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A number of well-developed theories shed light on the question, under what circumstances our beliefs enjoy epistemic justification. Yet, comparatively little is known about epistemic defeat--when new information causes the loss of epistemic justification. This book proposes and defends a detailed account of epistemic defeaters. The main kinds of defeaters are analyzed in detail and integrated into a general framework that aims to explain how beliefs lose justification. It is argued that defeaters introduce incompatibilities into a noetic system and thereby (...) prompt a structured re-evaluation process that makes a justified reinstatement of the defeated belief impossible. The account is then applied to the topic of disagreement, where it is used in an argument for conciliationism, as well as a new explanation for higher-order defeat. Throughout the book, the notion of defeat is the center of attention, while a number of new issues are discussed at the intersections of defeat and justification. Specifically, new problems are raised for broadly internalist accounts of defeat, a fully descriptive reliabilist account of defeat is provided, and the case for normative defeat is revisited. (shrink)

Introduction -- Part 1. The nature of order in world politics: the concept of order in world politics; does order exist in world politics?; how is order maintained in world politics?; order versus justice in world politics -- Part 2. Order in the contemporary international system: the balance of power and international order; international law and international order; diplomacy and international order; war and international order; the great powers and international order -- Part 3. Alternative paths to world order: alternatives (...) to the contemporary states system; the decline of the states system; the obsolescence of the states system; the reform of the states system; conclusion. (shrink)

A system of logic usually comprises a language for which a model-theory and a proof-theory are defined. The model-theory defines the semantic notion of model-theoretic logical consequence (⊨), while the proof-theory defines the proof- theoretic notion of logical consequence (or logical derivability, ⊢). If the system in question is sound and complete, then the two notions of logical consequence are extensionally equivalent. The concept of full formalization is a more restrictive one and requires in addition the preservation of the standard (...) meanings of the logical terms in all the admissible interpretations of the logical calculus, as it is proof-theoretically defined. Although classical first-order logic is sound and complete, its standard formalizations fall short to be full formalizations since they allow non-intended interpretations. This fact poses a challenge for the logical inferentialism program, whose main tenet is that the meanings of the logical terms are uniquely determined by the formal axioms or rules of inference that govern their use in a logical calculus, i.e., logical inferentialism requires a categorical calculus. This paper is the first part of a more elaborated study which will analyze the categoricity problem from its beginning until the most recent approaches. I will first start by describing the problem of a full formalization in the general framework in which Carnap (1934/1937, 1943) formulated it for classical logic. Then, in sections IV and V, I shall discuss the way in which the mathematicians B.A. Bernstein (1932) and E.V. Huntington (1933) have previously formulated and analyzed it in algebraic terms for propositional logic and, finally, I shall discuss some critical reactions Nagel (1943), Hempel (1943), Fitch (1944), and Church (1944) formulated to these approaches. (shrink)

Review the Istvan Kiraly V.'s book: Death and History.

This paper seeks to develop a phenomenological account of the disorientation of grief, specifically the relationship between disorientation and the breakdown in practical self-understanding at the heart of grief. I argue that this breakdown cannot be sufficiently understood as a breakdown of formerly shared practices and habitual patterns of navigating lived-in space that leaves the bereaved individual at a loss as to how to go on. Examining the experience of losing a loved person and a loved person-to-be, I instead propose (...) that this breakdown should be understood primarily in relation to a distinctive kind of futurity operative in disorientation,irrespectiveof the extent to which there is a breakdown of formerly shared practices and habitual patterns of navigating lived-in space. Drawing on the resources afforded by Heidegger’s phenomenology, I argue that it is acore characteristicof the experience of disorientation in grief that the grieving person can no longer meaningfully press ahead into aspecific futural self. This view comes with certain advantages over existing accounts of the temporality of grief for making sense of the disorientated relationship to futurity, which the appeal to Heideggerian resources makes possible. (shrink)

This paper seeks to develop a phenomenological account of the disorientation of grief, specifically the relationship between disorientation and the breakdown in practical self-understanding at the heart of grief. I argue that this breakdown cannot be sufficiently understood as a breakdown of formerly shared practices and habitual patterns of navigating lived-in space that leaves the bereaved individual at a loss as to how to go on. Examining the experience of losing a loved person and a loved person-to-be, I instead propose (...) that this breakdown should be understood primarily in relation to a distinctive kind of futurity operative in disorientation, irrespective of the extent to which there is a breakdown of formerly shared practices and habitual patterns of navigating lived-in space. Drawing on the resources afforded by Heidegger’s phenomenology, I argue that it is a core characteristic of the experience of disorientation in grief that the grieving person can no longer meaningfully press ahead into a specific futural self. This view comes with certain advantages over existing accounts of the temporality of grief for making sense of the disorientated relationship to futurity, which the appeal to Heideggerian resources makes possible. (shrink)

Background:Less sedated and more awake patients in the intensive care unit may cause ethical challenges.Research objectives:The purpose of this study is to describe ethical challenges registered nurses experience when patients refuse care and treatment.Research design:Narrative individual open interviews were conducted, and data were analysed using a phenomenological hermeneutic method developed for researching life experiences.Participants and research context:Three intensive care registered nurses from an intensive care unit at a university hospital in Norway were included.Ethical considerations:Norwegian Social Science Data Services approved the (...) study. Permission was obtained from the intensive care unit leader. The participants’ informed and voluntary consent was obtained in writing.Findings:Registered nurses experienced ethical challenges in the balance between situations of deciding on behalf of the patient, persuading the patient and letting the patient decide. Ethical challenges were related to patients being harmful to themselves, not keeping up personal hygiene and care or hindering critical treatment.Discussion:It is made apparent how professional ethics may be threatened by more pragmatic arguments. In recent years, registered nurses are faced with increasing ethical challenges to do no harm and maintain dignity.Conclusion:Ethically challenging situations are emerging, due to new targets including conscious and aware critical care patients, leaving an altered responsibility on the registered nurses. Reflection is required to adjust the course when personal and professional ideals no longer are in harmony with the reality in the clinical practice. RNs must maintain a strong integrity as authentic human beings to provide holistic nursing care. (shrink)

COnstantien Sandis speaks to Nassim Taleb about inductive knowledge,black swans, Hume, Popper, and Wittgenstein.

In this paper, I aim to defend Gadamer’s philosophical hermeneutics against what I call the radical hermeneutic critique, specifically the critique developed in Robert Bernasconi’s article “’You Don’t Know What I’m Talking About’: Alterity and the Hermeneutic Ideal” (1995). Key to this critique is the claim that Gadamer’s account does not rise to the ethical task of embracing the alterity of the Other, but instead reduces it to a projection of one’s self. The implication is therefore that Gadamer’s philosophical hermeneutics (...) has to be rejected on ethical grounds, as it does not appreciate but assimilates the alterity of the Other. In contrast to this radical hermeneutic critique, I argue that Gadamer’s philosophical hermeneutics can accommodate an Other that is not assimilated but appreciated, on the condition that the unique status of a dialogue with the Other as a person is no longer neglected, as Gadamer does, but instead truly distinguished from a dialogue with the Other as a text. (shrink)

History casts a spell on our minds more powerful than science or religion. It does not root us in the past at all. It rather flatters us with the belief in our ability to recreate the world in our image. It is a form of self-assertion that brooks no opposition or dissent and shelters us from the experience of time. So argues Constantin Fasolt in _The Limits of History_, an ambitious and pathbreaking study that conquers history's power by carrying (...) the fight into the center of its domain. Fasolt considers the work of Hermann Conring and Bartolus of Sassoferrato, two antipodes in early modern battles over the principles of European thought and action that ended with the triumph of historical consciousness. Proceeding according to the rules of normal historical analysis—gathering evidence, putting it in context, and analyzing its meaning—Fasolt uncovers limits that no kind of history can cross. He concludes that history is a ritual designed to maintain the modern faith in the autonomy of states and individuals. God wants it, the old crusaders would have said. The truth, Fasolt insists, only begins where that illusion ends. With its probing look at the ideological underpinnings of historical practice, _The Limits of History_ demonstrates that history presupposes highly political assumptions about free will, responsibility, and the relationship between the past and the present. A work of both intellectual history and historiography, it will prove invaluable to students of historical method, philosophy, political theory, and early modern European culture. (shrink)

At last available in paperback, this book anticipates and explains the post-structuralist turn to empiricism. Presenting a challenging reading of David Hume's philosophy, the work is invaluable for understanding the progress of Deleuze's thought.

First published in 2002. Routledge is an imprint of Taylor & Francis, an informa company.

Modern societies are characterized by a division of epistemic labor between laypeople and epistemic authorities. Authorities are often far more competent than laypeople and can thus, ideally, inform their beliefs. But how should laypeople rationally respond to an authority’s beliefs if they already have beliefs and reasons of their own concerning some subject matter? According to the standard view, the beliefs of epistemic authorities are just further, albeit weighty, pieces of evidence. In contrast, the Preemption View claims that, when one (...) discovers what an authority believes, it is not permissible to rely on any of one’s own reasons concerning the subject matter. The original version of this view, as proposed by Linda Zagzebski, has recently been severely criticized for recommending blind trust and for abandoning even minimal standards for critical thinking. In our paper, we defend a new version of the Preemption View—Defeatist Preemptionism—in a way that differs radically from Zagzebski’s. We argue that our view can be derived from certain widely accepted general epistemic principles. In particular, we claim that preemption can be identified as a special case of source sensitive defeat. Moreover, we argue that Defeatist Preemptionism does not lead to the undesirable consequences that critics ascribe to the Preemption View. The paper thus articulates the foundations and refinements of the Preemption View, such that it adequately captures the phenomenon of epistemic authority and the rational requirements related to it. (shrink)

Due to Gӧdel’s incompleteness results, the categoricity of a sufficiently rich mathematical theory and the semantic completeness of its underlying logic are two mutually exclusive ideals. For first- and second-order logics we obtain one of them with the cost of losing the other. In addition, in both these logics the rules of deduction for their quantifiers are non-categorical. In this paper I examine two recent arguments –Warren (2020), Murzi and Topey (2021)– for the idea that the natural deduction rules for (...) the first-order universal quantifier are categorical, i.e., they uniquely determine its semantic intended meaning. Both of them make use of McGee’s open-endedness requirement and the second one uses in addition Garson’s (2013) local models for defining the validity of these rules. I argue that the success of both these arguments is relative to their semantic or infinitary assumptions, which could be easily discharged if the introduction rule for the universal quantifier is taken to be an infinitary rule, i.e. non-compact. Consequently, I reconsider the use of the ω-rule and I show that the addition of the ω-rule to the standard formalizations of first-order logic is categorical. In addition, I argue that the open-endedness requirement does not make the first-order Peano Arithmetic categorical and I advance an argument for its categoricity based on the inferential conservativity requirement. (shrink)

Our technological lifeworld has become an info-computational media populated by data and algorithms, an artificial environment for life and shared experiences. In this chapter, I tried to sketch three new assumptions for bioethics – it is hardly possible to substantiate ethical guidelines or an idea of normativity in an aprioristic manner; moral status is a function of data entities, not something solely human; agency is plural and thus is shared or sometimes delegated – in order to chart a proposal for (...) a posthuman bioethics. Posthuman is perhaps not the best expression available, but it covers the idea of a shift from a world centered on self-contained and exclusively human agency to a more comprehensive and relational way of thinking. The “posthuman” label should be understood as a rebuttal of biocentrism and anthropocentrism by moving closer to conceptions we encounter in population ethics or in discourse about biosocial and technical systems. Posthuman bioethics is “environmentalist” without losing the humanistic stance. The question regarding how suitable an infocentric bioethics is in practice remains to be settled. The moral principles in bioethics could be reconceived as relying on these new assumptions, in a postindividualistic manner that accepts formal primacy of causal digital artifacts in affording actions in a world of ambient algorithmic intelligence. (shrink)

Accounts of human and animal action have been central to modern philosophy from Suarez and Hobbes in the sixteenth century to Wittgenstein and Anscombe in the mid-twentieth century via Locke, Hume, Kant, and Hegel, among many others. Philosophies of action have thus greatly influenced the course of both moral philosophy and the philosophy of mind. This book gathers together specialists from both the philosophy of action and the history of philosophy with the aim of re-assessing the wider philosophical impact of (...) action theory. It thereby explores how different notions of action, agency, reasons for action, motives, intention, purpose, and volition have affected modern philosophical understandings of topics as diverse as those of human nature, mental causation, responsibility, free will, moral motivation, rationality, normativity, choice and decision theory, criminal liability, weakness of will, and moral and social obligation. In so doing, it reinterprets the history of modern philosophy through the lens of action theory while also tracing the origins of contemporary questions in the philosophy of action back across half a millennium. This book was originally published as a special issue of Philosophical Explorations. (shrink)

Is knowledge-how, or “practical” knowledge, a species of knowledge-that, or “theoretical” knowledge? There is no comfortable position to take in the debate around this question. On the one hand, there are counterexamples against the anti-intellectualist thesis that practical knowledge is best analysed as an ability. They show that having an ability to ϕ is not necessary for knowing how to ϕ. On the other hand, the intellectualist analysis of practical knowledge as a subspecies of theoretical knowledge is threatened by its (...) own set of counterexamples, which convincingly establish that practical knowledge lacks many of the typical characteristics of theoretical knowledge. Most strikingly it does not even appear to require a belief. In this paper, I develop an account of practical knowledge that avoids these counterexamples. It also manages to preserve both the status of such knowledge as a cognitive achievement and its apparently close conceptual relation to abilities. I start with the counterexamples against the necessity of abilities for practical knowledge and show that they fail because they underestimate the cognitive demands of attempts. I then make use of the logic of dispositions to bridge the gap that counterexamples against the necessity of abilities for practical knowledge open. It is argued that, instead of the ability to ϕ, it is a specific disposition to have the ability to ϕ that constitutes practical knowledge about ϕ. The resulting theory is an anti-intellectualist position that preserves essential intellectualist motivations and thus should be satisfactory for proponents of both views. (shrink)

The two Heisenberg Uncertainties (UR) entail an incompatibility between the two pairs of conjugated variables E, t and p, q. But incompatibility comes in two kinds, exclusive of one another. There is incompatibility defineable as: (p → -q) & (q → -p) or defineable as [(p → -q) & (q → -p)] ↔ r. The former kind is unconditional, the latter conditional. The former, in accordance, is fact independent, and thus a matter of logic, the latter fact dependent, and thus (...) a matter of fact. The two types are therefore diametrically opposed. In spite of this, however, the existing derivations of the Uncertainties are shown here to entail both types of incompatibility simultaneously. ΔEΔt ≥ h is known to derive from the quantum relation E = hv plus the Fourier relation ΔvΔt ≥ 1. And the Fourier relation assigns a logical incompatibility between Δv = 0, Δt = 0. (Defining a repetitive phenomenon at an instant t → 0 is a self contradictory notion.) An incompatibility, therefore, which is fact independent and unconditional. How can one reconcile this with the fact that ΔEΔt exists if and only if h > 0, which latter supposition is a factual truth, entailing that a ΔE = 0, Δt = 0 incompatibility should itself be fact dependent? Are we to say that E and t are unconditionally incompatible (via ΔvΔt ≥ 1) on condition that E = hv is at all true? Hence, as presently standing, the UR express a self-contradicting type of incompatibility. To circumvent this undesirable result, I reinterpret E = hv as relating the energy with a period. Though only one such period. And not with frequency literally. (It is false that E = v. It is true that E = v times the quantum.) In this way, the literal concept of frequency does not enter as before, rendering ΔvΔt ≥ 1 inapplicable. So the above noted contradiction disappears. Nevertheless, the Uncertainties are derived. If energy is only to be defined over a period, momentum only over a distance (formerly a wavelength) resulting during such period, thus yielding quantized action of dimensions Et = pq, then energies will become indefinite at instants, momenta indefinite at points, leading, as demanded, to (symmetric!) ΔEΔt = ΔpΔq ≥ h's. (shrink)

The two Heisenberg Uncertainties (UR) entail an incompatibility between the two pairs of conjugated variables E, t and p, q. But incompatibility comes in two kinds, exclusive of one another. There is incompatibility defineable as: (p → − q) & (q→ − p) or defineable as [(p →− q) & (q →− p)] ↔ r. The former kind is unconditional, the latter conditional. The former, in accordance, is fact independent, and thus a matter of logic, the latter fact dependent, and (...) thus a matter of fact. The two types are therefore diametrically opposed. In spite of this, however, the existing derivations of the Uncertainties are shown here to entail both types of incompatibility simultaneously. Δ E Δ t ≥ h is known to derive from the quantum relation E = hν plus the Fourier relation Δ ν Δ t ≥ 1. And the Fourier relation assigns a logical incompatibility between Δ ν = 0, Δ t = 0. (Defining a repetitive phenomenon at an instant t → 0 is a self contradictory notion.) An incompatibility, therefore, which is fact independent and unconditional. How can one reconcile this with the fact that Δ EΔ t exists if and only if h > 0, which latter supposition is a factual truth, entailing that a Δ E = 0, Δ t = 0 incompatibility should itself be fact dependent? Are we to say that E and t are unconditionally incompatible (via Δ ν Δ t ≥ 1) on condition that E = hν is at all true? Hence, as presently standing, the UR express a self-contradicting type of incompatibility. To circumvent this undesirable result, I reinterpret E = hν as relating the energy with a period. Though only one such period. And not with frequency literally. (It is false that E = ν. It is true that E = ν times the quantum.) In this way, the literal concept of frequency does not enter as before, rendering Δ ν Δ t ≥ 1 inapplicable. So the above noted contradiction disappears. Nevertheless, the Uncertainties are derived. If energy is only to be defined over a period, momentum only over a distance (formerly a wavelength) resulting during such period, thus yielding quantized action of dimensions Et = pq, then energies will become indefinite at instants, momenta indefinite at points, leading, as demanded, to (symmetric!) Δ E Δ t = Δ p Δ q ≥ h’s. (shrink)

The classical response to Zeno’s paradoxes goes like this: ‘Motion cannot properly be defined within an instant. Only over a period’ (Vlastos.) I show that this ob-jection is exactly what it takes for Zeno to be right. If motion cannot be defined at an instant, even though the object is always moving at that instant, motion cannot be defined at all, for any longer period of time identical in content to that instant. The nonclassical response introduces discontinuity, to evade the (...) paradox of infinite proximity of any point of a distance with any ‘next’. But it introduces the wrong sort of discontinuity because, rather than assuming the discontinuity of motion, as Quantum Theory does, it assumes the discontinuity of space. Due then to the resulting spacetime disorder, though all else is certainly lost, the Tortoise now turns up at least as fast as Achilles and hence not even this much is rescued. Zeno rejects motion because he shows that a moving object must be where it is not. Hence motion, if to occur, must violate the Law of Contradiction (LNC). Applying the concept of quantum discontinuity, I produce an alternative. If an object is to move discontinuously between two boundary points, A and B, what actually obtains is, rather, that it is nowhere at all in-between A and B. And cannot therefore be at two places in-between A and B. And cannot therefore be where it is not. Thus, LNC is conserved. However, in these conditions, the Law of the Excluded Middle (LEM) fails. To mitigate the undesirability of this effect, I show that LEM fails because LNC holds. Thus, the resulting nonbivalent logic, which is also appropriate for quantized transitions of all kinds, will always turn up nonbivalent, because consistent. And this is not too bad, considering. (shrink)

The classical response to Zeno’s paradoxes goes like this: ‘Motion cannot properly be defined within an instant. Only over a period’ (Vlastos.) I show that this ob-jection is exactly what it takes for Zeno to be right. If motion cannot be defined at an instant, even though the object is always moving at that instant, motion cannot be defined at all, for any longer period of time identical in content to that instant. The nonclassical response introduces discontinuity, to evade the (...) paradox of infinite proximity of any point of a distance with any ‘next’. But it introduces the wrong sort of discontinuity because, rather than assuming the discontinuity of motion, as Quantum Theory does, it assumes the discontinuity of space. Due then to the resulting spacetime disorder, though all else is certainly lost, the Tortoise now turns up at least as fast as Achilles and hence not even this much is rescued. Zeno rejects motion because he shows that a moving object must be where it is not. Hence motion, if to occur, must violate the Law of Contradiction (LNC). Applying the concept of quantum discontinuity, I produce an alternative. If an object is to move discontinuously between two boundary points, A and B, what actually obtains is, rather, that it is nowhere at all in-between A and B. And cannot therefore be at two places in-between A and B. And cannot therefore be where it is not. Thus, LNC is conserved. However, in these conditions, the Law of the Excluded Middle (LEM) fails. To mitigate the undesirability of this effect, I show that LEM fails because LNC holds. Thus, the resulting nonbivalent logic, which is also appropriate for quantized transitions of all kinds, will always turn up nonbivalent, because consistent. And this is not too bad, considering. (shrink)

History casts a spell on our minds more powerful than science or religion. It does not root us in the past at all. It rather flatters us with the belief in our ability to recreate the world in our image. It is a form of self-assertion that brooks no opposition or dissent and shelters us from the experience of time. So argues Constantin Fasolt in The Limits of History , an ambitious and pathbreaking study that conquers history's power by (...) carrying the fight into the center of its domain. Fasolt considers the work of Hermann Conring (1606-81) and Bartolus of Sassoferrato (1313/14-57), two antipodes in early modern battles over the principles of European thought and action that ended with the triumph of historical consciousness. Proceeding according to the rules of normal historical analysis--gathering evidence, putting it in context, and analyzing its meaning--Fasolt uncovers limits that no kind of history can cross. He concludes that history is a ritual designed to maintain the modern faith in the autonomy of states and individuals. God wants it, the old crusaders would have said. The truth, Fasolt insists, only begins where that illusion ends. With its probing look at the ideological underpinnings of historical practice, The Limits of History demonstrates that history presupposes highly political assumptions about free will, responsibility, and the relationship between the past and the present. A work of both intellectual history and historiography, it will prove invaluable to students of historical method, philosophy, political theory, and early modern European culture. (shrink)

This paper documents a conversation between a philosopher and a human computer interaction researcher whose research has been enormously influenced by Wittgenstein. In particular, the in vivo use of categories in the design of communications and AI technologies are discussed, and how this meaning needs to evolve to allow creative design to flourish. The paper will be of interest to anyone concerned with philosophical tools in everyday action.

The deliberative conception of politics seems to be necessary for the legitimation of state power through democratic will‐formation and decision‐making. However, the author maintains that a complex theory of democracy cannot merely consist in procedural prerequisites for organizing the concomitant institutional settings. In particular, such a theory must comprise some substantive presuppositions, such as social and economic rights, in order to diminish existing material inequalities, especially those connected with social exploitation and domination. The author argues that a contemporary theory of (...) democracy should reflect on the autonomization of mechanisms of egoistic action challenging not only the democratic political order, but also the very reproduction preconditions of societies all over the world. In this perspective, the model of associative democracy, which is suggested nowadays as a sort of substantive correlative to the institutional proceduralism, could not significantly rejuvenate the traditional representative democracy. Instead, democracy could only be given a fresh impulse if democratic deliberation penetrates the currently forbidden field of capitalist production and social exploitation, the locus where social inequality and effective unfreedom are endlessly reproduced. (shrink)

We establish the existence uncountably many atoms in the subvariety lattice of the variety of involutive residuated lattices. The proof utilizes a construction used in the proof of the corresponding result for residuated lattices and is based on the fact that every residuated lattice with greatest element can be associated in a canonical way with an involutive residuated lattice.

In a unique collection of essays and poems written in English, French, German and Russian (with a seasoning of Greek and Japanese), Constantine Shapiro explores fascinating universal questions and connections. Why, he asks, do so many people feel unhappy? Why do so many go to the doctor? Why is there so much criminality? Why is there international unrest? What is a healthy soul? In his investigation of these and other "Big Questions," the author examines such disparate areas as Japanese kanji (...) ideograms and personal character, spiritual ontology and science, Zeus and Christianity, objectivity and subjectivity in time and space, the nature of experience, and the flow of consciousness. The modern reader will find much to ponder in this second, expanded edition of Selected Writings. Also included are facsimiles of the typescript of Two Mathematical Problems and the Japanese translation of Transzendentale Musiklehre. (shrink)

The question whether quantum discontinuity can or cannot provide an answer to Zeno’s Paradoxes is reopened. It is observed that what is usually understood by the term “discontinuity”, namely, Einstein’s conception of the photon as described by himself and all others, is unsuitable to the task because, essentially,it reduces to the trivial ‘discontinuity’ of objects scattered in space. By contrast, quantization of energy levels, which are not in space but can only alternate in time, provide the right sort of discontinuity (...) required. Discrete quantized orbits, corresponding to eigen-frequencies, are irreducible, and nothing is allowed to stand in-between them in satisfaction of the quantum postulate, furnishing the requisite, and so far missing, immediate nextness of a point to a certain other. In this way, Zeno’s Runner need not postpone his first step indefinitely, always waiting upon an infinity of preceding steps, before it can be taken. There is now a point that is next to a point and so a step on that point, which is the first step. It follows that, if one kind of discontinuity, Einstein’s, is incapable of offerring an answer to Zeno, while another kind can, the two are discrepant. One of them, the former, is not a kind of discontinuity properly so called at all, though evidently the consequence of one. (shrink)

The question whether quantum discontinuity can or cannot provide an answer to Zeno’s Paradoxes is reopened. It is observed that what is usually understood by the term “discontinuity”, namely, Einstein’s conception of the photon as described by himself and all others, is unsuitable to the task because, essentially,it reduces to the trivial ‘discontinuity’ of objects scattered in space. By contrast, quantization of energy levels, which are not in space but can only alternate in time, provide the right sort of discontinuity (...) required. Discrete quantized orbits, corresponding to eigen-frequencies, are irreducible, and nothing is allowed to stand in-between them in satisfaction of the quantum postulate, furnishing the requisite, and so far missing, immediate nextness of a point to a certain other. In this way, Zeno’s Runner need not postpone his first step indefinitely, always waiting upon an infinity of preceding steps, before it can be taken. There is now a point that is next to a point and so a step on that point, which is the first step. It follows that, if one kind of discontinuity, Einstein’s, is incapable of offerring an answer to Zeno, while another kind can, the two are discrepant. One of them, the former, is not a kind of discontinuity properly so called at all, though evidently the consequence of one. (shrink)