1. If A is strongly amorphous , then its power set P is dually Dedekind infinite, i. e., every function from P onto P is injective. 2. The class of “inexhaustible” sets is not closed under supersets unless AC holds.

Gödel’s philosophical conceptions bear striking similarities to Cantor’s. Although there is no conclusive evidence that Gödel deliberately used or adhered to Cantor’s views, one can successfully reconstruct and see his “Cantorianism” at work in many parts of his thought. In this paper, I aim to describe the most prominent conceptual intersections between Cantor’s and Gödel’s thought, particularly on such matters as the nature and existence of mathematical entities (sets), concepts, Platonism, the Absolute Infinite, the progress and inexhaustibility of mathematics.

This essay locates the problem of dirty hands within virtue ethics – specifically Alasdair MacIntyre’s neo-Aristotelian thesis in After Virtue. It demonstrates that, contra contemporary expositions of this problem, MacIntyre’s thesis provides us with a more nuanced account of tragedy and DH in ordinary life, in its conventional understanding as a stark, rare and momentary conflict in which moral wrongdoing is inescapable. The essay then utilizes elements from MacIntyre’s thesis as a theoretical premise for Machiavelli’s thought so as to set (...) the foundations for a nascent but richer framework of DH in politics and move beyond the standard, ‘static’ conceptualization of the problem within this context. In developing a dynamic account of DH, I conceive of politics as a distinct practice and way of life, with its own demands and standards of excellence, and draw on Machiavelli’s thought to sketch some of these. The dynamic account uncovers an inexhaustible tension between two ways of life, each with its own demands and standards of excellence: a virtuous politician should become partially vicious and no longer innocent. Understood in dynamic terms, DH in politics involves a paradox of character, not just a paradox of action as the standard, ‘static’ DH thesis suggests. (shrink)

How philosophy transformed human knowledge and the world we live in Philosophical investigation is the root of all human knowledge. Developing new concepts, reinterpreting old truths, and reconceptualizing fundamental questions, philosophy has progressed—and driven human progress—for more than two millennia. In short, we live in a world philosophy made. In this concise history of philosophy's world-shaping impact, Scott Soames demonstrates that the modern world—including its science, technology, and politics—simply would not be possible without the accomplishments of philosophy. Firmly rebutting the (...) misconception of philosophy as ivory-tower thinking, Soames traces its essential contributions to fields as diverse as law and logic, psychology and economics, relativity and rational decision theory. Beginning with the giants of ancient Greek philosophy, The World Philosophy Made chronicles the achievements of the great thinkers, from the medieval and early modern eras to the present. It explores how philosophy has shaped our language, science, mathematics, religion, culture, morality, education, and politics, as well as our understanding of ourselves. Philosophy's idea of rational inquiry as the key to theoretical knowledge and practical wisdom has transformed the world in which we live. From the laws that govern society to the digital technology that permeates modern life, philosophy has opened up new possibilities and set us on more productive paths. The World Philosophy Made explains and illuminates as never before the inexhaustible richness of philosophy and its influence on our individual and collective lives. (shrink)

In The Priority of the Person, world-class philosopher David Walsh advances the argument set forth in his highly original philosophic meditation Politics of the Person as the Politics of Being (2015), that "person" is the central category of modern political thought and philosophy. This book is divided into three main parts. Beginning with the political discovery of the inexhaustibility of persons, it then explores the philosophic differentiation of the idea of the "person," and finally traces its historical emergence through art, (...) science, and faith. Walsh argues that, although the roots of the idea of "person" are found in the Greek concept of the mind and in the Christian conception of the soul, this notion is ultimately a distinctly modern achievement, because it is only the modern turn toward interiority that illuminated the unique nature of persons as each being a world unto him or herself. As Walsh shows, it is precisely this feature of persons that makes it possible for us to know and communicate with others, for we can only give and receive one another as persons. In this way alone can we became friends and, in friendship, build community. In showing how the person is modernity's central preoccupation, and in demonstrating how it is only as persons that we can truly give ourselves to others and thus develop real community, David Walsh's The Priority of the Person makes an important contribution to current discussions in both political theory and philosophy. It will also appeal to students and scholars of theology and literature, and any groups interested in the person and personalism. (shrink)

The Meditations of Marcus Aurelius are treasured today--as they have been over the centuries--as an inexhaustible source of wisdom. And as one of the three most important expressions of Stoicism, this is an essential text for everyone interested in ancient religion and philosophy. Yet the clarity and ease of the work's style are deceptive. Pierre Hadot, eminent historian of ancient thought, uncovers new levels of meaning and expands our understanding of its underlying philosophy. Written by the Roman emperor for (...) his own private guidance and self-admonition, the Meditations set forth principles for living a good and just life. Hadot probes Marcus Aurelius's guidelines and convictions and discerns the hitherto unperceived conceptual system that grounds them. Abundantly quoting the Meditations to illustrate his analysis, the author allows Marcus Aurelius to speak directly to the reader. And Hadot unfolds for us the philosophical context of the Meditations, commenting on the philosophers Marcus Aurelius read and giving special attention to the teachings of Epictetus, whose disciple he was. The soul, the guiding principle within us, is in Marcus Aurelius's Stoic philosophy an inviolable stronghold of freedom, the "inner citadel." This spirited and engaging study of his thought offers a fresh picture of the fascinating philosopher-emperor, a fuller understanding of the tradition and doctrines of Stoicism, and rich insight on the culture of the Roman empire in the second century. Pierre Hadot has been working on Marcus Aurelius for more than twenty years; in this book he distills his analysis and conclusions with extraordinary lucidity for the general reader. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Save Us From Being Saved:Girard's Critical SoteriologySimon J. Taylor (bio)One of the most striking things about the work of René Girard is that it is an overarching critique of what it is to be saved. The paradox that Girard presents us with is that "salvation" is something from which we must be saved. This combination of salvation as something that must be avoided and something we desperately need appears (...) to have the characteristic of a "double bind," about which Girard has written so much.1 Are we then, even within Girard's own thesis, bound to the mimetic doubling of need and revulsion that can end only in the expulsion or death of a common victim? It is this question that I hope to explore in this article.Girard's Critique Of SalvationTo set out Girard's critique of salvation, it is first necessary to see that what he has come to call the victimage mechanism ("mécanisme victimaire") is, in fact, salvation. Girard's thesis states that desire is mimetic, based on the imitation of another's desire. "Desire is always an imitation of another desire, desire for the same object, and therefore, an inexhaustible source of conflicts and rivalries" (Girard 1978, 39). As the rivalry produced by this desire intensifies, it spreads and produces conflict and violence on a large scale. On an unlimited scale, Girard identifies this with the Hobbesian "war of all against all."2 And it is at this point that the victimage mechanism kicks in. It is at this point that the identification of a scapegoat and the expulsion or killing of the scapegoat produce unity and restore peace. It is this, in Girard's originary scene, that leads to the simultaneous origins of society and religion. In restoring unity to the group, and hence founding the community, the victim becomes sacred, a god. This in itself is an indicator of the salvific power that the scapegoat effects in the community. In order that the violence, once diverted, should not break out again, two things are necessary: ritual, specifically sacrifice, and myth. Ritual reenacts the founding murder upon which society is based, enabling the [End Page 21] participants to transfer their violence to another victim. Myth is the story told by the persecutors about the events that led to the creation of their community, that is, the founding murder. Thus, religion "dehumanizes violence; it protects man from his own violence by taking it out of his hands, transforming it into a transcendent and ever-present danger to be kept in check by the appropriate rites appropriately observed and by a modest and prudent demeanor.... religion protects man as long as its ultimate foundations are not revealed" (Girard 1978, 134–35). Salvation has happened, and a cult arises to disburse the effects of this salvation.It is important that we take this account of religion with full seriousness and do not regard it simply as the prologue to Girard's later promotion, even rehabilitation, of the Judaeo-Christian tradition. This is salvific: religion protects us from our own violence; it has mechanisms for bringing to an end outbreaks of that violence. Elsewhere Girard can speak of "the negative usefulness of religion as a social prop against anarchy and chaos" (Girard 1997, 162–63). This is salvation: violence is dealt with and human society can dwell in relative peace. Religion, on this account, is not "pie-in-the-sky," a transcendent deferral of gratification. It has real benefits to society today; it is attractive, elegant, and simple. What more could we want from religion?But Girard will not allow us to rest content with this account of religion. He is blunt when he declares that "most religions are based on a lie, which is the supposed guilt of the victim" (Girard 1993c, 32). Nietzsche wrote that "The sacrificial animal does not share the spectators' ideas about sacrifice, but one has never let it have its say,"3 and this is precisely the origin of Girard's critique of the salvation offered by religions based on scapegoating. Girard is attempting to let the sacrificial animal have its say. All too often, these... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:The Aesthetics of Asian Art:The Study of Montien Boonma in the Undergraduate Education ClassroomMary Ann Maslak (bio)John Dewey, in his Experience and Nature, expounds on the developmental nature of human experience premised on the concept of qualitative propinquity—the integration and harmonization with the environment exemplifies the essence of experience. This principal line of reasoning shapes his fundamental argument in Art as Experience, one of Dewey's most significant educational works (...) used today.1 In it he argues that the roots of experience lie in the commonplace occurrences in the course of human life. The integration of the meaning and value drawn from previous and present circumstances results in an "experience," which, in turn, constitutes a foundation of learning for the individual. Dewey's ideas are not only appropriate but also useful in the field of education. In what ways may future teachers experience art? How does the experiential process of examining art contribute to future teachers' pedagogical tools that can be used to incorporate art into the elementary school classroom?This article demonstrates how an experiential study of a work of art in an undergraduate education course—part of the St. John's University/Lincoln Center Institute for the Arts in Education (LCIAE) Higher Education Collaborative program in New York City—contributes to the pedagogical repertoire of process-based learning for teachers and their future students. The article begins by providing a theoretical background for the work, followed by the definitions of three concepts that are pertinent to this piece: aesthetic education, the experiential approach to learning, and process-based instruction.2 The article then focuses on the specific aims of the university's arts course, the LCIAE Partnership Program, and data and analyses from the process-based aesthetic learning for the students enrolled in the course. It ends with conclusions about aesthetic education. [End Page 67]Theoretical BackgroundAssuming the sociocultural nature of learning suggested by Dewey demands consideration of the layers of experience, it is necessary to discern the learners' emerging thoughts in light of various experiences. As a teacher, I subscribe to this belief and wanted to note how I could employ scaffolding practices to the aesthetic education learning experience for my undergraduate students. I aimed to accomplish this goal by scaffolding aesthetic experiences for students from their personal points of understanding, the university's classroom setting, and the context of museum exhibitions.Scaffolding instruction has a lengthy and persuasive history that provided the underpinning for my approach to the course.3 The intent of scaffolding is to help a learner undertake a task or goal that is beyond the learner's present level.4 Effective scaffolding encompasses a number of key features within cognitive and emotional domains.5 Cognitive elements include (1) selecting challenging activities for students; (2) monitoring their efforts; (3) using guiding questions to develop self-regulation; and (4) using indirect questioning strategies to extend and enhance the learning experience. Emotional support includes (1) structuring experiences to discourage frustration and encourage confidence; and (2) creating an atmosphere of comfort. While not all these features are applicable to scaffolding activities in an art education course of a university's teacher education program, I was mindful of their importance in structuring this course.Given that this course was my undergraduate students' initial exposure to aesthetic education, scaffolding served as an appropriate systematic pedagogical approach. In the words of Maxine Greene, aesthetic education is an intentional experience "designed to nurture appreciative, reflective, cultural, participatory engagements with the arts" by enabling the student to observe, analyze, and interpret meaning.6 In other words, it is the interactive study of the nature and expression of beauty. It seeks to enhance perception and affect, cognitive development, and personal growth by assuming that works of art provide an inexhaustible resource for exploration and reflection by the learner and an understanding based on the cultural context in which the artwork was created. Greene's philosophy, developed from the work of Dewey, posits that understanding a work of art is based on the active connection to and association with the piece. Therefore, the approach to its study is of the utmost importance.The experiential approach to learning is a... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Criticism, Imagination, and the Subjectivization of AestheticsRoger W. H. SavageThe growing discontent with reductivist practices signals a new current in contemporary criticism's understanding of music, literature and art. George Levine's unease with critics who are unable or unwilling to account for their continuing preoccupation with literary texts they expose as "imperialist, sexist, homophobic and racist" illumines the contradiction fueling the reduction of aesthetics to ideology.1 Cultural studies that deploy (...) literature as evidence of the aesthetics' socio-historical substance mask literature's capacity to break open new perspectives on reality by assuming that literary works are politically complicit with the aesthetics' strategic "mystification of the status quo" (A&I, p. 3). Criticism's indifference to its philosophical presuppositions exacerbates the paradox of denouncing a body of works that constitute criticism's aesthetic and intellectual heritage. According to Mario Valdés, literary studies' coming of age mandates that criticism take account of a tradition nurtured by a succession of philosophers including Hans-Georg Gadamer and Paul Ricoeur.2 For Valdés, the post-structuralist realization that literary texts are indeterminate and inexhaustible prohibits replacing the work of art with critical commentaries on it; criticism's collective and determining role belongs to a shared community of commentary whose history and thought is a record of the changing interpretations and understandings of literary texts' meanings. [End Page 164]Valdés's claim extends to the field of contemporary music criticism, where the fashion of denouncing aesthetics as socially pernicious turns against traditional musicology's institutional authority. By demystifying absolute music (instrumental music devoid of programmatic associations), a self-proclaimed critical musicology revolts against traditional musicology's perceived political and ideological agenda. Critical musicology militates against the aesthetic conceit that absolute music transcends its social construction. Yet, by overlooking the philosophical presuppositions that set music's autonomy against practical affairs, new musicology accedes to the schema it recoils against.The tradition nurtured by Gadamer's philosophical hermeneutics and Ricoeur's hermeneutic phenomenology represents a critical current whose significance has been overshadowed by postmodernist investments in decoding music's social and political content. Gadamer's critique of the subjectivization of aesthetics and Ricoeur's meditations on the imagination's capacity for invention offer an alternativeto contemporary music criticism's reaction against the principle of music's aesthetic autonomy. Gadamer and Ricoeur question art's formal separation from reality, which belongs to the history of Kant's radical subjectivization of aesthetics. Gadamer's critique of art's aesthetic differentiation prepares the ground for revealing how socially informed analyses conform to the schema Kant initiates by divorcing judgments of taste from their surrounding cultural ethos. Gadamer argues that, by discrediting theoretical knowledge that does not rely on the methodology of the natural sciences, the transcendental function Kant ascribes to aesthetic judgment lays the foundation for differentiating between art's aesthetic constitution and a concept of truth that accommodates the standard of the natural sciences. Through reducing the "sensus communis to a subjective principle," Kant legitimates his critique of aesthetic judgments by denying taste any importanceas a mode of knowledge.3Ricoeur's hermeneutical reflections on imagination complement Gadamer's critique of a differentiating consciousness that abstracts art works from their cultural worlds. For Ricoeur, imagination is productive when the fictions that works create affect our understanding of ourselves and our world by re-describing reality. Aesthetics' alignment with ideology encounters a limit in the power works evince by unfolding different ways of seeing or hearing reality. Ricoeur's reflections on imagination stand in stark contrast to the idea that individual works represent a form of cultural capital in the struggle for social position [End Page 165] and power. Contemporary critical practices' failure to account for the philosophical separation of judgments of taste from knowledge of reality precipitates the impasse criticism encounters when it identifies aesthetics with ideology at the expense of a work's capacity to affect reality in productive ways. The dispersal and potential disappearance of music's aesthetic character into the recesses of cultural and political analysis keeps step with the conceptual narrowing imposed by a restrictive sociological critique. Demystifying music's ideological representations of gender, race and identity purges romantic... (shrink)

Mark Nelson argues that we have no positive epistemic duties. His case rests on the evidential inexhaustibility of sensory and propositional evidence—what he calls their ‘infinite justificational fecundity’. It is argued here that Nelson’s reflections on the richness of sensory and propositional evidence do make it doubtful that we ever have an epistemic duty to add any particular beliefs to our belief set, but that they fail to establish that we have no positive epistemic duties whatsoever. A theory of epistemic (...) obligation based on Kant’s idea of an imperfect duty is outlined. It is suggested that such a theory is consistent with the inexhaustibility of sensory and propositional evidence. Finally, one feature of our epistemic practice suggestive of the existence of imperfect epistemic duties is identified and promoted. (shrink)

A truism of art history is that the lifespan of artworks can exceed their original social spaces: Artworks can sometimes be successfully transplanted into completely different settings where they continue to be valued. Does their potential to outlive their original context have to do with a specific feature of artworks’ ontology? Or with how human brains are wired? Or is it a mere function of their historical and social circumstances? I argue that David Summers’s magisterial _Real Spaces: World Art History (...) and the Rise of Western Modernism_ contains elements of a different answer. This answer focuses on what in artworks’ appearance causes them to be effective under various circumstances, without subscribing to the view that art is an irreducibly and inexhaustibly complex source of meaning. In delineating the Summersian perspective, I contrast it with the position (exemplified by the recent work of the archaeologist David Wengrow and the anthropologist Philippe Descola) that treats the art object as a document of underlying structures or currents. (shrink)

Contacts of the two logicians are listed, and all Gödel's written mentions of Tarski's work are quoted. Why did Gödel almost never mention Tarski's definition of truth in his notes and papers? This puzzle of Gödel's silence, proposed by Feferman, is not merely biographical or psychological but has interesting connections to Gödel's philosophical views.No satisfactory answer is given by the three “standard” explanations: no need to repeat the work already done; Tarski's achievement was obvious to Gödel; Gödel's exceptional caution. In (...) fact, , Tarski had done the work, but Gödel almost never mentioned the achievement; , the obviousness is no explanation for the omission of Tarski's work in contexts in which an application of the definition of satisfaction was useful, and even necessary; , the point was not just caution: if Gödel had felt the need to mention the program of scientific semantics he could easily have done that in his manuscripts, or in conversations.Three ideas, detectable in Gödel's approach, can help us understand Gödel's silence: the idea of truth as the intuitive provability in the most general sense; defining it set-theoretically would contribute nothing. the idea of truth as an inexhaustible idea in the sense of Kant; “truth in general” is a category that must be applicable to all kinds of sentential expressions; also, while for Gödel language was secondary, Tarski's definition is focused on language. the idea of logic as the universal language, in Hintikka's sense, as opposed to the perception of logic as a reinterpretable calculus; hence the thesis that semantics is inexpressible. Gödel always remains a Platonist who asks a natural question: what does really happen in the realm of abstract objects? (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Stewart R. Sutherland LANGUAGE AND INTERPRETATION IN CRIME AND PUNISHMENT OF some novels it is possible to argue with justification that the problems of interpretation and understanding begin on the first page. Of Dostoyevsky's Crime and Punishment it is possible to contend that the problems of interpretation and understanding begin on the title page. The terms "crime" and "punishment" are overtly moral. The novel is read in the context (...) of a title which exhibits its values where they may be seen by all. What significance must we attach to the choice of such a title? It is possible that the title gives to us a characterization of the content of the novel which, retrospectively, we can only regard as either self-consciously ironical, or perhaps pathetically so, but how are we to decide whether this is more than a possibility? Having raised this question, I propose to set it aside in the hope of returning to it equipped with the experience derived from a number of literary and philosophical digressions. The literary and conceptual style of critical comment upon a novel often betrays much that is definitive of the response being elucidated. If one compares and contrasts Philip Rahv's essay on Crime and Punishment with the appropriate chapter of Mochulsky's Dostoevsky,2 the implication of this general claim may be clarified. My intention here is to use largely ostensive means to suggest the radically different accounts of the novel given to us by these two writers. That the conclusions are different is hardly in need of statement. What is worth examining is the character of the language and of the arguments used to substantiate those conclusions. The point of my discussion will rest not upon assessing the accuracy or legitimacy of particular judgments, but rather upon delineating the general contours of the intellectual contexts in which these judgments are offered. 223 224Philosophy and Literature Rahv's focus is on the character or psyche of Raskolnikov. The novel is seen largely in the form of a psychological study of the mind of the murderer. He reminds us that a central problem for Raskolnikov, for the reader, and for Dostoyevsky himself, is the identification of Raskolnikov's motive—not that there is an absence of motives, indeed quite the reverse: "He is soon lost in the maze of his own motivation" (p. 20). His mind spins incessantly upon an axis the nature of which is nowhere made unambiguously apparent. Nor, it seems, can we replace our search for a motive, or the sufficient reason for action, by identifying a supreme assertion of will: again, on the contrary, we are told, "Dostoyevsky is the first novelist to have fully accepted and dramatized the principle of uncertainty or indeterminacy in the presentation of character" (p. 21). Despite that, however, the success of the novel rests upon Dostoyevsky's skills in allying the insight "... that human consciousness is inexhaustible and incalculable" (p. 19), to "his capacity to combine [seeming contradictions] in a single brain and a single psyche, while staving off the danger of incoherence at one end and of the specious reconciliation at the other..." (p. 38). The mood ofRahv's treatment of the novel is dominated in this way by the language of (largely popularized) psychology—"pathological depression," "nameless guilt-feelings," "wishful fantasy"—and not surprisingly, he sees his own preoccupations as mirroring those of Dostoyevsky: "What Dostoyevsky has done in revising Raskolnikov's justification is to convert into a theory of human nature what is in Hegel not a psychological theory at all but a theory of men as subjects and objects of history" (p. 34). Mochulsky, on the other hand, breathes a very different atmosphere. From the outset he talks of "the spiritual existence of Dostoyevsky's heroes," and he says of a man who wills a murder or one who "betrays himself to the powers of dark necessity" (pp. 302-303), "The murderer has stepped beyond something more than the moral law: the very basis ofthe spiritual worlditself" (p. 303). "The whole tragedy of man-godhood," he tells us, "is expressed in these few words: 'It seems to me,' Raskolnikov says, 'truly great people experience an immense... (shrink)

The articles in the present issue are the result of a study day on William Desmond’s recent book, Ethics and the Between, held at the K.U.Leuven's Institute of Philosophy. This important book certainly deserves a thorough discussion and for many reasons. It is the manifestation of an ambition that reminds us of past periods in the history of philosophy. These days not so many philosophers venture to set up a body of work — three volumes — in the tradition of (...) the great philosophical systems that once tried to cover and understand the whole of reality. When reading Ethics and the Between, Hegel’s system comes to mind not only because we know that William Desmond is a Hegel scholar but also because of its ambition to give everything a place within an encompassing system.Of course Desmond takes care to explain where the difference lies between his metaxological approach and an insufficiently open and pluralist dialectics. His is an attempt to rethink the notion of system so that it is more hospitable to otherness as other, to the rich inexhaustibility of being. He wants to be true to the equivocities of our being and not to give in to the seduction of mediated univocity, as some forms of dialectics do. What is at stake in his work is not forgetting the surplus, the excess, the otherness of the good which suggests a transcendence of each dialectical completion in terms of self-determination.Desmond’s oeuvre is not only unusual today because of its systematic ambition. A second characteristic marks off more specifically his book on ethics: the attempt to write an ethics dominated by the idea of the good, raising the question of the relation between metaphysics and ethics, between being and being good. Desmond seems to tell us that something has been forgotten: the good which continues to be there incognito, present even in philosophies that forget it. This seems to be Desmond’s way of situating himself in an age characterized by the subtle ethos of valuelessness, the perplexity about the inherent value of being, of being as good; an age oscillating between foundationalist unity and postmodern plurality.If it were my task to give an introduction to Desmond’s thought I would have to pay attention to the richness of the concept of metaxu, the between, the way it seems to encompass the whole and keep it together, while at the same time being excessive. I would have to speak about the fourfold meaning of being and the way it shapes different ways of being ethical that dwell more and more intensively with what is at play in the ethos, that is to say in the metaxu.But it is not my task to do this; others have made the attempt more extensively in the papers reproduced here. I doubt whether they will be sufficient to bring to light all the rich dimensions of this book that is written at the same tempo as William speaks when discussing and, I suppose, when he goes running in the morning. (shrink)

he notion of linguistic adequacy (the adequacy of sentences to express or describe) is explicated in terms of a set theoretical model of the communication situation. Roughly: a message is adequate to the degree it answers the receiver's questions. Adequacy is distinguished from openness, in such a way that a message can be both completely adequate in a communication event and also “inexhaustibly open”;. Using this explication it is possible to translate and clarify several familiar philosophical theses concerning the adequacy (...) of language. The view that “humanistic”; language differs from scientific language is clarified and criticized. Seven types of empirical test of the explication are discussed. (shrink)

Study, with text of the Yogasūtra of Patañjali, text on Yoga philosophy.

O objetivo deste artigo é apresentar a analogia básica entre arquitetura e retórica antiga a partir dos tratados De Architectura, de Vitrúvio, e o De Oratore, de Cícero. A analogia se verifica na definição do artífice, dos gêneros e partes das técnicas e dos fins de cada uma delas. Para tanto, tomaram-se como referência as fontes do tratado vitruviano, que menciona a influência de Varrão na gramática, de Lucrécio na filosofia e de Cícero no método oratório. A analogia com Cícero (...) serve tanto para conferir autoridade ao modelo de arquiteto vitruviano quanto para diferenciar a arquitetura das demais ciências, sobretudo a filosofia, objeto de debate na obra ciceroniana. (shrink)

As is well known, any preorder R on a set U induces an Alexandrov topology on U. In some interesting cases related to data mining an Alexandrov topology can be transformed into different types of logico-algebraic models. In some cases, (pre)topological operators provided by Pointless Topology may define a topological space on U even if R is not a preorder. If this is the case, then we call R a crypto-preorder. The paper studies the conditions under which a relation R (...) is a crypto-preorder and how to transform a crypto-preorder into a proper preorder. (shrink)

We investigate the logical structure of intuitionistic Kripke-Platek set theory , and show that the first-order logic of is intuitionistic first-order logic IQC.

Introduction. The system of axioms for set theory to be exhibited in this paper is a modification of the axiom system due to von Neumann. In particular it adopts the principal idea of von Neumann, that the elimination of the undefined notion of a property (“definite Eigenschaft”), which occurs in the original axiom system of Zermelo, can be accomplished in such a way as to make the resulting axiom system elementary, in the sense of being formalizable in the logical calculus (...) of first order, which contains no other bound variables than individual variables and no accessory rule of inference (as, for instance, a scheme of complete induction).The purpose of modifying the von Neumann system is to remain nearer to the structure of the original Zermelo system and to utilize at the same time some of the set-theoretic concepts of the Schröder logic and of Principia mathematica which have become familiar to logicians. As will be seen, a considerable simplification results from this arrangement.The theory is not set up as a pure formalism, but rather in the usual manner of elementary axiom theory, where we have to deal with propositions which are understood to have a meaning, and where the reference to the domain of facts to be axiomatized is suggested by the names for the kinds of individuals and for the fundamental predicates.On the other hand, from the formulation of the axioms and the methods used in making inferences from them, it will be obvious that the theory can be formalized by means of the logical calculus of first order (“Prädikatenkalkul” or “engere Funktionenkalkül”) with the addition of the formalism of equality and the ι-symbol for “descriptions” (in the sense of Whitehead and Russell). (shrink)

PurposeDrawing upon the stimulus-organism-response model, this paper aims to investigate the effects of stores’ environmental components on Chinese consumers’ emotions and intentions to purchase luxury brands.Design/Methodology/ApproachData were collected from Chinese consumers who have purchased luxury brands from retail stores. Partial least squares-structural equation modeling and fuzzy-set qualitative comparative analysis were employed for data analysis.FindingsThe results of PLS-SEM indicated that three dimensions of the store environment directly and significantly influenced Chinese consumers’ emotions. However, fsQCA revealed greater heterogeneity among respondents by highlighting (...) stores’ environmental components and Chinese consumers’ emotions.Originality/ValueThis study makes significant contributions to the marketing literature by examining the S-O-R model alongside fsQCA approach to conditionally predict outcomes in a luxury brand context. The present study may be one of the first to examine the effects of stores’ environmental components on Chinese consumers’ emotions and purchase intentions, applying both symmetrical test and asymmetrical test approaches to determine significant components of the store environment as well as component combinations that predict Chinese consumers’ emotions and purchase intentions. (shrink)

Recent work has defended “Euclidean” theories of set size, in which Cantor’s Principle (two sets have equally many elements if and only if there is a one-to-one correspondence between them) is abandoned in favor of the Part-Whole Principle (if A is a proper subset of B then A is smaller than B). It has also been suggested that Gödel’s argument for the unique correctness of Cantor’s Principle is inadequate. Here we see from simple examples, not that Euclidean theories of set (...) size are wrong, but that they must be either very weak and narrow or largely arbitrary and misleading. (shrink)

A Cantorian argument that there is no set of all truths. There is, for the same reason, no possible world as a maximal set of propositions. And omniscience is logically impossible.

This third edition, now available in paperback, is a follow up to the author's classic Boolean-Valued Models and Independence Proofs in Set Theory. It provides an exposition of some of the most important results in set theory obtained in the 20th century: the independence of the continuum hypothesis and the axiom of choice.

In previous works, an ontology of properties for quantum mechanics has been proposed, according to which quantum systems are bundles of properties with no principle of individuality. The aim of the present article is to show that, since quasi-set theory is particularly suited for dealing with aggregates of items that do not belong to the traditional category of individual, it supplies an adequate meta-language to speak of the proposed ontology of properties and its structure.

Boolean-valued models of set theory were independently introduced by Scott, Solovay and Vopěnka in 1965, offering a natural and rich alternative for describing forcing. The original method was adapted by Takeuti, Titani, Kozawa and Ozawa to lattice-valued models of set theory. After this, Löwe and Tarafder proposed a class of algebras based on a certain kind of implication which satisfy several axioms of ZF. From this class, they found a specific 3-valued model called PS3 which satisfies all the axioms of (...) ZF, and can be expanded with a paraconsistent negation *, thus obtaining a paraconsistent model of ZF. The logic (PS3 ,*) coincides (up to language) with da Costa and D'Ottaviano logic J3, a 3-valued paraconsistent logic that have been proposed independently in the literature by several authors and with different motivations such as CluNs, LFI1 and MPT. We propose in this paper a family of algebraic models of ZFC based on LPT0, another linguistic variant of J3 introduced by us in 2016. The semantics of LPT0, as well as of its first-order version QLPT0, is given by twist structures defined over Boolean agebras. From this, it is possible to adapt the standard Boolean-valued models of (classical) ZFC to twist-valued models of an expansion of ZFC by adding a paraconsistent negation. We argue that the implication operator of LPT0 is more suitable for a paraconsistent set theory than the implication of PS3, since it allows for genuinely inconsistent sets w such that [(w = w)] = 1/2 . This implication is not a 'reasonable implication' as defined by Löwe and Tarafder. This suggests that 'reasonable implication algebras' are just one way to define a paraconsistent set theory. Our twist-valued models are adapted to provide a class of twist-valued models for (PS3,*), thus generalizing Löwe and Tarafder result. It is shown that they are in fact models of ZFC (not only of ZF). (shrink)

In this paper we extend to non-classical set theories the standard strategy of proving independence using Boolean-valued models. This extension is provided by means of a new technique that, combining algebras (by taking their product), is able to provide product-algebra-valued models of set theories. In this paper we also provide applications of this new technique by showing that: (1) we can import the classical independence results to non-classical set theory (as an example we prove the independence of $\mathsf {CH}$ ); (...) and (2) we can provide new independence results. We end by discussing the role of non-classical algebra-valued models for the debate between universists and multiversists and by arguing that non-classical models should be included as legitimate members of the multiverse. (shrink)

Quasi-set theory Q is an alternative set-theory designed to deal mathematically with collections of indistinguishable objects. The intended interpretation for those objects is the indistinguishable particles of non-relativistic quantum mechanics, under one specific interpretation of that theory. The notion of cardinal of a collection in Q is treated by the concept of quasi-cardinal, which in the usual formulations of the theory is introduced as a primitive symbol, since the usual means of cardinal definition fail for collections of indistinguishable objects. In (...) a recent work, Domenech and Holik have proposed a definition of quasi-cardinality in Q. They claimed their definition of quasi-cardinal not only avoids the introduction of that notion as a primitive one, but also that it may be seen as a first step in the search for a version of Q that allows for a greater representative power. According to them, some physical systems can not be represented in the usual formulations of the theory, when the quasi-cardinal is considered as primitive. In this paper, we discuss their proposal and aims, and also, it is presented a modification from their definition we believe is simpler and more general. (shrink)

Multiverse Views in set theory advocate the claim that there are many universes of sets, no-one of which is canonical, and have risen to prominence over the last few years. One motivating factor is that such positions are often argued to account very elegantly for technical practice. While there is much discussion of the technical aspects of these views, in this paper I analyse a radical form of Multiversism on largely philosophical grounds. Of particular importance will be an account of (...) reference on the Multiversist conception, and the relativism that it implies. I argue that analysis of this central issue in the Philosophy of Mathematics indicates that Radical Multiversism must be algebraic, and cannot be viewed as an attempt to provide an account of reference without a softening of the position. (shrink)

Benacerraf’s 1965 multiple-reductions argument depends on what I call ‘deferential logicism’: his necessary condition for number-set identity is most plausible against a background Quineanism that allows autonomy of the natural number concept. Steinhart’s ‘folkist’ sufficient condition on number-set identity, by contrast, puts that autonomy at the center — but fails for not taking the folk perspective seriously enough. Learning from both sides, we explore new conditions on number-set identity, elaborating a suggestion from Wright.

This paper discusses some of the ways in which Martin-Löf type theory differs from set theory. The discussion concentrates on conceptual, rather than technical, differences. It revolves around four topics: sets versus types; syntax; functions; and identity. The difference between sets and types is spelt out as the difference between unified pluralities and kinds, or sorts. A detailed comparison is then offered of the syntax of the two languages. Emphasis is put on the distinction between proposition and judgement, drawn by (...) type theory, but not by set theory. Unlike set theory, type theory treats the notion of function as primitive. It is shown that certain inconveniences pertaining to function application that afflicts the set-theoretical account of functions are thus avoided. Finally, the distinction, drawn in type theory, between judgemental and propositional identity is discussed. It is argued that the criterion of identity for a domain cannot be formulated in terms of propositional identity. It follows that the axiom of extensionality cannot be taken as a statement of the criterion of identity for sets. (shrink)

We study the degrees of unsolvability of sets which are cohesive . We answer a question raised by the first author in 1972 by showing that there is a cohesive set A whose degree a satisfies a' = 0″ and hence is not high. We characterize the jumps of the degrees of r-cohesive sets, and we show that the degrees of r-cohesive sets coincide with those of the cohesive sets. We obtain analogous results for strongly hyperimmune and strongly hyperhyperimmune sets (...) in place of r-cohesive and cohesive sets, respectively. We show that every strongly hyperimmune set whose degree contains either a Boolean combination of ∑2 sets or a 1-generic set is of high degree. We also study primitive recursive analogues of these notions and in this case we characterize the corresponding degrees exactly. MSC: 03D30, 03D55. (shrink)

We present a novel treatment of set theory in a four-valued paraconsistent and paracomplete logic, i.e., a logic in which propositions can be both true and false, and neither true nor false. Our approach is a significant departure from previous research in paraconsistent set theory, which has almost exclusively been motivated by a desire to avoid Russell’s paradox and fulfil naive comprehension. Instead, we prioritise setting up a system with a clear ontology of non-classical sets, which can be used to (...) reason informally about incomplete and inconsistent phenomena, and is sufficiently similar to ${\mathrm {ZFC}}$ to enable the development of interesting mathematics.We propose an axiomatic system ${\mathrm {BZFC}}$, obtained by analysing the ${\mathrm {ZFC}}$ -axioms and translating them to a four-valued setting in a careful manner, avoiding many of the obstacles encountered by other attempted formalizations. We introduce the anti-classicality axiom postulating the existence of non-classical sets, and prove a surprising results stating that the existence of a single non-classical set is sufficient to produce any other type of non-classical set.Our theory is naturally bi-interpretable with ${\mathrm {ZFC}}$, and provides a philosophically satisfying view in which non-classical sets can be seen as a natural extension of classical ones, in a similar way to the non-well-founded sets of Peter Aczel [1].Finally, we provide an interesting application concerning Tarski semantics, showing that the classical definition of the satisfaction relation yields a logic precisely reflecting the non-classicality in the meta-theory. (shrink)

Continuity as appears to us immediately by intuition differs from its current formalization, the arithmetical continuum or equivalently the set of real numbers used in modern mathematical analysis. Motivated by the known mathematical and physical problems arising from this formalization of the continuum, our aim in this paper is twofold. Firstly, by interpreting Chaitin’s variant of Gödel’s first incompleteness theorem as an inherent uncertainty or fuzziness of the arithmetical continuum, a formal set-theoretic entropy is assigned to the arithmetical continuum. Secondly, (...) by analyzing Noether’s theorem on symmetries and conserved quantities, we argue that whenever the four dimensional space-time continuum containing a single, stationary, asymptotically flat black hole is modeled by the arithmetical continuum in the mathematical formulation of general relativity, the hidden set-theoretic entropy of this latter structure reveals itself as the entropy of the black hole, indicating that this apparently physical quantity might have a pure set-theoretic origin, too. (shrink)

Quasi-set theory is a theory for dealing with collections of indistinguishable objects. In this paper we discuss some logical and philosophical questions involved with such a theory. The analysis of these questions enable us to provide the first grounds of a possible new view of physical reality, founded on an ontology of non-individuals, to which quasi-set theory may constitute the logical basis.