It is shown that—over Bishop's constructive mathematics—the indecomposability of ℝ is equivalent to the statement that all functions from a complete metric space into a metric space are sequentially nondiscontinuous. Furthermore we prove that the indecomposability of ℝ \ {0} is equivalent to the negation of the disjunctive version of Markov's Principle. These results contribute to the programme of Constructive Reverse Mathematics.

In this paper we show that the compactness of a Loeb space depends on its cardinality, the nonstandard universe it belongs to and the underlying model of set theory we live in. In $\S1$ we prove that Loeb spaces are compact under various assumptions, and in $\S2$ we prove that Loeb spaces are not compact under various other assumptions. The results in $\S1$ and $\S2$ give a quite complete answer to a question of D. Ross (...) in [9], [11] and [12]. (shrink)

We characterize Maharam spectra of Loeb probability spaces and give some applications of the results.

We characterize Maharam spectra of Loeb probability spaces and give some applications of the results.

It is shown that for compact metric spaces the following statements are pairwise equivalent: “X is Loeb”, “X is separable”, “X has a we ordered dense subset”, “X is second countable”, and “X has a dense set G = ∪{Gn : n ∈ ω}, ∣Gn∣ < ω, with limn→∞ diam = 0”. Further, it is shown that the statement: “Compact metric spaces are weakly Loeb” is not provable in ZF0 , the Zermelo-Fraenkel set theory without the (...) axiom of regularity, and that the countable axiom of choice for families of finite sets CACfin does not imply the statement “Compact metric spaces are separable”. (shrink)

This volume will thus appeal to advanced students and scholars not just in the history of early modern philosophy but in epistemology and other core areas of ...

Moral realism, the view that there are moral facts that are independent of our beliefs about them, has many defenders. But much less has been said about realism concerning other sorts of value. One of these, gastronomic realism is likely to seem implausible on its face. This paper argues, however, that much of the reasoning used to defend moral realism is about as well suited for defending gastronomic realism. Although these considerations do not directly undermine moral realism, they do suggest (...) that the two views should stand or fall together. And they rob moral realists of one ad hominem argument that often emerges in their debate with irrealists, that the irrealists cannot justify their widespread practice of taking their own moral values seriously. 2012 APA, all rights reserved). (shrink)

The distinguished philosopher Louis Loeb examines the epistemological framework of Scottish philosopher David Hume, as employed in his celebrated work A Treatise of Human Nature. Loeb's project is to advance an integrated interpretation of Hume's accounts of belief and justification. His thesis is that Hume, in his Treatise, has a "stability-based" theory of justification which posits that his belief is justified if it is the result of a belief producing mechanism that engenders stable beliefs. But Loeb argues (...) that the striking corollary to this theory is that no belief generating mechanism is fully stable - or fully justified - for a fully reflective person. This carefully argued and original interpretation of Hume makes sense of seemingly contradictory ideas and will provoke serious discussion among Hume scholars. (shrink)

Experimental moral philosophy emerged as a methodology in the last decade of the twentieth century, as a branch of the larger experimental philosophy (X-Phi) approach. Experimental moral philosophy is the empirical study of moral intuitions, judgments, and behaviors. Like other forms of experimental philosophy, it involves gathering data using experimental methods and using these data to substantiate, undermine, or revise philosophical theories. In this case, the theories in question concern the nature of moral reasoning and judgment; the extent and sources (...) of moral obligations; the nature of a good person and a good life; even the scope and nature of moral theory itself. This entry begins with a brief look at the historical uses of empirical data in moral theory and goes on to ask what, if anything, is distinctive about experimental moral philosophy—how should we distinguish it from related work in empirical moral psychology? After discussing some strategies for answering this question, the entry examines two of the main projects within experimental moral philosophy, and then discusses some of the most prominent areas of research within the field. As we will see, in some cases experimental moral philosophy has opened up new avenues of investigation, while in other cases it has influenced longstanding debates within moral theory. (shrink)

We introduce and study several nonstandard representations of Banach-valued operators defined on the space of Bochner integrable functions. They will be less restrictive than the usual standard representation. In particular, without any hypothesis, we shall find a representation whose kernel belongs to a space of “extended Bochner integrable functions”, introduced by Zimmer by using Loeb measures.

Experimental moral philosophy began to emerge as a methodology inthe last decade of the twentieth century, a branch of the largerexperimental philosophy approach. From the beginning,it has been embroiled in controversy on a number of fronts. Somedoubt that it is philosophy at all. Others acknowledge that it isphilosophy but think that it has produced modest results at best andconfusion at worst. Still others think it represents an important advance., Before the research program can be evaluated, we should have someconception of (...) its scope. But controversy surrounds questionsabout its boundaries as well. Uncontroversially, the distinctionbetween experimental and non-experimental philosophy is not identicalto the distinction between a posteriori and a prioriphilosophy. Experimental evidence is a proper subset of empiricalevidence, which is itself a subset of a posteriorievidence. Can any more be said?, L1: A child raised in a particular linguistic community almostinevitably ends up speaking an idiolect of the local language despitelack of sufficient explicit instruction, lack of extensive negativefeedback for mistakes, and grammatical mistakes by caretakers., character, moral | cognitivism vs. non-cognitivism, moral | emotion | ethics: virtue | intuition | moral anti-realism | moral psychology: empirical approaches | moral realism | moral relativism | well-being, Copyright © 2014 byMark Alfano Don Loeb, View this site from another server:. (shrink)

Since the mid-1970s, scholars have recognized that the skeptical interpretation of Hume's central argument about induction is problematic. The science of human nature presupposes that inductive inference is justified and there are endorsements of induction throughout "Treatise" Book I. The recent suggestion that I.iii.6 is confined to the psychology of inductive inference cannot account for the epistemic flavor of its claims that neither a genuine demonstration nor a non-question-begging inductive argument can establish the uniformity principle. For Hume, that inductive inference (...) is justified is part of the data to be explained. Bad argument is therefore excluded as the cause of inductive inference; and there is no good argument to cause it. Does this reinstate the problem of induction, undermining Hume's own assumption that induction is justified? It does so only if justification must derive from "reason", from the availability of a cogent argument. Hume rejects this internalist thesis; induction's favorable epistemic status derives from features of custom, the mechanism that generates inductive beliefs. Hume is attracted to this externalist posture because it provides a direct explanation of the epistemic achievements of children and non-human animals — creatures that must rely on custom unsupplemented by argument. (shrink)

We give an overview of the role of equicontinuity of sequences of real-valued functions on [0,1] and related notions in classical mathematics, intuitionistic mathematics, Bishop’s constructive mathematics, and Russian recursive mathematics. We then study the logical strength of theorems concerning these notions within the programme of Constructive Reverse Mathematics. It appears that many of these theorems, like a version of Ascoli’s Lemma, are equivalent to fan-theoretic principles.

Hume's claim that a state is a belief is often intertwined—though without his remarking on this fact—with epistemic approval of the state. This requires explanation. Beliefs, in Hume's view, are steady dispositions (not lively ideas), nature's provision for a steady influence on the will and action. Hume's epistemic distinctions call attention to circumstances in which the presence of conflicting beliefs undermine a belief's influence and thereby its natural function. On one version of this interpretation, to say that a belief is (...) justified, ceteris paribus, is to say that for all that has been shown the belief would be steady in its influence under suitable reflection. On a second version, it is to say that prima facie justification is an intrinsic property of the state, in virtue of its steadiness. These versions generate different understandings of the relationship between Parts iii and iv of Book I of the Treatise. (shrink)

Hume's claim that a state is a belief is often intertwined—though without his remarking on this fact—with epistemic approval of the state. This requires explanation. Beliefs, in Hume's view, are steady dispositions (not lively ideas), nature's provision for a steady influence on the will and action. Hume's epistemic distinctions call attention to circumstances in which the presence of conflicting beliefs undermine a belief's influence and thereby its natural function. On one version of this interpretation, to say that a belief is (...) justified, ceteris paribus, is to say that for all that has been shown the belief would be steady in its influence under suitable reflection. On a second version, it is to say that prima facie justification is an intrinsic property of the state, in virtue of its steadiness. These versions generate different understandings of the relationship between Parts iii and iv of Book I of the Treatise. (shrink)

The eternal recurrence of the same. Simmel's critique ; Awareness ; Evidence ; Significance ; Coherence -- Demon or god? Deathbed revelation ; Daimonic prophecy ; Dionysian doctrine ; Diagnostic test -- The dwarf and the gateway. The gateway to Hades ; The dwarf's interpretation ; Zarathustra's cross-examination ; The inescapable cycle ; Crossing the gateway ; No time until rebirth ; The ancient memory ; Midnight swan song -- The great noon. Two conclusions ; Tragic end and analeptic satyr (...) play ; Zarathustra's hour ; Noon crucifixion ; Seventh-day convalescence ; Last temptation ; Third-day resurrection -- The laughing lions. Revaluation of values ; Dawn reunion ; Morning consecration ; Call to arms ; Final farewell and last will ; Zarathustra's great destiny -- The shepherd and the serpent. The eternally recurring human ; The future human ; Zarathustra the dragon-slayer ; The decapitation ; The heavy hammer ; No longer shepherd -- Circulus vitiosus deus. Impotence and revenge ; Backward-willing ; Self-redemption ; The third transformation ; The child spirit ; No longer human ; Zarathustra's dying gift -- Post-Zarathustra. Nietzsche and Zarathustra ; Reversing the bad conscience ; Atheism and the death of God ; Countering the ascetic ideal. (shrink)

Gilbert Harman and Judith Thomson have argued that moral facts cannot explain our moral beliefs, claiming that such facts could not play a causal role in the formation of those beliefs. This paper shows these arguments to be misguided, for they would require that we abandon any number of intuitively plausible explanations in non‐moral contexts as well. But abandoning the causal strand in the argument over moral explanations does not spell immediate victory for the moral realist, since it must still (...) be shown that moral facts do figure in our best global explanatory theory. (shrink)

David Hume's A Treatise of Human Nature is famous for its extreme skepticism. Louis Loeb argues that Hume's destructive conclusions have in fact obscured a constructive stage that Hume abandons prematurely. Working within a philosophical tradition that values tranquillity, Hume favors an epistemology that links justification with settled belief. Hume appeals to psychological stability to support his own epistemological assessments, both favorable regarding causal inference, and unfavorable regarding imaginative propensities. The theory's success in explaining Hume's epistemic distinctions gives way (...) to pessimism, since Hume contends that reflection on beliefs is deeply destabilizing. So much the worse, Hume concludes, for placing a premium on reflection. Hume endorses and defends the position that stable beliefs of unreflective persons are justified, though they would not survive reflection. At the same time, Hume relishes the paradox that unreflective beliefs enjoy a preferred epistemic status and strains to establish it. Loeb introduces a series of amendments to the Treatise that secures a more positive result for justified belief while maintaining Hume's fundamental principles. In his review of Hume's applications of his epistemology, Loeb uncovers a stratum of psychological doctrine beyond associationism, a theory of conditions in which beliefs are felt to conflict and of the resolution of this uneasiness or dissonance. This theory of mental conflict is also essential to Hume's strategy for integrating empiricism about meaning with his naturalism. However, Hume fails to provide a general account of the conditions in which conflicting beliefs lead to persisting instability, so his theory is incomplete. Loeb explores Hume's concern with stability in reference to his discussions of belief, education, the probability of causes, unphilosophical probability, the belief in body, sympathy and moral judgment, and the passions, among other topics. (shrink)

In this study of Nietzsche's Thus Spoke Zarathustra, Paul S. Loeb proposes a fresh account of the relation between the book's literary and philosophical aspects and argues that the book's narrative is designed to embody and exhibit the truth of eternal recurrence. Loeb shows how Nietzsche constructed a unified and complete plot in which the protagonist dies, experiences a deathbed revelation of his endlessly repeating life, and then returns to his identical life so as to recollect this revelation (...) and gain a power over time that advances him beyond the human. Through close textual analysis and careful attention to Nietzsche's use of Platonic, biblical, and Wagnerian themes, Loeb explains how this novel design is the key to solving the many riddles of Thus Spoke Zarathustra - including its controversial fourth part, its obscure concept of the Übermensch, and its relation to Nietzsche's Genealogy of Morals. (shrink)

In his Treatise, Hume confronted the tensions between his project of uncovering the causal operations of the human mind and the extreme skeptical tendencies of his system. Louis Loeb argues that Hume overreaches, and he advances a controversial interpretation of Hume's epistemological framework that shows how Hume could have avoided the more destructive positions in his work.

We introduce a mathematical framework for quantifying fine-tuning in general physical settings. In particular, we identify two distinct perspectives on fine-tuning, namely, a local and a global perspective --- and develop corresponding measures. These measures apply broadly to settings characterized by an arbitrary number of observables whose values are dependent on an arbitrary number of parameters. We illustrate our formalism by quantifying fine-tuning as it arises in two pertinent astrophysical settings: in models where a significant fraction of the dark matter (...) in the universe is in the form of primordial black holes, and in scenarios that derive the fraction of protons in habitable dark-matter halos from underlying models of cosmic inflation. (shrink)

Many of the earliest books, particularly those dating back to the 1900s and before, are now extremely scarce and increasingly expensive. We are republishing these classic works in affordable, high quality, modern editions, using the original text and artwork.

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)

The paper [Tarski: Les fondements de la géométrie des corps, Annales de la Société Polonaise de Mathématiques, pp. 29—34, 1929] is in many ways remarkable. We address three historico-philosophical issues that force themselves upon the reader. First we argue that in this paper Tarski did not live up to his own methodological ideals, but displayed instead a much more pragmatic approach. Second we show that Leśniewski's philosophy and systems do not play the significant role that one may be tempted to (...) assign to them at first glance. Especially the role of background logic must be at least partially allocated to Russell's systems of Principia mathematica. This analysis leads us, third, to a threefold distinction of the technical ways in which the domain of discourse comes to be embodied in a theory. Having all of this in place, we discuss why we have to reject the argument in [Gruszczyński and Pietruszczak: Full development of Tarski's Geometry of Solids, The Bulletin of Symbolic Logic, vol. 4, no. 4, pp. 481—540] according to which Tarski has made a certain mistake. (shrink)

In recent years it has become fashionable to criticize a family of approaches to moral thinking, loosely collected under the heading, “moral theory.” Unfortunately, these criticisms have often lacked a sharp focus. To do better, we will have to sort out the various elements of moral theory, and carefully consider them one at a time. One element which has heretofore received too little attention is generality. Demands for generality have become so central to the way many philosophers do ethics that (...) we hardly notice them. Yet they sometimes exert a powerful influence on our thinking, pressing us to treat more general moral positions, such as consequentialism, as superior to more specific ones, like those which incorporate agent-centered restrictions or prerogatives. Although few moral philosophers accept these demands for generality explicitly, and no one has articulated them very clearly, many are in their grip to some significant degree. (shrink)

It is often said that our moral experience, broadly construed to include our ways of thinking and talking about morality, has a certain objective-seeming character to it, and that this supports a presumption in favor of objectivist theories and against anti-objectivist theories like Mackie’s error theory. In this paper, I argue that our experience of morality does not support objectivist moral theories in this way. I begin by arguing that our moral experience does not have the uniformly objective-seeming character it (...) is typically claimed to have. I go on to argue that even if moral experience were to presuppose or display morality as a realm of fact, we would still need a reason for taking that to support theories according to which it is such a realm. I consider what I take to be the four most promising ways of attempting to supply such a reason: inference to the best explanation, epistemic conservatism, the Principle of Credulity, and the method of wide reflective equilibrium. In each case, I argue, the strategy in question does not support a presumption in favor of objectivist moral theories. (shrink)