We establish the following results: 1. In ZF (i.e., Zermelo-Fraenkel set theory minus the Axiom of Choice AC), for every set I and for every ordinal number α ≥ ω, the following statements are equivalent: (a) The Tychonoff product of| α| many non-empty finite discrete subsets of I is compact. (b) The union of| α| many non-empty finite subsets of I is well orderable. 2. The statement: For every infinite set I, every closed subset of the Tychonoff product [0, 1] (...) I which consists of functions with finite support is compact, is not provable in ZF set theory. 3. The statement: For every set I, the principle of dependent choices relativised to I implies the Tychonoff product of countably many non-empty finite discrete subsets of I is compact, is not provable in ZF⁰ (i.e., ZF minus the Axiom of Regularity). 4. The statement: For every set I, every ℵ₀-sized family of non-empty finite subsets of I has a choice function implies the Tychonoff product of ℵ₀ many non-empty finite discrete subsets of I is compact, is not provable in ZF⁰. (shrink)

A Hausdorff space is called effectively Hausdorff if there exists a function F—called a Hausdorff operator—such that, for every with,, where U and V are disjoint open neighborhoods of x and y, respectively. Among other results, we establish the following in, i.e., in Zermelo–Fraenkel set theory without the Axiom of Choice (): is equivalent to “For every set X, the Cantor cube is effectively Hausdorff”. This enhances the result of Howard, Keremedis, Rubin and Rubin [13] that is equivalent to “Hausdorff (...) spaces are effectively Hausdorff” in. The Boolean Prime Ideal Theorem and the statement “For every infinite set X, the Stone space of the Boolean algebra is effectively Hausdorff” are mutually independent. In particular, the latter statement is not provable in. The Axiom of Choice for non‐empty subsets of () is equivalent to each of “Separable Hausdorff spaces are effectively Hausdorff” and “The Cantor cube is effectively Hausdorff”. The Principle of Dependent Choices in conjunction with the Axiom of Choice for continuum sized families of non‐empty subsets of does not imply the axiom of choice for partitions of. The latter independence result fills the gap in information in Howard and Rubin's book “Consequences of the Axiom of Choice”. The axiom of countable choice for non‐empty subsets of is equivalent to each of “Denumerable Hausdorff spaces are effectively Hausdorff”, “Denumerable T3 spaces are completely normal” and “Denumerable Tychonoff spaces are Urysohn”. (shrink)

Un autore che esprime la propria visione del mondo attraverso immagini può essere considerato un filosofo? La filosofia è un dialogo tra quanti si confrontano entro gli orizzonti che il pensiero ha sedimentato nel tempo. Chi si esprime attraverso l’esclusivo esercizio delle arti visive o della musica non può essere considerato filosofo, questo nonostante l’artista esprima la propria visione del mondo al pari del filosofo. Ciò non impedisce a un autore che crei immagini o azioni o suoni di esp...

Previous studies have offered indications that the way pre-adolescents structure their perceptions of their teacher’s interaction in terms of Agency and Communion differs from adolescents. The purpose of this study was to delineate previous findings by thoroughly examining the structure of pre-adolescents’ perceptions of their teacher’s interpersonal behaviour, and by investigating the extent to which this structure relates to pre-adolescents’ learning outcomes. A mixed methods research design was implemented including a qualitative instrument adaptation procedure followed by a quantitative large-scale administration. (...) Participants for the latter were a random sample of 504 pre-adolescents and 36 teachers. Results showed that participants placed significantly more emphasis on their teachers’ Communion behavioural dimension and this emphasis was illustrated on the association of perceived teacher interaction and students’ learning outcomes. Implications for theory, research and practice are discussed. (shrink)

We study within the framework of Zermelo-Fraenkel set theory ZF the role that the axiom of choice plays in the theory of Lindelöf metric spaces. We show that in ZF the weak choice principles: Every Lindelöf metric space is separable and Every Lindelöf metric space is second countable are equivalent. We also prove that a Lindelöf metric space is hereditarily separable iff it is hereditarily Lindelöf iff it hold as well the axiom of choice restricted to countable sets and to (...) topologies of Lindelöf metric spaces as the countable union theorem restricted to Lindelöf metric spaces. (shrink)

We show that the both assertions “in every vector space B over a finite element field every subspace V ⊆ B has a complementary subspace S” and “for every family [MATHEMATICAL SCRIPT CAPITAL A] of disjoint odd sized sets there exists a subfamily ℱ={Fj:j ϵω} with a choice function” together imply the axiom of choice AC. We also show that AC is equivalent to the statement “in every vector space over ℚ every generating set includes a basis”.

I undertake a metaphysical investigation of Saul Kripke's modern classic, Naming and Necessity . The general problem of my study may be expressed as follows: What is the metaphysical justification of the validity and existence of the pertinent classes of truths, the necessary a posteriori and the contingent a priori, according to the Kripke Paradigm? My approach is meant to disclose the logical and ontological principles underlying Kripke's arguments for the necessary a posteriori and the contingent a priori respectively. The (...) results of my study are to a certain extent negative: They attempt to show that the classes of the necessary a posteriori and the contingent a priori statements cannot possibly be valid. If the general argument of this thesis is sound, then, on this ground, the realist conception of metaphysical essentialism is rejected. The positive thesis of this study is the articulation of certain ways to frame the necessary a posteriori and the contingent a priori which avoid the problems of realistic essentialism and which suggest a certain transcendental reading of modality. According to the Kripke Paradigm, any de dicto modal status derives ultimately from the de re modality inherent in the object designated, as the object is characterised by contingent and necessary properties on the ontological level. Thus, the Kripke Paradigm is primarily a thesis in de re realist essentialism. The final verdict in this study is that the Kripke Paradigm cannot sustain the realistic conception of de re metaphysical essentialism. If we should adopt a transcendental reading of modality, then certain portions of the Kripke Paradigm are valid. I do not delineate or possess the details of a comprehensive doctrine of transcendental metaphysic. Nevertheless, the observations I make should suffice to bring about the rough orientation of how I conceive the notion of transcendental modality. (shrink)

From the early 1930s to the early 1960s many scholars, whether liberalminded or socialist ideologues, Marxist or scientific positivists, classical scholars or political theorists and historians, have shown a widespread consensus in discrediting and assailing the man and political philosopher Plato. Such an extensive assault led the ‘Platonic Legend’ to an unprecedented crisis. Philosophically, it was a reaction to the undisguised Platonolatry coming from Oxford and the school of the British Idealists. Ideologically, the appropriation of Plato by Nazi apologists fostered (...) further this vehement indictment. But a lot of other causes worked to the same effect. The general anguish and humanistic anxiety on the eve ofWorldWar II and the postwar traumas led scholars to reconsider the meaning of history and historicism, the psychology of the masses and the ethical responsibility of the citizen, the role of propaganda and state education. Such complementary elements converged in sustained anti-Platonic polemics, which in turn provoked a vigorous defence. Here an attempt is made to offer a preliminary survey of this complex debate and to provide a general intellectual framework in terms of which that controversy can be further explored. (shrink)

This article critiques the theological and moral foundations that undergird the approach of the document For the Life of the World: Toward a Social Ethos of the Orthodox Church to the question of the Church's role in the public sphere. The article's focus is essentially two-fold. First, it strives to clarify For the Life of the World's hermeneutical method through a consideration of its frequent appeals to the authority of the Orthodox tradition. Second, the article seeks to understand the document's (...) tacit reliance upon a Eucharistic model of ecclesiology and its connection to the recurring theme of the ‘transfiguring of the world’. (shrink)

We find some characterizations of the Axiom of Choice in terms of certain families of open sets in T1 spaces.

We show that the statement “separable, countably compact, regular spaces are Baire” is deducible from a strictly weaker form than AC, namely, CAC . We also find some characterizations of the axiom of dependent choices.

In the realm of Lindelöf metric spaces the following results are obtained in : (i) If is a Lindelöf metric space then it is both densely Lindelöf and almost Lindelöf. In addition, under the countable axiom of choice, the three notions coincide. (ii) The statement “every separable metric space is almost Lindelöf” implies that every infinite subset of has a countably infinite subset). (iii) The statement “every almost Lindelöf metric space is quasi totally bounded implies. (iv) The proposition “every quasi (...) totally bounded metric space is separable” lies, in the deductive hierarchy of choice principles, strictly between the countable union theorem and. Likewise, the statement “every pre‐Lindelöf (or Lindelöf) metric space is separable” lies strictly between and. (shrink)

In the realm of metric spaces the role of choice principles is investigated.

This is the second paper of an essay in two parts. The first paper (Part I) is a critical discussion of Mark Risjord's conception of nursing knowledge where I argued against the conception of nursing knowledge as a kind of nursing science. The aim of the present paper (Part II) is to explicate and substantiate the thesis of nursing as a kind of concrete philosophy. My strategy is to elaborate upon certain themes from Wittgenstein's Tractatus in order to canvass a (...) general scheme of philosophy based on a distinction between reality and the world. This distinction will be employed in the appropriation of certain significant features of nursing and nursing knowledge. By elaborating on the contrast between the abstract and the concrete, I will suggest that nursing may be seen as a kind of concrete philosophy, being primarily concerned with reality (and secondarily with the world). This thesis, I will argue, implies that philosophy is the kind of theory that is essential to nursing (which is not so much a theory than a certain kind of activity). (shrink)

We show that (i) it is consistent with that there are infinite sets X on which every metric is discrete; (ii) the notion of real infinite is strictly stronger than that of metrically infinite; (iii) a set X is metrically infinite if and only if it is weakly Dedekind‐infinite if and only if the cardinality of the set of all metrically finite subsets of X is strictly less than the size of ; and (iv) an infinite set X is weakly (...) Dedekind‐infinite if and only if has infinite towers if and only if X has countable partitions. (shrink)

This article reviews Bergson’s proposed solution to the classical mind-body problem, confronting his system with previous alternatives, but mainly Kant's. To this end, it offers an analysis of Matter and memory’s main argument, aimed at discerning its consequences for a competing definition of knowledge.

We show that the countable axiom of choice CAC strictly implies the statements “Lindelöf metric spaces are second countable” “Lindelöf metric spaces are separable”. We also show that CAC is equivalent to the statement: “If is a Lindelöf topological space with respect to the base ℬ, then is Lindelöf”.

An overview is provided of how the concept of the thought experiment has developed and changed for the natural sciences in the course of the 20th century. First, we discuss the existing definitions of the term 'thought experiment' and the origin of the thought experimentation method, identifying it in Greek Presocratics epoch. Second, only in the end of the 19th century showed up the first systematic enquiry on thought experiments by Ernst Mach's work. After the Mach's work, a negative attitude (...) towards thought experiments came in the beginning of the 20th century, which went on until the Thomas Kuhn's and Karl Popper's work on thought experiments. Only from the mid-1980s did thought experiments begin to be considered relevant to scientific enterprise. Finally, we show the existing empirical and 'functional' theories which have developed about the nature and purpose of thought experiments. (shrink)

This article is an investigation into the field of neurotheology, a controversial field which has attracted criticism from both the scientific and religious community and which is often quite divided among its own practitioners. Regretfully, but not too unexpectedly, science has gotten entangled with ideology, as we shall see, with proponents on all sides of the spectrum using findings from the laboratory in support of their own philosophical positions. We will begin by exploring some definitions of the field and then (...) move on to examine the claims made by the different scientists working in this area, before examining what conclusions we can draw for spirituality. (shrink)

La pobreza tiene una base material, la penuria de recursos. Pero su naturaleza es moral, consiste en el reconocimiento de que ciertas situaciones son acreedoras de acciones preventivas o remediadoras. Las sociedades vienen colocando tradicionalmente el umbral de pobreza en el punto por debajo del cual esta en peligro la subsistencia. Pero la conciencia reformista del mundo occidental ha llegado a servirse de la idea de pobreza para señalar los fallos de su compromiso de igualdad social. No se trata tanto (...) de una benévola elevación del baremo, como de una reconceptualización acorde con la ideología política dominante. (shrink)

The present article proceeds from the mainstream view that the conceptual framework underpinning adversarial systems of criminal adjudication, i.e. a mixture of common-sense philosophy and probabilistic analysis, is unsustainable. In order to provide fact-finders with an operable structure of justification, we need to turn to epistemology once again. The article proceeds in three parts. First, I examine the structural features of justification and how various theories have attempted to overcome Agrippa’s trilemma. Second, I put Inferential Contextualism to the test and (...) show that a defeasible structure of justification allocating epistemic rights and duties to all participants of an inquiry manages to dissolve the problem of scepticism. Third, I show that our epistemic practice already embodies a contextualist mechanism. Our problem was not that our Standard of Proof is inoperable but that it was not adequately conceptualized. Contextualism provides the framework to articulate the abovementioned practice and to treat ‘reasonable doubts’ as a mechanism which we can now describe in detail. The seemingly insurmountable problem with our efforts to define the concept “reasonable doubts” was the fact that we have been conflating the surface features of this mechanism and its internal structure, i.e. the rules for its use. (shrink)

Arthur Danto è figura di spicco della critica d’arte contemporanea nordamericana. I suoi saggi, le opinioni espresse su The Nation (è dal 1984 che occupa le stesse colonne una volta affidate a Clement Greenberg) o su Artforum hanno una forte presa sulla scena artistica. Danto è una figura anomala nell’ambito della critica d’arte in quanto, pur essendo un filosofo, frequenta gli studi degli artisti e le gallerie, comprese quelle più radical. La filosofia e la critica d’arte utilizzano linguagg...

The riddle of the Cleitophon is a creature of nineteenth-century German scholarship which premised that Plato had developed a profound philosophical system. Thus, having no intrinsic purpose to serve in the context of the development of Plato's philosophy, Cleitophon was disallowed as spurious. Documenting the reception of this minor dialogue provides insights into the pluralism and the perplexities of modern Platonic exegesis. The more recent idea of a genre of literary fiction helps to restore cleitophon to its place in the (...) Platonic corpus. S.R. Slings's edition of this dialogue, with introduction, translation and commentary reconsiders both Plato's dialogical engagements and intellectual commitments, utilizes a two-dimensional approach to the text that avoids exaggerated conjectures, and effectively defends the authenticity of Cleitophon. (shrink)

PREFACE Strange or inappropriate as it may appear, we propose to start this little book by stating as flatly and unrhetorically as possible a fundamental ...

Over the past few years, several proposals aimed at procuring human pluripotent (embryonic-like) stem cells without involving the destruction of a human embryo have been proposed and widely discussed. This article focuses on a basic aspect of the debate, namely the plausibility of one or more of these new proposals being able to meet the ethical requirements that those who regard the human embryo as sacred have tried to impose on stem cells research in the last ten years. The thesis (...) of the article is that focusing the discussion only on the sources of stem cells has prevented a full understanding of the foundation, meaning and scope of these ethical requirements. To substantiate this thesis, the article takes into consideration two issues: the first has to do with the potential of the cells obtained through some of the new approaches (iPS included), the second (and decisive) with the argument of the ‘indirect complicity’, applied to the use of ‘contaminated’ knowledge. (shrink)