This book presents an English translation of a classic Russian text on duality theory for Heyting algebras. Written by Georgian mathematician Leo Esakia, the text proved popular among Russian-speaking logicians. This translation helps make the ideas accessible to a wider audience and pays tribute to an influential mind in mathematical logic. The book discusses the theory of Heyting algebras and closure algebras, as well as the corresponding intuitionistic and modal logics. The author introduces the key notion of a hybrid (...) that “crossbreeds” topology and order, resulting in the structures now known as Esakia spaces. The main theorems include a duality between the categories of closure algebras and of hybrids, and a duality between the categories of Heyting algebras and of so-called strict hybrids. Esakia’s book was originally published in 1985. It was the first of a planned two-volume monograph on Heyting algebras. But after the collapse of the Soviet Union, the publishing house closed and the project died with it. Fortunately, this important work now lives on in this accessible translation. The Appendix of the book discusses the planned contents of the lost second volume. (shrink)

In the pioneering article and two papers, written jointly with McKinsey, Tarski developed the so-called algebraic and topological frameworks for the Intuitionistic Logic and the Lewis modal system. In this paper, we present an outline of modern systems with a topological tinge. We consider topological interpretation of basic systems GL and G of the provability logic in terms of the Cantor derivative and the Hausdorff residue.

In this paper we define an augmentation mHC of the Heyting propositional calculus HC by a modal operator ?. This modalized Heyting calculus mHC is a weakening of the Proof-Intuitionistic Logic KM of Kuznetsov and Muravitsky. In Section 2 we present a short selection of attractive (algebraic, relational, topological and categorical) features of mHC. In Section 3 we establish some close connections between mHC and certain normal extension K4.Grz of the modal system K4. We define a translation of mHC into (...) K4.Grz and prove that this translation is exact, i. e. theorem-preserving and deducibility-invariant. We have established (however, in this note we do not present a proof of this) that the lattice of all extensions of mHC is isomorphic to the lattice of normal extensions of K4.Grz (a generalization of the Kuznetsov and Muravitsky theorem). (shrink)

We present some results on algebraic and modal analysis of polynomial distortions of the standard provability predicate in Peano Arithmetic PA, and investigate three provability-like modal systems related to the Gödel–Löb modal system GL. We also present a short review of relational and topological semantics for these systems, and describe the dual category of algebraic models of our main modal system.

A class of toposes is introduced and studied, suitable for semantical analysis of an extension of the Heyting predicate calculus admitting Gödel's provability interpretation.

A description of the free cyclic algebra over the variety of Solovay algebras, as well as over its pyramid locally finite subvarieties is given.

Hamkins and Löwe proved that the modal logic of forcing is S4.2 . In this paper, we consider its modal companion, the intermediate logic KC and relate it to the fatal Heyting algebra H ZFC of forcing persistent sentences. This Heyting algebra is equationally generic for the class of fatal Heyting algebras. Motivated by these results, we further analyse the class of fatal Heyting algebras.

We consider two topological interpretations of the modal diamond—as the closure operator (C-semantics) and as the derived set operator (d-semantics). We call the logics arising from these interpretations C-logics and d-logics, respectively. We axiomatize a number of subclasses of the class of nodec spaces with respect to both semantics, and characterize exactly which of these classes are modally definable. It is demonstrated that the d-semantics is more expressive than the C-semantics. In particular, we show that the d-logics of the six (...) classes of spaces considered in the paper are pairwise distinct, while the C-logics of some of them coincide. (shrink)

This book closely examines our commonly held beliefs about human suffering, and offers unique insights into God's role in why we suffer. Dr. Perri critically examines what it means to be human from a Judeo-Christian perspective, and extrapolates from the work of Carl Gustav Jung showing a deeply complex development of human transcendence in human suffering. On an interpersonal level, Dr. Perri elaborates on the work of Martin Buber and Emanuel Levinas and shows how our suffering can be shared and (...) lessened by our unconditionality and presence for the other. Scriptural passages are quoted from both the Hebrew and New Testament texts, and are used to depict the suffering of humankind as seen in the face of Job. It was this face, according to Perri, that Jesus saw in the faces of those He came to redeem, and that ultimately brought Him to the crucifix of redemption. Dr. Perri challenges Sartre's position that our existence precedes our essence, and draws on the work of Abraham Joshua Heschel and Christian theology in this regard. He further elaborates a neurodevelopmental paradigm that creates essence as the basis of our freedom to choose. This book can be read and studied by both professionals in the helping and healing professions, and lay persons seeking to better understand our suffering, and how to better help those who suffer. Dr. Perri offers specific ways in how to be present for those who suffer, how to read and receive the hidden language of human suffering, and how to use silence as a communication skill. (shrink)

The draft Code of Conduct for Research is an important initiative towards assisting the scientific community in complying with the provisions of the Protection of Personal Information Act 4 of 2013 (POPIA). However, its approach towards cross-border data sharing should be reconsidered to clarify the ambiguities inherent in the legal requirements for the cross-border sharing of health data in the POPIA. These ambiguities include the concept of ‘transfer of information’, the application of adequacy as a legal mechanism for transfer, the (...) nature of consent for cross-border sharing and the scope of the recipient third party. We suggest that the draft Code of Conduct for Research can be improved by: Explaining or defining the concept of ‘transfer of information’ and when it applies to cross-border sharing in research Clarifying the application of adequacy as a legal mechanism for transfer vis-à-vis the other alternatives Expanding on the interpretation and application of consent as a legal mechanism for cross-border transfers Expanding the category of persons who may be recipients of personal information in a third country. (shrink)

Language, like other human traits, could only have evolved during one or more stages of development. We enlist the theoretical framework of human life history to account for certain aspects of linguistic evolution, with special reference to initial phases in the process. It is hypothesized that selection operated at several developmental stages, the earlier ones producing new behaviors that were reinforced by additional, and possibly more powerful, forms of selection during later stages, especially adolescence and early adulthood. Peer commentaries have (...) provided opportunities to explain human life history more comprehensively, and to add details to our account of spoken language. We made no attempt to explain syntax in the target article, but we propose here that selection for “vocal plumage” may have increased our species’ capacity for utterance complexity, a development that would have benefited all levels of language. (shrink)

In this timely book L.W. Sumner addresses these issues within the wider context of palliative care for patients in the dying process.

The genetic modification of pigs as a source of transplantable organs is one of several possible solutions to the chronic organ shortage. This paper describes existing ethical tensions in xenotransplantation (XTx) that argue against pursuing it. Recommendations for lifelong infectious disease surveillance and notification of close contacts of recipients are in tension with the rights of human research subjects. Parental/guardian consent for pediatric xenograft recipients is in tension with a child’s right to an open future. Individual consent to transplant is (...) in tension with public health threats that include zoonotic diseases. XTx amplifies concerns about justice in organ transplantation and could exacerbate existing inequities. The prevention of infectious disease in source animals is in tension with the best practices of animal care and animal welfare, requiring isolation, ethologically inappropriate housing, and invasive reproductive procedures that would severely impact the well-being of intelligent social creatures like pigs. (shrink)

Rationality requires us to respond to apparent normative reasons. Given the independence of appearance and reality, why think that apparent normative reasons necessarily provide real normative reasons? And if they do not, why think that mistakes of rationality are necessarily real mistakes? This paper gives a novel answer to these questions. I argue first that in the moral domain, there are objective duties of respect that we violate whenever we do what appears to violate our first-order duties. The existence of (...) these duties of respect, I argue, ensures that apparent moral reasons are exceptions to the independence of appearance and reality. I then extend these arguments to the domain of overall reason. Just as there are objective duties of respect for moral reasons that explain moral blameworthiness, so there are objective duties of respect for reasons that explain blameworthiness in the court of overall reason. The existence of these duties ensures that apparent reasons are exceptions to the independence of appearance and reality. (shrink)

Bundle theory takes objects to be bundles of properties. Some bundle theorists take objects to be bundles of instantiated universals, and some take objects to be bundles of tropes. Tropes are instances of properties: some take instantiated universals to be tropes, while others deny the existence of universals and take tropes to be ontologically fundamental. Historically, the bundling relation has been taken to be a primitive relation, not analyzable in terms of or ontologically reducible to some other relation, and has (...) been variously characterized as, e.g., “compresence,” “concurrence,” or “consubstantiation.” Bertrand Russell (1940) defends compresence of universals, and Hector-Neri Castañeda (1974) defends consubstantiation of universals. John Bacon (1995) defends concurrent tropes and Keith Campbell (1990) defends compresent tropes. Jonathan Schaffer (2001) bucks this trend, endorsing compresence understood as co-location in spacetime, but this brings with it undesireable consequences such as the impossibility of distinguishing between objects (such as electrons or other microentities) with the same location. Mereological bundle theory improves upon traditional bundle theory by taking the primitive relation of bundling to be the more familiar relation of fusing or composing, such that objects are fusions of properties or fusions of property instances. Hence, mereological bundle theorists endorse a property mereology: a mereology where properties or property instances can be parts of objects. An advantage of the approach derives from the fact that standard mereologies take composition to be primitive or define it using a different primitive mereological notion (such as primitive parthood). Thus, taking the basic primitive of bundle theory to be composition can reduce the need for.. (shrink)

Adequate epistemic justification is best conceived as the appearance, over time, of knowledge to the subject. ‘Appearance’ is intended literally, not as a synonym for belief. It is argued through consideration of examples that this account gets the extension of ‘adequately justified belief’ at least roughly correct. A more theoretical reason is then offered to regard justification as the appearance of knowledge: If we have a knowledge norm for assertion, we do our best to comply with this norm when we (...) express as assertions only beliefs that appear to us to be knowledge. If we are doing our best, there is little point in further sanctions. So a norm of knowledge for assertion would lead to a secondary norm of justified belief as the appearance of knowledge, marking a point at which our assertions may be corrected but should not be blamed. (shrink)

This book is a major work in the history of ethics, and provides the first study of early modern British philosophy in several decades. Professor Darwall discerns two distinct traditions feeding into the moral philosophy of the seventeenth and eighteenth centuries. On the one hand, there is the empirical, naturalist tradition, comprising Hobbes, Locke, Cumberland, Hutcheson, and Hume, which argues that obligation is the practical force that empirical discoveries acquire in the process of deliberation. On the other hand, there is (...) a group including Cudworth, Shaftesbury, Butler, and in some moments Locke, which views obligation as inconceivable without autonomy and which seeks to develop a theory of the will as self-determining. (shrink)

A substantial portion of Adam Smith's discussion of Stoicism in TMS VII is dedicated to the Stoic “principle of suicide,” according to which suicide is sometimes morally required. While scholars agree that Stoicism exercised considerable influence over Smith, no recent work has explored his views on suicide, despite the central role it plays in his treatment of Stoicism. I argue that Smith opposes the principle of suicide on both epistemic and moral grounds, providing an important critique of Stoicism. I also (...) show how Smith departs from other early modern accounts of suicide by offering a notably humane analysis of this tragic act. I conclude by considering what Smith's discussion of Stoicism and suicide tells us about his moral theory, and I forward a reading of Smith as a skeptic. (shrink)

The object of this paper is to show why recent research in the psychology of deductive and probabilistic reasoning does not have.