quantum logic, the main steps in the thought process leading to the 1936 Birkhoff–von Neumann paper are reconstructed. The reconstruction makes it clear why Birkhoff and von Neumann rejected the notion of quantum logic as the projection lattice of an infinite dimensional complex Hilbert space and why they postulated in their 1936 paper (...) that the quantum propositional system should be isomorphic to an abstract projective geometry. Looking at the paper now I see, that I forgot to say this, which should be said somewhere in the first §: That while common logics did apply to quantum mechanics, if the notion of simultaneous measurability is introduced as an auxiliary notion, we wished to construct a logical system, which applies directly to quantum mechanics – without any extraneous secondary notions like simultaneous measurability. And in order to have such a consequent, one-piece system of logics, we must change the classical class calculus of logics. (J. von Neumann to G. Birkhoff, November 21, 1935). (shrink)

A finitely alternative normal tense logic \ is a normal tense logic characterized by frames in which every point has at most n future alternatives and m past alternatives. The structure of the lattice \\) is described. There are \ logics in \\) without the finite model property, and only one pretabular logic in \\). There are \ logics in \\) which are not finitely axiomatizable. For \, there are \ logics in \\) without the FMP, and infinitely many (...) pretabular extensions of \. (shrink)

We give a characterization of the fixed points and of the lattices of fixed points of fuzzy Galois connections. It is shown that fixed points are naturally interpreted as concepts in the sense of traditional logic.

Since the 1960s, many artists have incorporated ecological concerns into their work, an endeavor that has required new strategies in art-making. To explore recent American manifestations of these interests, the David and Alfred Smart Museum commissioned new projects from artists Mark Dion, Peter Fend, and Dan Peterman, each focusing on interrelationships between particular organisms—human beings-and a specific group of sites—a museum building, a river landscape, and a university campus. The results, exhibited at the Smart Museum during the summer of 2000, (...) evoke the varied scales, from the microscopic to the global, at which human actions affect the environment. This catalog documents each of the artists' projects through an array of images and words: Smart Museum Associate Curator Stephanie Smith provides an introduction and brief overviews of the three projects, each of the artists contribute statements, and photographers Susan Anderson and Tom van Eynde document—in over 100 images—the processes and projects that comprise the exhibition. (shrink)

We shall show that certain natural and interesting intervals in the lattice of varieties of representable relation algebras embed the lattice of all subsets of the natural numbers, and therefore must have a very complicated lattice-theoretic structure.

Since the pioneering work of Birkhoff and von Neumann, quantum logic has been interpreted as the logic of (closed) subspaces of a Hilbert space. There is a progression from the usual Boolean logic of subsets to the "quantum logic" of subspaces of a general vector space--which is then specialized to the closed subspaces of a Hilbert space. But there is a "dual" progression. The notion of a partition (or quotient set or equivalence relation) is dual (in a category-theoretic sense) to (...) the notion of a subset. Hence the Boolean logic of subsets has a dual logic of partitions. Then the dual progression is from that logic of partitions to the quantum logic of direct-sum decompositions (i.e., the vector space version of a set partition) of a general vector space--which can then be specialized to the direct-sum decompositions of a Hilbert space. This allows the logic to express measurement by any self-adjoint operators rather than just the projection operators associated with subspaces. In this introductory paper, the focus is on the quantum logic of direct-sum decompositions of a finite-dimensional vector space (including such a Hilbert space). The primary special case examined is finite vector spaces over ℤ₂ where the pedagogical model of quantum mechanics over sets (QM/Sets) is formulated. In the Appendix, the combinatorics of direct-sum decompositions of finite vector spaces over GF(q) is analyzed with computations for the case of QM/Sets where q=2. (shrink)

The duality between congruence lattices of semilattices, and algebraic subsets of an algebraic lattice, is extended to include semilattices with operators. For a set G of operators on a semilattice S, we have \ \cong^{d} {{\rm S}_{p}}}\), where L is the ideal lattice of S, and H is a corresponding set of adjoint maps on L. This duality is used to find some representations of lattices as congruence lattices of semilattices with operators. It is also shown that these (...) congruence lattices satisfy the Jónsson–Kiefer property. (shrink)

We study the lattice of extensions of four-valued Belnap–Dunn logic, called super-Belnap logics by analogy with superintuitionistic logics. We describe the global structure of this lattice by splitting it into several subintervals, and prove some new completeness theorems for super-Belnap logics. The crucial technical tool for this purpose will be the so-called antiaxiomatic (or explosive) part operator. The antiaxiomatic (or explosive) extensions of Belnap–Dunn logic turn out to be of particular interest owing to their connection to graph theory: (...) the lattice of finitary antiaxiomatic extensions of Belnap–Dunn logic is isomorphic to the lattice of upsets in the homomorphism order on finite graphs (with loops allowed). In particular, there is a continuum of finitary super-Belnap logics. Moreover, a non-finitary super-Belnap logic can be constructed with the help of this isomorphism. As algebraic corollaries we obtain the existence of a continuum of antivarieties of De Morgan algebras and the existence of a prevariety of De Morgan algebras which is not a quasivariety. (shrink)

orbit and far beyond it. Indeed, the immense, painstaking, indefatigable and ever-improving effort of Husserl to find ever-deeper and more reliable foundations for the philosophical enterprise (as well as his constant critical re-thinking and perfecting of the approach and so called "method" in order to perform this task and thus cover in this source-excavation an ever more far-reaching groundwork) stands out and maintains itself as an inepuisable reservoir for philosophical reflec tion in which all the above-mentioned work has either its (...) core or its source. In fact, in his undertaking to re-think the entire philosophical enterprise as such and to recreate philosophy upon what he sought to be at least a satisfactorily legitimated basis, Husserl, through his already systematised and "authorized" work, and his courses, and later on in his spontaneous reflection (which did not find its way into a definitive corpus but was nevertheless sufficiently coherent with his previously established body of thought to be considered a continuation of it), uncovers perspectives upon the universe of man and projects their new philosophical thematisation that brings together all the attempts by philosophers (e. g., Merleau-Ponty, who drew upon this material and found there his own inspiration) who succeeded him with foundational intentions; it also gives a core of philosophical ideas and insights for the youngergenerationofphilosophers today. (shrink)

We prove new results on common cause closedness of quantum probability spaces, where by a quantum probability space is meant the projection lattice of a non-commutative von Neumann algebra together with a countably additive probability measure on the lattice. Common cause closedness is the feature that for every correlation between a pair of commuting projections there exists in the lattice a third projection commuting with both of the correlated projections and which is a Reichenbachian common (...) cause of the correlation. The main result we prove is that a quantum probability space is common cause closed if and only if it has at most one measure theoretic atom. This result improves earlier ones published in [1]. The result is discussed from the perspective of status of the Common Cause Principle. Open problems on common cause closedness of general probability spaces (L, ϕ) are formulated, where L is an orthomodular bounded lattice and ϕ is a probability measure on L. (shrink)

Most material below is ranked around the splittings of lattices of normal modal logics. These splittings are generated by nite subdirect irreducible modal algebras. The actual computation of the splittings is often a rather delicate task. Rened model structures are very useful to this purpose, as well as they are in many other respects. E.g. the analysis of various lattices of extensions, like ES5, ES4:3 etc becomes rather simple, if rened structures are used. But this point will not be touched (...) here. The variety T BA , which corresponds to S4, is congruence- distributive. Hence any every tabular extension L S4 has nite many extensions only. Around 1975 it has been proved by several authors, that also the converse is true: If L S4 has nitely many extensions only, then L is necessarely tabular. These and other results will be extended to much richer lattices. For simplicity we state the results explicitely only for the lattice N of modal logics with one modal operator. But most of them carry over literally to the lattice Nk of k-ramied normal modal logics . It is easy to construct examples of 2-ramied modal logics which are P OST-complete but not tabular. Whether this will be possible for normal modal logic seems to be an open problem. In view of results below it seems very unlikely that such examples exist. (shrink)

Most donated organs in the United States come from brain dead donors, while a small percentage come from patients who die in “controlled,” or expected, circumstances, typically after the family or surrogate makes a decision to withdraw life support. The number of organs available for transplant could be substantially if donations were permitted in “uncontrolled” circumstances–that is, from people who die unexpectedly, often outside the hospital. According to projections from the Institute of Medicine, establishing programs permitting “uncontrolled donation after (...) circulatory determination of death,” or uDCDD, throughout the United States has the potential to provide 22,000 more donation opportunities annually. In contrast, U.S. controlled donation after circulatory determination of death, or cDCDD, cases have increased progressively over the past decade from 87 to 848 donors, but currently account for only 10.6 percent of all deceased donors. Following the IOM recommendations, several projects exploring the feasibility of uDCDD were funded by the federal government, including a grant from the Health Resources and Services Administration that supported a pilot project in New York City in which the authors of this article participated.A key feature of our protocol, and indeed of many uDCDD protocols, is the initiation of preservation methods such as chest compressions and extracorporeal membrane oxygenation shortly after death in order to perfuse and preserve the donor's organs. Critics of uDCDD argue that the means of determining death deviates from generally ascribed principles. They assert that reinstituting circulation in order to preserve organs has the effect of “undoing” the prior determination of death. The result is that cDCDD is widely accepted and practiced routinely even though it only marginally increases the number of organs available for transplantation, and uDCDD is widely considered unacceptable despite being ethically embraced and proven to significantly increase organ donation opportunities in other countries. This article explores the evolution of this counterintuitive state of affairs and calls for a policy that, in line with the IOM report, allows for both cDCDD and uDCDD protocols. (shrink)

Modal logics are studied in their algebraic disguise of varieties of so-called modal algebras. This enables us to apply strong results of a universal algebraic nature, notably those obtained by B. Jonsson. It is shown that the degree of incompleteness with respect to Kripke semantics of any modal logic containing the axiom □ p → p or containing an axiom of the form $\square^mp \leftrightarrow\square^{m + 1}p$ for some natural number m is 2 ℵ 0 . Furthermore, we show that (...) there exists an immediate predecessor of classical logic (axiomatized by $p \leftrightarrow \square p$ ) which is not characterized by any finite algebra. The existence of modal logics having 2 ℵ 0 immediate predecessors is established. In contrast with these results we prove that the lattice of extensions of S4 behaves much better: a logic extending S4 is characterized by a finite algebra iff it has finitely many extensions and any such logic has only finitely many immediate predecessors, all of which are characterized by a finite algebra. (shrink)

Let K be the least normal modal logic and BK its Belnapian version, which enriches K with ‘strong negation’. We carry out a systematic study of the lattice of logics containing BK based on: • introducing the classes of so-called explosive, complete and classical Belnapian modal logics; • assigning to every normal modal logic three special conservative extensions in these classes; • associating with every Belnapian modal logic its explosive, complete and classical counterparts. We investigate the relationships between special (...) extensions and counterparts, provide certain handy characterisations and suggest a useful decomposition of the lattice of logics containing BK. (shrink)

In the present paper we continue the investigation of the lattice of subvarieties of the variety of ${\sqrt{\prime}}$ quasi-MV algebras, already started in [6]. Beside some general results on the structure of such a lattice, the main contribution of this work is the solution of a long-standing open problem concerning these algebras: namely, we show that the variety generated by the standard disk algebra D r is not finitely based, and we provide an infinite equational basis for the (...) same variety. (shrink)

We study the structure of algebraic τ-closed subsets of an algebraic lattice L, where τ is some Browerian binary relation on L, in the special case when the lattice of such subsets is an atomistic lattice. This gives an approach to investigate the atomistic lattices of congruence-Noetherian quasivarieties.

If $\mathbf{S}$ is a semilattice with operators, then there is an implicational theory $\mathscr{Q}$ such that the congruence lattice $\operatorname{Con}$ is isomorphic to the lattice of all implicational theories containing $\mathscr{Q}$.

The present paper investigates the groups of automorphisms for some lattices of modal logics. The main results are the following. The lattice of normal extensions of S4.3, NExtS4.3, has exactly two automorphisms, NExtK.alt1 has continuously many automorphisms. Moreover, any automorphism of NExtS4 fixes all logics of finite codimension. We also obtain the following characterization of pretabular logics containing S4: a logic properly extends a pretabular logic of NExtS4 iff its lattice of extensions is finite and linear.

In the present paper we continue the investigation of the lattice of subvarieties of the variety of √′ P quasi-MV algebras, already started in [6]. Beside some general results on the structure of such a lattice, the main contribution of this work is the solution of a long-standing open problem concerning these algebras: namely, we show that the variety generated by the standard disk algebra D r is not finitely based, and we provide an infinite equational basis for (...) the same variety. (shrink)

A decade ago, Isham and Butterfield proposed a topos-theoretic approach to quantum mechanics, which meanwhile has been extended by Döring and Isham so as to provide a new mathematical foundation for all of physics. Last year, three of the present authors redeveloped and refined these ideas by combining the C*-algebraic approach to quantum theory with the so-called internal language of topos theory (Heunen et al. in arXiv:0709.4364). The goal of the present paper is to illustrate our abstract setup through the (...) concrete example of the C*-algebra M n (ℂ) of complex n×n matrices. This leads to an explicit expression for the pointfree quantum phase space Σ n and the associated logical structure and Gelfand transform of an n-level system. We also determine the pertinent non-probabilisitic state-proposition pairing (or valuation) and give a very natural topos-theoretic reformulation of the Kochen–Specker Theorem.In our approach, the nondistributive lattice ℘(M n (ℂ)) of projections in M n (ℂ) (which forms the basis of the traditional quantum logic of Birkhoff and von Neumann) is replaced by a specific distributive lattice $\mathcal{O}(\Sigma_{n})$ of functions from the poset $\mathcal{C}(M_{n}(\mathbb{C}))$ of all unital commutative C*-subalgebras C of M n (ℂ) to ℘(M n (ℂ)). The lattice $\mathcal{O}(\Sigma_{n})$ is essentially the (pointfree) topology of the quantum phase space Σ n , and as such defines a Heyting algebra. Each element of $\mathcal{O}(\Sigma_{n})$ corresponds to a “Bohrified” proposition, in the sense that to each classical context $C\in\mathcal{C}(M_{n}(\mathbb{C}))$ it associates a yes-no question (i.e. an element of the Boolean lattice ℘(C) of projections in C), rather than being a single projection as in standard quantum logic. Distributivity is recovered at the expense of the law of the excluded middle (Tertium Non Datur), whose demise is in our opinion to be welcomed, not just in intuitionistic logic in the spirit of Brouwer, but also in quantum logic in the spirit of von Neumann. (shrink)

In Culture, Genre, and Literary Vocation, Michael Davitt Bell charts the important and often overlooked connection between literary culture and authors' careers. Bell's influential essays on nineteenth-century American writers—originally written for such landmark projects as The Columbia Literary History of the United States and The Cambridge History of American Literature—are gathered here with a major new essay on Richard Wright. Throughout, Bell revisits issues of genre with an eye toward the unexpected details of authors' lives, and invites us to reconsider (...) the hidden functions that terms such as "romanticism" and "realism" served for authors and their critics. Whether tracing the demands of the market or the expectations of readers, Bell examines the intimate relationship between literary production and culture; each essay closely links the milieu in which American writers worked with the trajectory of their storied careers. (shrink)