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)

"In the gorgeous and rugged terrain of Jewish thought, there is no higher mountain to climb than Maimonides, and no more slippery or exhilarating ascent. Halbertal has made it all the way to the top, and his survey of the whole of the Maimonidean landscape is trustworthy and masterful. This is the richest and most intellectually sophisticated book on Maimonides I have ever read."--Leon Wieseltier "In this learned and penetrating work, Halbertal offers us a Maimonides who draws on the dominant (...) Greco-Islamic thought of his time while creating a system of thought that is fully Jewish. He shows us how the early "Commentary on the Mishnah" links up with the "Mishneh Torah" and with the "Guide of the Perplexed," written at the end of his life, to form an unexpected and radical intellectual unity. Beautifully written, "Maimonides" brings out both Maimonides's intellectual success and the paradoxical critical approaches to him after his death."--David J. Wasserstein, Vanderbilt University "Insightful and learned. Halbertal is perhaps the leading philosopher of Jewish law today. His book on Maimonides, like his other writings, reflects wide erudition and is written clearly and sharply."--Warren Zev Harvey, professor emeritus, Hebrew University of Jerusalem "Displaying the marvelous depth and clarity that mark all his work, Halbertal explains in abundant detail the transformations that Maimonides sought to effect in the Jewish world. He provides incisive interpretations of both legal and philosophical writings, yet he is also a biographer, binding together Maimonides's life, self-perception, and intellectual agenda. This is an exceptionally rich book, one that offers fresh perspectives for experts and a highly accessible introduction for general readers."--David Shatz, Yeshiva University "An outstanding and thrilling portrait of Maimonides. Halbertal's analytic lucidity and psychological depth are singular, and his talents are abundantly apparent on every page. This is an extraordinary book."--Menachem Lorberbaum, Tel Aviv University. (shrink)

In the latter part of the fifteenth century, a Jewish translator, working together with a Slavic amanuensis, translated into the East Slavic language of the Grand Duchy of Lithuania three medieval Hebrew translations of Arabic philosophical texts: the Logical Terminology, a short work on logic attributed to Maimonides (but probably by a different medieval Jewish author); and two sections of the Muslim theologian Al-Ghazali's famous Intentions of the Philosophers. Highlighting the unexpected role played by Jewish translators as agents of cultural (...) transmission in the heady messianic atmosphere leading up to the year 1492, these texts drew the attention of the Orthodox Church authorities as being in the possession of the enlightened heretical sect known as the Judaizers, which had emerged in Novgorod and spread to Moscow. Reflecting three and even four layers of translation, Prof. Moshe Taube's new triple-language critical edition of the Logika of the Judaizers displays the Slavic texts alongside the Hebrew translations on which they are based and accompanies them with a modern English translation. In his comprehensive introduction and commentary, Taube surveys earlier scholarly efforts to identify the provenance and purport of the translations, discusses the linguistic and textological issues raised by these early Ruthenian texts, puts forward the likely dissimilar motivations of the Jewish translator and the Christians who commissioned the work, and reveals the translator's probable identity. The present publication, a long-awaited desideratum, will be of interest not only to historians of the Great Duchy of Lithuania and of the principality of Muscovy, but also to scholars of Jewish history and of the history of philosophy and science, as well as to linguists studying the history of the Belorussian, Ukrainian and Russian languages. (shrink)

This volume of essays examines key themes in Jewish philosophy and ethics from the rigorous perspective of philosophical analysis. The first set of essays takes up the challenge of living a Jewish life, and includes essays on pleasure, joy, human suffering, Jewish ritual practice and the philosophical life. The second set of essays analyzes the value and meaning of autonomy, human freedom and tolerance in Jewish thought, crucial themes in western political thought and life. Other essays in the volume examine (...) the many meanings of Jewish texts, and such crucial issues in applied Jewish ethics as ecology, medical ethics, and justified homicide. Finally, a number of essays plumb the depths of one of the most influential and creative Jewish thinkers of the twentieth century, Rabbi Joseph B. Soloveitchik. Taken as a whole, this volume advances the engagement of classical Jewish themes with Anglo-American philosophy, shedding new light both on the Jewish tradition, and on the western philosophical enterprise. (shrink)

In Judaism Straight Up, Moshe Koppel explores the central differences between traditional societies--including traditional Judaism--and contemporary cosmopolitan ones. He explains everything you always wanted to know about the subtleties of Jewish morality, tradition, and belief, and how these have unfolded to beat cosmopolitanism at its own game: advancing cooperation, fairness, and freedom. Written with incisiveness and droll wit--and a scientific sensibility that draws on economics, game theory, and other disciplines--Judaism Straight Up reveals the secret of Jewish traditionalism's endurance."--Page [4] of (...) cover. (shrink)