Since the time, about a century ago, when DeMorgan, Boole and Jevons, inaugurated the study of the logic of numerically definite reasoning, no one has been concerned to establish the validity rules for a very general type of numerically definite inference which is a strong analogue of the classical syllogism. The reader will readily agree that the traditional rules of syllogistic inference cannot even begin to decide whether the following proportionally quantified syllogism is a valid argument: at most 4/7 p (...) are at most 2/3 not-m and at least 3/5 s are precisely 7/8 m, hence, at most 1/3 s are p. Likely reasons, of broad methodological interest, why this readily definable type of formal inference has been neglected will presently be proposed. Among the masters of twentieth century logic, Hilbert and Bernays have, to be sure, meticulously shown how such numerically definite propositions as “at least 7 individuals have the property p” can be expressed in the predicate calculus with the use of the identity relation. They do not, since their concern with such propositions is indirect, proceed to establish a systematic calculus of classes whose “size” or extent is numerically definite. As we shall always notice, traditional logic and even ordinary Boolean algebra prefer to generalize about the laws governing the interrelationship of classes in inference systems without making any hypotheses concerning class “size” or extent except by the classical quantifier “some”, which quantifier admits the measure or extent of a class only in terms of the proposition that a given class contains or does not contain at least one member. Such a restricted admission of the relevance for inference of class size is among the fundamental reasons for the neglect of proportional quantifiers, i.e., quantifiers where numerical value is any rational number r such that 0/100 ≤ r ≤ 100/100. It can, of course, also be pointed out that the traditional syllogism and ordinary Boolean algebra do not i.e. measure, numerically define, the degree of inclusion of one class in another. Inclusion is taken either as total or partial, with no discrimination of the degree of partial inclusion. (shrink)

“Just as philosophy begins with doubt, so also a life that may be called human begins with irony” (Kierkegaard The Concept of Irony with Continual Reference to Socrates, pg.6, thesis XV) What has proportion to do with humor or irony? And what do either of these have to do with being human? Jokes, laughter, and funniness connote excess, exaggeration, incongruity, dissonance, etc., the opposite of proportion--balance, symmetry, Aristotle’s golden mean. Yet, The Philosopher maintains, the wit has found the ideal moderate (...) position between the deficient boor who seems constitutionally incapable of laughing, and the excessive buffoon who doesn’t know how to laugh at the right time, in the right way, or for the right reasons--buffoonish laughter is indiscriminate. The title, paraphrasing Hume, is consistent with Kierkegaard’s valorization of irony, as philosophical doubt and the confusion induced through Socratic Irony both call for a special sort of proportion: ontological proportion. Put simply, one ought to proportion one’s being, as well as others’, as though along a shifting spectrum rather than essentially and hierarchically. The latter fosters a serious attitude that takes polarities as the rule with no exceptions; black or white, good or evil, with us or against us, one of us or one of them, always seeking the ideal of purity. We see this disproportion in authoritarians who manifestly lack a sense of humor--Hitler’s joke courts, Turkey’s president Erdoğan attempting legal punishment against a comedian who mocked him, and Trump’s thin-skinned inability to laugh at his own expense, a monumental character flaw for the most powerful person in the world. Their views are serious, rigid, closed, and dogmatic about themselves, others, and for each of them excepting Trump, the presumed absolute nature of values--Trump’s unironic pretense toward any value is merely a means to service his own absolute end, himself. The playfully ironic person can cope with reality, with ambiguity and, borrowing from Maria Lugones, “ontological confusion,” the not-fully-settled identity. (shrink)

For various domains in proportional reasoning cognitive development is characterized as a progression through a series of increasingly complex rules. A multiplicative relationship between two task features, such as weight and distance information of blocks placed at both sides of the fulcrum of a balance scale, appears difficult to discover. During development, children change their beliefs about the balance scale several times: from a focus on the weight dimension (Rule I) to occasionally considering the distance dimension (Rule (...) II), guessing (Rule III), and applying multiplication (Rule IV; Siegler, 1981). Because of the detailed empirical findings the balance scale task has become a benchmark task for computational models of proportional reasoning. In this article, we present a large empirical study (N = 420) of which the findings provide a challenge for computational models. The effect of feedback and the effect of individually adapted training items on rule transition were tested for children using Rule I or Rule II. Presenting adapted training items initiates belief revision for Rule I but not for Rule II. The experience of making mistakes (by providing feedback) induces a change for both Rule I and Rule II. However, a delayed posttest shows that these changes are preserved after 2 weeks only for children using Rule I. We conclude that the transition from Rule I to Rule II differs from the transition from Rule II to a more complex rule. Concerning these empirical findings, we will review performance of computational models and the implications for a future belief revision model. It is one Thing, to show a Man that he is in an Error, and another, to put him in possession of Truth. John Locke. (shrink)

Sixty years of democratic representation in Germany allow us to study the working of a specific type of electoral system, namely a mixed system combining proportional and majoritarian rules, in great detail.

We consider a decision board with representatives who vote on proposals on behalf of their constituencies. We look for decision rules that realize utilitarian and (welfarist) egalitarian ideals. We set up a simple model and obtain roughly the following results. If the interests of people from the same constituency are uncorrelated, then a weighted rule with square root weights does best in terms of both ideals. If there are perfect correlations, then the utilitarian ideal requires proportional weights, whereas (...) the egalitarian ideal requires equal weights. We investigate correlations that are in between these extremes and provide analytic arguments to connect our results to Barberà and Jackson (J Polit Econ 114(2):317–339, 2006) and to Banzhaf voting power. (shrink)

How rich should the 1% be? And, most importantly, when does the distance in economic resources between the richest citizens and ‘us’, the average citizenry, become a concern for justice? This volume explores how excessive economic inequality gives the best-off considerably more political influence than average citizens, thereby violating political equality. It argues that the gap between the best-off and the worst-off should not be reduced because it is good, but rather as an inescapable instrument to protect citizens from the (...) risk of material domination. For this reason, it defends the ‘principle of proportionality’: economic inequality should not exceed a certain range or proportion to enable both the best-off and the worst-off to be co-authors of the legal, political, and socioeconomic rules that govern the ‘social’ relations in which they are involved. Further, the book discusses material domination and explains how money influences politics and what are the remedies for this phenomenon; how social justice should face and harmonise power, poverty, efficiency, individual merit, and economic liberties; and, most importantly, how to determine income and wealth limit ratios in a liberal democracy. A thoughtful investigation on the interdependencies of money and justice and their influence our socio-political systems, this volume will be of great interest to students and researchers of political theory, political philosophy, economics and development, economics theory and philosophy, and social policy. (shrink)

We study how to partition a set of agents in a stable way when each coalition in the partition has to share a unit of a perfectly divisible good, and each agent has symmetric single-peaked preferences on the unit interval of his potential shares. A rule on the set of preference profiles consists of a partition function and a solution. Given a preference profile, a partition is selected and as many units of the good as the number of coalitions (...) in the partition are allocated, where each unit is shared among all agents belonging to the same coalition according to the solution. A rule is stable at a preference profile if no agent strictly prefers to leave his coalition to join another coalition and all members of the receiving coalition want to admit him. We show that the proportional solution and all sequential dictator solutions admit stable partition functions. We also show that stability is a strong requirement that becomes easily incompatible with other desirable properties like efficiency, strategy-proofness, anonymity, and non-envyness. (shrink)

Among the most complicated issues in the law of delict, in the case of multiple debtors, is to determine the scope of each co-debtor’s liability. The rule of proportional liability clearly favours debtors more than the aggrieved party. And, on the contrary, the solidary liability best suits the interests of the aggrieved party as the risk of co-debtor’s insolvency is transferred to the debtors. Furthermore, in the latter case, the debtors who attempt to allocate the scope of their (...) liability among themselves and fail to reach an agreement have to undergo two legal procedures: the first concerning the compensation to the creditor, and the second concerning the determination of the scope of their liability among themselves. Adequate balancing of the interests of the two parties (the debtor and the creditor) lies among the major tasks of tort law. Therefore, he law should seek specific rules instrumental in determining the scope of the co-debtor’s liability in the case of multiple debtors. The application of the rule of solidary liability is based on the assumption that individuals whose joint actions inflicted damage must compensate it together. (shrink)

Greek philosophical tradition, not only the Aristotelian one, is strongly associated with proportion (Eco, 1993: 90). This principle of symmetry is generalisable; forasmuch as it is used as a normative rule in figurative arts. Nonetheless, the proportion for Ancient Greeks does not only describe a mathematical relation, but also represents a metaphysical principle. Thus, beauty is the measurement of the elements of the external form (in the case of tragedy, the meter, the symmetry of the parts, the number of (...) syllables, the number of verses) as well as the aesthetic harmony which the play transmits as total to the spectators (in the same case, the content must be neither superior nor inferior to the chosen form; otherwise the form would give the impression of grotesque or inadequate). (shrink)

Judges and juries often make factual decisions even if the facts are disputed and there is no clear-cut evidence available. Despite this common state of uncertainty, verdicts are thought of as having clear winners and losers––either the plaintiff wins and receives a full remedy, or the defendant wins and the plaintiff gets nothing. In private disputes, factfinders base their binary factual determinations on the preponderance of the evidence. There are, however, several doctrines that allow for partial remedy, discounted by the (...) probability that the facts support the plaintiff’s case, given the available evidence (proportional liability). This Article offers a general theory for proportional liability in private law. It identifies three types of factual uncertainty—mutual uncertainty, unilateral uncertainty, and institutional uncertainty—and shows that legal economists should support proportional liability when the state of uncertainty is shared by the parties and the court (mutual uncertainty), and they should adopt an all-or-nothing rule whenever the information is observable but unverifiable (institutional uncertainty). In cases where one party holds private information (unilateral uncertainty), proportional liability is sometimes, but not always, superior to an all-or-nothing rule. (shrink)

Estimation and evaluation of individualized treatment rules have been studied extensively, but real-world treatment resource constraints have received limited attention in existing methods. We investigate a setting in which treatment is intervened upon based on covariates to optimize the mean counterfactual outcome under treatment cost constraints when the treatment cost is random. In a particularly interesting special case, an instrumental variable corresponding to encouragement to treatment is intervened upon with constraints on the proportion receiving treatment. For such settings, we first (...) develop a method to estimate optimal individualized treatment rules. We further construct an asymptotically efficient plug-in estimator of the corresponding average treatment effect relative to a given reference rule. (shrink)

In this study I analyse the performance of a democratic decision-making rule: the weighted majority rule. It assigns to each voter a number of votes that is proportional to her stakes in the decision. It has been shown that, for collective decisions with two options, the weighted majority rule in combination with self-interested voters maximises the common good when the latter is understood in terms of either the sum-total or prioritarian sum of the voters’ well-being. The (...) main result of my study is that this argument for the weighted majority rule — that it maximises the common good — can be improved along the following three main lines. (1) The argument can be adapted to other criteria of the common good, such as sufficientarian, maximin, leximin or non-welfarist criteria. I propose a generic argument for the collective optimality of the weighted majority rule that works for all of these criteria. (2) The assumption of self-interested voters can be relaxed. First, common-interest voters can be accommodated. Second, even if voters are less than fully competent in judging their self-interest or the common interest, the weighted majority rule is weakly collectively optimal, that is, it almost certainly maximises the common good given large numbers of voters. Third, even for smaller groups of voters, the weighted majority rule still has some attractive features. (3) The scope of the argument can be extended to decisions with more than two options. I state the conditions under which the weighted majority rule maximises the common good even in multi-option contexts. I also analyse the possibility and the detrimental effects of strategic voting. Furthermore, I argue that self-interested voters have reason to accept the weighted majority rule. (shrink)

To investigate the impact of framing on rule-breaking behavior in social dilemmas, we incorporated a rule in a one-shot resource game with two framing-treatments: One frame was a give-some dilemma (i.e., a variant of a public goods game) and the other frame a take-some dilemma (i.e., a variant of a commons dilemma game). In each frame, all participants were part of one single collective sharing a common good. Each participant was initially equipped with one of five different endowments (...) of points from which they must give/could take amounts to/from the common good. The exact amounts of contributions/withdrawals were regulated by the rule. Participants decided whether to cooperate and comply with the rule or to break the rule to their own advantage and at the expense of the collective (i.e., giving lower/taking higher amounts). Results of an online-experiment with 202 participants showed a significantly higher proportion of individuals breaking the rule in the take-some frame than in the give-some frame. In addition, endowment size influenced the proportion of rule-breaking behavior in the take-some frame. However, the average amounts of points not given/taken too much were not different between the frames. (shrink)

To investigate the impact of framing on rule-breaking in social dilemmas, we incorporated a rule in a 1-shot resource game with 2 framing treatments: in one frame, we offered a give-some dilemma (i.e., a variant of a public goods game), and in the other frame, a take-some dilemma (i.e., a variant of a commons dilemma game). In each frame, all participants were part of 1 single collective sharing a common good. Each participant was initially equipped with 1 of (...) 5 different endowments of points from which they must give/were allowed to take amounts to/from the common good. The rule established outcome equality between participants by prescribing the exact amounts of what to give/take to/from the common good, which was finally divided equally among participants. Participants decided whether to cooperate and comply with the rule or to break the rule to their own advantage and to the detriment of the collective (i.e., giving lower/taking higher amounts). The results of an online experiment with 202 participants showed a significantly higher proportion of individuals breaking the rule in the take-some dilemma than in the give-some dilemma. In addition, endowment size influenced the proportion of rule-breaking behavior in the take-some dilemma. However, the average amounts of points not given/taken too much were not different between the 2 dilemma types. (shrink)

We develop a utilitarian framework to assess different decision rules for the European Council of Ministers. The proposals to be decided on are conceptualized as utility vectors and a probability distribution is assumed over the utilities. We first show what decision rules yield the highest expected utilities for different means of the probability distri- bution. For proposals with high mean utility, simple bench- mark rules (such as majority voting with proportional weights) tend to outperform rules that have been proposed (...) in the political arena. For proposals with low mean utility, it is the other way round. We then compare the expected utilities for smaller and larger countries and look for Pareto- dominance relations. Finally, we provide an extension of the model, discuss its restrictions, and compare our approach with assessments of decision rules that are based on the Penrose measure of voting power. (shrink)

We show that, when the number of participating agents n tends to infinity, all classical social choice rules are asymptotically strategy -proof with the proportion of manipulable profiles being of order O.

According to a given quota q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q$$\end{document}, a candidate a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a$$\end{document} is beaten by another candidate b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b$$\end{document} if at least a proportion of q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q$$\end{document} individuals prefer b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b$$\end{document} to a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a$$\end{document}. (...) The q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q$$\end{document}-majority efficiency of a voting rule is the probability that the rule selects a candidate who is never beaten under the q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q$$\end{document}-majority, given that such a candidate exits. Closed form representations are obtained for the q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q$$\end{document}-majority efficiency of positional rules in three-candidate elections. It turns out that the q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q$$\end{document}-majority efficiency is: significantly greater for sequential rules than for simple positional rules; and very close to the q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q$$\end{document}-Condorcet efficiency, the conditional probability that a positional rule will elect the candidate who beats all others under the q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q$$\end{document}-majority, when one exists. (shrink)

The information explosion and digital modes of learning often combine to inform the quest for the best ways of transforming information in digital form for pedagogical purposes. This quest has become more urgent and pervasive with the ‘turn’ to online learning in the context of COVID-19. This can result in linear, asynchronous, transmission-based modes of teaching and learning which commodify, package and deliver knowledge for individual ‘customers’. The primary concerns in such models are often technical and economic – technology as (...) a cost-effective ‘solution’ to educational challenges. In this paper I argue for the importance of dialogic learning space in teaching and learning by means of Information and Communication Technologies, whether in the form of fully online learning, blended learning or face-to-face encounters using ICT affordances. Although the 20th Century theorists Mikhail Bakhtin (1895–1975) and Hans-Georg Gadamer (1900–2002) produced their seminal works before the advent of ICTs, they were both concerned with the quality and authenticity of human engagement with texts and with other persons and contexts. Besides a shared interest in dialogue as an ontological feature of human life and being, they both used spatiotemporal concepts for understanding and interpreting texts. The article draws on Gadamer’s notions of dialogue and horizon, and Bakhtin’s notions of dialogue and chronotope, to conceptualize dialogic possibilities for online education. Its purpose is to provide a framework, grounded in Bakhtin’s and Gadamer’s ideas, for a dialogic approach to online teaching and learning in higher education. (shrink)

Jesus of Nazareth, like Socrates, left nothing behind written by himself. Yet, the records of his teaching indicate a rich interest in dialogic pedagogy, reflected in his use of the parable, primarily an oral genre, as a dialogic provocation. Working at the interface of pedagogy, theology and philosophy, this article explores the parable of the Good Samaritan from the perspective of dialogic pedagogy. It employs an analytical approach termed diacognition, developed from the notions of dialogue, position and cognition, to analyse (...) the moves within the parable and the teaching situation in which it is located. The article explores how Jesus engages the dialogue of and around the parable to position and reposition his interlocutor, provoking a re-cognition of what it means to love one’s neighbour. It concludes by reflecting on the implications of this analysis for the relation of meaning to knowing and doing. (shrink)

Dialogue is a seminal concept within the work of the Brazilian adult education theorist, Paulo Freire, and the Russian literary critic and philosopher, Mikhail Bakhtin. While there are commonalities in their understanding of dialogue, they differ in their treatment of dialectic. This paper addresses commonalities and dissonances within a Bakhtin-Freire dialogue on the notions of dialogue and dialectic. It then teases out some of the implications for education theory and practice in relation to two South African contexts of learning that (...) facilitate the access to education of disadvantaged groups, one in higher education and the other in early childhood education. (shrink)

This article examines the relationship between online dispute resolution (ODR) and trust. We discuss what trust is, why trust is important, and how trust develops. Our claim is that efforts to implement online dispute resolution on a site or service in a manner that promotes trust need to consider ODR as just one tool in a broader toolbox of trust-building tools and techniques. These techniques are amongst others marketing, education, trust seals, and transparency. By evaluating ODR in its proper context (...) as one component of a larger trust strategy, we can more accurately set expectations for its results and position our projects for success. (shrink)

Externality games are studied in Grafe et al. (1998, Math. Methods Op. Res. 48, 71). We define a generalization of this class of games and show, using the methodology in Izquierdo and Rafels (1996, 2001, Working paper, Univ Barcelona; Games Econ. Behav. 36, 174), some properties of the new class of generalized externality games. They include, among others, the algebraic structure of the game, convexity, and their implications for the study of cooperative solutions. Also the proportional rule is (...) characterized for this class of games. (shrink)

The article presents proofs of the context freeness of a family of typelogical grammars, namely all grammars that are based on a uni- ormultimodal logic of pure residuation, possibly enriched with thestructural rules of Permutation and Expansion for binary modes.