This is an introduction to set theory and logic that starts completely from scratch. The text is accompanied by many methodological remarks and explanations.

The hope of this work is that these notes will help to convince more mathematicians that non-standard analysis can make a useful addition to the tools of their trade.

We address simple voting games as mathematical objects in their own right, and study structures made up of these objects, rather than focusing on SVGs primarily as co-operative games. To this end it is convenient to employ the conceptual framework and language of category theory. This enables us to uncover the underlying unity of the basic operations involving SVGs.

I propose a twofold classification of the main considerations underlying the choice of an election procedure: political criteria on the one hand, and social-choice criteria on the other. I formulate political dichotomies, each combination of which narrows down the choice of procedure to a sub-class of the class of all procedures. I discuss what social-choice theory has to offer in each of these.

LS Penrose was the first to propose a measure of voting power (which later came to be known as ‘the [absolute] Banzhaf index’). His limit theorem – which is implicit in Penrose (1952) and for which he gave no rigorous proof – says that, in simple weighted voting games, if the number of voters increases indefinitely while the quota is pegged at half the total weight, then – under certain conditions – the ratio between the voting powers (as measured by (...) him) of any two voters converges to the ratio between their weights. We conjecture that the theorem holds, under rather general conditions, for large classes of variously defined weighted voting games, other values of the quota, and other measures of voting power. We provide proofs for some special cases. (shrink)

In this account, we explain the meaning of a priori voting power and outline how it is measured. We distinguish two intuitive notions as to what voting power means, leading to two approaches to measuring it. We discuss some philosophical and pragmatic objections, according to which a priori (as distinct from actual) voting power is worthless or inapplicable.

If K is an index of relative voting power for simple voting games, the bicameral postulate requires that the distribution of K -power within a voting assembly, as measured by the ratios of the powers of the voters, be independent of whether the assembly is viewed as a separate legislature or as one chamber of a bicameral system, provided that there are no voters common to both chambers. We argue that a reasonable index – if it is to be used (...) as a tool for analysing abstract, ‘uninhabited’ decision rules – should satisfy this postulate. We show that, among known indices, only the Banzhaf measure does so. Moreover, the Shapley–Shubik, Deegan–Packel and Johnston indices sometimes witness a reversal under these circumstances, with voter x ‘less powerful’ than y when measured in the simple voting game G1 , but ‘more powerful’ than y when G1 is ‘bicamerally joined’ with a second chamber G2 . Thus these three indices violate a weaker, and correspondingly more compelling, form of the bicameral postulate. It is also shown that these indices are not always co-monotonic with the Banzhaf index and that as a result they infringe another intuitively plausible condition – the price monotonicity condition. We discuss implications of these findings, in light of recent work showing that only the Shapley–Shubik index, among known measures, satisfies another compelling principle known as the bloc postulate. We also propose a distinction between two separate aspects of voting power: power as share in a fixed purse (P-power) and power as influence (I-power). (shrink)

We consider a singular event of the following form: in a simple voting game, a particular division of the voters resulted in a positive outcome. We propose a plausible measure that quantifies the causal contribution of any given voter to the outcome. This measure is based on a conceptual analysis due to Braham [1], but differs from his solution to the problem of measuring causality of singular events.

In the voting-power literature the rules of decision of the US Congress and the UN Security Council are widely misreported as though abstention amounts to a `no' vote. The hypothesis (proposed elsewhere) that this is due to a specific cause, theory-laden observation, is tested here by examining accounts of these rules in introductory textbooks on American Government and International Relations, where that putative cause does not apply. Our examination does not lead to a conclusive outcome regarding the hypothesis, but reveals (...) that the rules in question are also widely misreported in these textbooks. A second hypothesis---that the widespread misreporting is explicable by the relative rarity and unimportance of abstention in the two bodies concerned---is also tested and found to be untenable. (shrink)

We analyse and evaluate the qualified majority (QM) decision rules for the Council of Ministers of the EU that are included in the Draft Constitution for Europe proposed by the European Convention [5]. We use a method similar to the one we used in [9] for the QM prescriptions made in the Treaty of Nice.

We analyse and evaluate the qualified majority (QM) decision rule for the Council of Ministers of the EU adopted at the EU Inter-Governmental Conference, Brussels, 18 June 2004 [1]. We compare this rule with the QM rule prescribed in the Treaty of Nice, and the rule included in the original draft Constitution proposed by the European Convention in July 2003. We use a method similar to the one we used in [3] and [4].

We analyse and assess the qualified majority (QM) decision rule for the Council of Ministers of the EU, adopted at the Council of the European Union, Brussels, 23 June 2007. This rule is essentially the same as that adopted at the Inter-Governmental Conference, Brussels, 18 June 2004. We compare this rule with the QM rule prescribed in the Treaty of Nice, and the scientifically-based rule known as the ‘Jagelonian Compromise’.

L S Penrose’s Limit Theorem – which is implicit in Penrose [7, p. 72] and for which he gave no rigorous proof – says that, in simple weighted voting games, if the number of voters increases indefinitely and the relative quota is pegged, then – under certain conditions – the ratio between the voting powers of any two voters converges to the ratio between their weights. Lindner and Machover [4] prove some special cases of Penrose’s Limit Theorem. They give (...) a simple counter-example showing that the theorem does not hold in general even under the conditions assumed by Penrose; but they conjecture, in effect, that under rather general conditions it holds ‘almost always’ – that is with probability 1 – for large classes of weighted voting games, for various values of the quota, and with respect to several measures of voting power. We use simulation to test this conjecture. It is corroborated with respect to the Penrose–Banzhaf index for a quota of 50% but not for other values; with respect to the Shapley–Shubik index the conjecture is corroborated for all values of the quota. (shrink)

The Mesillas Yesharim / Path of the Just was the masterpiece of Rabbi Moshe Chaim Luzzatto, the great mystic, philosopher, sage, and saint. For centuries, this classic text for better living has been every man's primer for ideal life. Wherever there were Jews there was a well-thumbed Mesillas Yesharim. In this book, Rabbi Twerski applies the text and themes of Mesillas Yesharim to the everyday challenges of the 90s. He shows us how we can succeed in the quest for a (...) better life and in bettering ourselves. (shrink)

My question in the chapter is this: could (and should) the role of the physician be construed as that of a friend to the patient? I begin by briefly discussing the “friendship model” of the physician-patient relationship—according to which physicians and patients could, and perhaps should, be friends—as well as its history and limitations. Given these limitations, I focus on the more one-sided idea that the physician could, and perhaps should, be a friend to the patient (a “physician-qua-friend model” of (...) the physician-patient relationship). I show that given recent developments in our understanding of the physician-patient relationship, this idea is far from asinine. I then make the case that the most plausible conception of the physician-qua-friend model incorporates the following components: (a) a common goal, that is, one that physician and patient share; (b) certain forms of equality between physician and patient; (c) an ideal of a caring physician. Finally, I show how the model can be instantiated in a certain type of physician-patient interaction, namely, in physician-assisted dying. (shrink)

Annotation. In this volume, the third in his classic series of texts surveying the history of art theory, Moshe Barasch traces the hidden patterns and interlocking themes in the study of art, from Impressionism to Abstract Art. Barasch details the immense social changes in the creation, presentation, and reception of art which have set the history of art theory on a vertiginous new course: the decreased relevance of workshops and art schools; the replacement of the treatise by the critical review; (...) and the interrelation of new modes of scientific inquiry with artistic theory and praxis. The consequent changes in the ways in which critics as well as artists conceptualized paintings and sculptures were radical, marked by an obsession with intense, immediate sensory experiences, psychological reflection on the effects of art, and a magnetic pull to the exotic and alien, making for the most exciting and fertile period in the history of art criticism. (shrink)

Answers to the questions of what justifies conscientious objection in medicine in general and which specific objections should be respected have proven to be elusive. In this paper, I develop a new framework for conscientious objection in medicine that is based on the idea that conscience can express true moral claims. I draw on one of the historical roots, found in Adam Smith’s impartial spectator account, of the idea that an agent’s conscience can determine the correct moral norms, even if (...) the agent’s society has endorsed different norms. In particular, I argue that when a medical professional is reasoning from the standpoint of an impartial spectator, his or her claims of conscience are true, or at least approximate moral truth to the greatest degree possible for creatures like us, and should thus be respected. In addition to providing a justification for conscientious objection in medicine by appealing to the potential truth of the objection, the account advances the debate regarding the integrity and toleration justifications for conscientious objection, since the standard of the impartial spectator specifies the boundaries of legitimate appeals to moral integrity and toleration. The impartial spectator also provides a standpoint of shared deliberation and public reasons, from which a conscientious objector can make their case in terms that other people who adopt this standpoint can and should accept, thus offering a standard fitting to liberal democracies. (shrink)

Physicians frequently ask whether they should give patients what they want, usually when there are considerations pointing against doing so, such as medicine’s values and physicians’ obligations. It has been argued that the source of medicine’s values and physicians’ obligations lies in what has been dubbed “the internal morality of medicine”: medicine is a practice with an end and norms that are definitive of this practice and that determine what physicians ought to do qua physicians. In this paper, I defend (...) the claim that medicine requires a morality that is internal to its practice, while rejecting the prevalent characterization of this morality and offering an alternative one. My approach to the internal morality of medicine is constructivist in nature: the norms of medicine are constructed by medical professionals, other professionals, and patients, given medicine’s end of “benefitting patients in need of prima facie medical treatment and care.” I make the case that patients should be involved in the construction of medicine’s morality not only because they have knowledge that is relevant to the internal morality of medicine—namely, their own values and preferences—but also because medicine is an inherently relational enterprise: in medicine the relationship between physician and patient is a constitutive component of the craft itself. The framework I propose provides an authoritative morality for medicine, while allowing for the incorporation, into that very morality, of qualified deference to patient values. (shrink)

Two problems seem to plague Adam Smith’s account of sympathy and approbation in The Theory of Moral Sentiments (TMS). First, Smith’s account of sympathy at the beginning of TMS appears to be inconsistent with the account of sympathy at the end of TMS. In particular, it seems that Smith did not appreciate the distinction between ‘self-oriented sympathy’ and ‘other-oriented sympathy’, that is, between imagining being oneself in the actor’s situation and imagining being the actor in the actor’s situation. Second, Smith’s (...) account of approbation, according to which a sentiment of approval arises when there is recognition of concordance between the spectator’s sympathetic passion and the actor’s original passion, seems to face the following problem: since the spectator attains both his own sympathetic passion and the actor’s original passion by sympathizing with the actor, the sympathetic passion of the spectator and the original passion of the actor will necessarily be identical. Therefore, Smith’s account of approbation requires that the spectator utilize both self-oriented and other-oriented sympathy (‘the double-sympathy model of approbation’). I offer a novel developmental account of sympathy in TMS that renders Smith’s account of sympathy consistent and allows for the utilization of the double-sympathy model of approbation. (shrink)

This book describes Carmeli's cosmological general and special relativity theory, along with Einstein's general and special relativity. These theories are discussed in the context of Moshe Carmeli's original research, in which velocity is introduced as an additional independent dimension. Four- and five-dimensional spaces are considered, and the five-dimensional braneworld theory is presented. The Tully-Fisher law is obtained directly from the theory, and thus it is found that there is no necessity to assume the existence of dark matter in the halo (...) of galaxies, nor in galaxy clusters.The book gives the derivation of the Lorentz transformation, which is used in both Einstein's special relativity and Carmeli's cosmological special relativity theory. The text also provides the mathematical theory of curved space?time geometry, which is necessary to describe both Einstein's general relativity and Carmeli's cosmological general relativity. A comparison between the dynamical and kinematic aspects of the expansion of the universe is made. Comparison is also made between the Friedmann-Robertson-Walker theory and the Carmeli theory. And neither is it necessary to assume the existence of dark matter to correctly describe the expansion of the cosmos. (shrink)

The literature on conscientious objection in medicine presents two key problems that remain unresolved: Which conscientious objections in medicine are justified, if it is not feasible for individual medical practitioners to conclusively demonstrate the genuineness or reasonableness of their objections? How does one respect both medical practitioners’ claims of conscience and patients’ interests, without leaving practitioners complicit in perceived or actual wrongdoing? My aim in this paper is to offer a new framework for conscientious objections in medicine, which, by bringing (...) medical professionals’ conscientious objection into the public realm, solves the justification and complicity problems. In particular, I will argue that: an “Uber Conscientious Objection in Medicine Committee” —which includes representatives from the medical community and from other professions, as well as from various religions and from the patient population—should assess various well-known conscientious objections in medicine in terms of public reason and decide which conscientious objections should be permitted, without hearing out individual conscientious objectors; medical practitioners should advertise their conscientious objections, ahead of time, in an online database that would be easily accessible to the public, without being required, in most cases, to refer patients to non-objecting practitioners. (shrink)

I offer a novel two-stage reconstruction of Hume’s general-point-of-view account, modeled in part on his qualified-judges account in ‘Of the Standard of Taste.’ In particular, I argue that the general point of view needs to be jointly constructed by spectators who have sympathized with (at least some of) the agents in (at least some of) the actor’s circles of influence. The upshot of the account is two-fold. First, Hume’s later thought developed in such a way that it can rectify the (...) problems inherent in his Treatise account of the general point of view. Second, the proposed account provides the grounds for an adequate and well-motivated modest ideal observer theory of the standard of virtue. (shrink)