Two competing accounts of value incomparability have been put forward in the recent literature. According to the standard account, developed most famously by Joseph Raz, ‘incomparability’ means determinate failure of the three classic value relations ( better than , worse than , and equally good ): two value-bearers are incomparable with respect to a value V if and only if (i) it is false that x is better than y with respect to V , (ii) it is false that x (...) is worse than y with respect to V and (iii) it is false that x and y are equally good with respect to V . Most philosophers have followed Raz in adopting this account of incomparability. Recently, however, John Broome has advocated an alternative view, on which value incomparability is explained in terms of vagueness or indeterminacy . In this paper I aim to further Broome’s view in two ways. Firstly, I want to supply independent reasons for thinking that the phenomenon of value incomparability is indeed a matter of the indeterminacy inherent in our comparative predicates. Secondly, I attempt to defend Broome’s account by warding off several objections that worry him, due mainly to Erik Carlson and Ruth Chang. (shrink)

Recently, some of the leading proponents of the view that there is widespread incommensurability among goods have suggested that the incommensurability of some goods is a constitutive feature of the goods themselves. So, for example, a friendship and a million dollars are incommensurable because it is part of what it is to be a friendship that it be incommensurable with money. According to these ‘constitutive incommensurabilists’ incommensurability follows from the very nature of certain goods. In this paper, I examine this (...) idea and argue that constitutive incommensurabilists have mistaken for constitutive incommensurability a particular emphatic kind of comparability. This examination involves an account of ‘higher’ and ‘lower’ goods and an explanation of how goods of different ‘types’ figure in practical conflict. (shrink)

Abstract: The paper provides a general account of value relations. It takes its departure in a special type of value relation, parity, which according to Ruth Chang is a form of evaluative comparability that differs from the three standard forms of comparability: betterness, worseness and equal goodness. Recently, Joshua Gert has suggested that the notion of parity can be accounted for if value comparisons are interpreted as normative assessments of preference. While Gert's basic idea is attractive, the way he develops (...) it is flawed: His modeling of values by intervals of permissible preference strengths is inadequate. Instead, I provide an alternative modeling in terms of intersections of rationally permissible preference orderings. This yields a general taxonomy of all binary value relations. The paper concludes with some implications of this approach for rational choice. (shrink)

This article discusses the possibility of a rationally justified choice between two options neither of which is better than the other while they are not equally good either (‘3NT’). Joseph Raz regards such options as incomparable and argues that reason cannot guide the choice between them. Ruth Chang, by contrast, tries to show that many cases of putative incomparability are instead cases of parity—a fourth value relation of comparability, in addition to the three standard value relations ‘better than’, ‘worse than’ (...) and ‘equally good as’. It follows, she argues, that many choice situations in which rationally justified choice seems precluded are in fact situations within the reach of practical reason. This article has three aims: (1) it challenges Chang’s argument for the possibility of parity; (2) it demonstrates that, even if parity would exist, its problematic implications for practical reason would not differ from those of Raz’s incomparability; (3) it discusses the underlying cause of hard cases of comparison: the fact that none of the three standard value relations applies (‘3NT’). It will be shown that the problematic implications for the rational justification of the choice are due to 3NT itself, irrespective of whether 3NT is explained as incomparability or parity. (shrink)

Overview of the main issues about value pluralism.

Perfect being theism is the view that the perfect being exists and the property being-perfect is the property being-God. According to the strong analysis of perfection, a being is perfect just in case it exemplifies all perfections. On the other hand, the weak analysis of perfection claims that a being is perfect just in case it exemplifies the best possible combination of compatible perfections. Strong perfect being theism accepts the former analysis while weak perfect being theism accepts the latter. In (...) this paper, I argue that there are good reasons to reject both versions of perfect being theism. On the one hand, strong perfect being theism is false if there are incompatible perfections; I argue that there are. On the other hand, if either no comparison can be made between sets of perfections, or they are equally good, then there is no best possible set of perfections. I argue for the antecedent of this conditional statement, concluding that weak perfect being theism is false. In the absence of other analyses of perfection, I conclude that we have reason to reject perfect being theism. (shrink)

Multidimensional concepts are everywhere, and they are important. Examples include moral value, welfare, scientific confirmation, democracy, and biodiversity. How, if at all, can we aggregate the underlying dimensions of a multidimensional concept F to yield verdicts about which things are Fer than which overall? Social choice theory can be used to model and investigate this aggregation problem. Here, we focus on a particularly thorny problem made salient by this social choice-theoretic framework: the underlying dimensions of a given concept might be (...) measurable on different types of scales—e.g., some ordinal and some cardinal. An underappreciated impossibility theorem due to Anna Khmelnitskaya shows that seemingly plausible constraints on aggregation across scale types are inconsistent. This impossibility threatens to render the notion of overall Fness incoherent. We attempt to defuse this threat, arguing that the impossibility depends on an overly restrictive conception of measurement and of how measurement constrains aggregation. Adopting a more flexible—and, we think, more perspicuous—conception of measurement opens an array of possibilities for aggregation across disparate scale types. (shrink)

Recent developments in theories for responsible innovation have focused on the importance of actively accounting for values in our technological designs. Leading among these theories is that of Value Sensitive Design which attempts to guide the design process on the basis of evaluative analysis. However, values often come into conflict and VSD has been criticized for not providing a proper method to resolve such inevitable conflicts. This paper examines three such methods and argues that although each has its merits, they (...) all fail to account for a common source of value conflicts known as value incommensurability. Drawing on literature from the field of axiology, this paper argues that by incorporating the evaluative relation of ‘parity’ each of these three methods, and the VSD framework in general, will be able to properly understand the relation which holds between conflicting design options stemming from the incommensurable of values and be able to guide designers in making rational decision in the face of such conflicts. (shrink)

According to a widely held view, rationality demands that the preferences of a person be transitive. The transitivity assumption is an axiom in standard theories of rational choice. It is also prima facie very plausible. I argue here that transitivity is not a necessary condition of rationality; it is a constraint only in some cases. The argument presented here is based on the non-linearity of differential utility functions. This paper has four parts. First, I present an argument against the transitivity (...) assumption (I, II). Second, I discuss some objections; this will also bring out some features of the view defended here, like the essentially comparative nature of choice (III). Then, I discuss certain proposals that promise to avoid my conclusion (IV). Finally, I go into some implications concerning the nature of rational decision making (V). To accept some intransitivities as rational does not make our life easier; it can even lead to hard choices. However, it does not lead to skepticism about rational choice. (shrink)

Normative reasons are of constant importance to us as agents trying to navigate through life. For this reason it is natural and vital to ask philosophical questions about reasons and the normative realm. This thesis explores various issues concerning reasons and normativity. The thesis consists of five free-standingpapers and an extended introduction. The aim of the extended introduction is not merely to situate the papers within a wider philosophical context but also to provide an overview of some of the central (...) research questions concerning reasons and normativity. The introduction is divided into five chapters. Chapter 1 introduces and outlines the dissertation. Chapter 2 concerns the issue of what a reason is and what different types of reasons there are, such as pro tanto reasons and overall reasons. Chapter 3 discusses the frequently employed weighing metaphor, i.e. that reasons have a certain weight and can be weighed against the weight of other reasons. Chapter 4 covers how reasons relate to other normative notions such as ought, value, and obligation. For example, it discusses whether it is the case that for something to be valuable just is for there to be reasons to favor it. Chapter 5 concludes the introduction by briefly exploring how it all relates.The thesis can be said to be embedded in a recent research trend within philosophy of normativity that has a distinguished focus on reasons.The five papers deal with various issues concerning reasons and normativity. Paper I argues for a novel and theoretically parsimonious way to understand background conditions for values and reasons. Paper II explores an interpretation of the distinction between subjectivism and objectivism about reasons and value so as to accommodate that the distinction does not commit either party to certain first order claims about what reasons there are or what makes objects valuable. Paper III discusses the dictum ‘ought implies can’and how to properly interpret the ‘can’. Paper IV defends principles according to which what we ought and have reasons to do transmit from ends to the necessary means of that action. Paper V analyzes how the Fitting Attitudes analysis of value should best understand degrees of value. (shrink)

This thesis explores whether the three standard value relations, “better than”, “worse than” and “equally as good”, exhaust the possibilities in which things can relate with respect to their value. Or more precisely, whether there are examples in which one of these relations is not instantiated. There are cases in which it is not obvious that one of these relations does obtain; these are referred to as “hard cases of comparison”. These hard cases of comparison become interesting, since if it (...) not the case that the standard three value relations obtains in these cases then the three standard relations do not exhaust the possibility of instantiated value relations. It is argued that for some of the hard cases of comparison, the standard relations determinately obtain. For some it is indeterminate, due to vagueness, which of the three relations obtains, but it is determinate that one of them obtains. Thereafter it is argued that the influential Collapsing Argument fails in ruling out other accounts of the hard cases of comparison. Since one cannot depend on the Collapsing Argument in order to conclude that all items are related by the standard three relations, the investigation continues. It is argued that none of the hard cases of comparison are cases of incomparability. Furthermore, none of them are cases of a fourth basic positive value relation such as “parity”. Consequently, for all the hard cases of comparison one of the standard value relations holds, although sometimes we do not know which one and sometimes it is indeterminate which one holds. This means that there is no reason to assume that one of the standard three value relations does not hold between items we are comparing. This is followed by a brief discussion about the normative consequences of this result. The thesis ends with four different appendices in which related topics are discussed. (shrink)

In decision-making based on multiple criteria, situations may arise where agents find their options to be neither better than, worse than nor equal to each other with respect to the relevant criteria. How, if at all, can a justified choice be made between such options? Are the options incomparable? This article explores a hypothetical case that illustrates how such a situation can arise in an environmental context; more specifically, it considers the deliberations of an imagined 'ethics committee' as it struggles (...) to decide whether or not a deep-sea mining project should be allowed to proceed. I argue that the case is best understood as involving options that are comparable in the sense of being 'on a par'. Working out from a discussion of Ruth Chang's 'self-governance' theory of choice in cases of parity, I suggest that, in the environmental context, the idea of choices expressing a moral identity may lead us in the direction of a plausible solution to these particularly challenging cases. (shrink)

A review of some major topics of debate in normative decision theory from circa 2007 to 2019. Topics discussed include the ongoing debate between causal and evidential decision theory, decision instability, risk-weighted expected utility theory, decision-making with incomplete preferences, and decision-making with imprecise credences.

What makes a choice hard? I discuss and criticize three common answers and then make a proposal of my own. Paradigmatic hard choices are not hard because of our ignorance, the incommensurability of values, or the incomparability of the alternatives. They are hard because the alternatives are on a par; they are comparable, but one is not better than the other, and yet nor are they equally good. So understood, hard choices open up a new way of thinking about what (...) it is to be a rational agent. (shrink)

This paper defends a novel account of how to determine the intrinsic value of possible worlds. Section 1 argues that a highly intuitive and widely accepted account leads to undesirable consequences. Section 2 takes the first of two steps towards a novel account by clarifying and defending a view about value-contribution that is based on some of W. D. Ross’ claims about the value of pleasure. Section 3 takes the second step by clarifying and defending a view about value-suppression that (...) is based on Ross’ claims about the interplay between prima-facie duties. Section 4 states and defends the account that I call Rossian Totalism. According to this account, the atoms of intrinsic value within a world only sometimes contribute their intrinsic value to the value of that world. (shrink)

This article explores the main similarities and differences between Derek Parfit’s notion of imprecise comparability and a related notion I have proposed of parity. I argue that the main difference between imprecise comparability and parity can be understood by reference to ‘the standard view’. The standard view claims that 1) differences between cardinally ranked items can always be measured by a scale of units of the relevant value, and 2) all rankings proceed in terms of the trichotomy of ‘better than’, (...) ‘worse than’, and ‘equally good’. Imprecise comparability, which can be understood in terms of the more familiar notions of cardinality and incommensurability, rejects only the first claim while parity rejects both claims of the standard view. -/- I then argue that insofar as those attracted to imprecise comparability assume that all rankings are trichotomous, as Parfit appears to, the view should be rejected. This is because imprecise equality is not a form of equality but is a sui generis ‘fourth’ basic way in which items can be ranked. We should, I argue, understand imprecise equality as parity, and imprecise comparability as entailing ‘tetrachotomy’ – that if two items are comparable, one must better than, worse than, equal to, or on a par with the other. Thus those attracted to the idea that cardinality can be imprecise should abandon trichotomy and accept parity and tetrachotomy instead. -/- Finally, I illustrate the difference between Parfit’s trichotomous notion of imprecise comparability and parity by examining how each notion might be employed in different solutions to the problem posed by the Repugnant Conclusion in population ethics. I suggest that parity provides the arguably more ecumenical solution to the problem. (shrink)

This introductory article describes the phenomena of incommensurability and incomparability, how they are related, and why they are important. Since incomparability is the more significant phenomenon, the paper takes that as its focus. It gives a detailed account of what incomparability is, investigates the relation between the incomparability of values and the incomparability of alternatives for choice, distinguishes incomparability from the related phenomena of parity, indeterminacy, and noncomparability, and, finally, defends a view about practical justification that vindicates the importance of (...) incomparability — assuming it exists — for practical reason. (shrink)

According to the fitting-attitude analysis of value , to be valuable is to be a fitting object of a pro-attitude. In earlier publications, setting off from this format of analysis, I proposed a modelling of value relations which makes room for incommensurability in value. In this paper, I first recapitulate the value modelling and then move on to suggest adopting a structurally similar analysis of probability. Indeed, many probability theorists from Poisson onwards did adopt an analysis of this kind. This (...) move allows to formally model probability and probability relations in essentially the same way as value and value relations. One of the advantages of the model is that we get a new account of Keynesian incommensurable probabilities, which goes beyond Keynes in distinguishing between different types of incommensurability. It also becomes possible to draw a clear distinction between incommensurability and vagueness in probability comparisons. (shrink)

Most people believe that death is bad for the one who dies. Much attention has been paid to the Epicurean puzzle about death that the rests on a tension between that belief and another—that death is the end of one’s existence. But there is nearby puzzle about death that philosophers have largely left untouched. This puzzle rests on a tension between the belief that death is bad for the one who dies and the belief that that death is not the (...) end of one’s existence. Many philosophers have responded to the Epicurean puzzle with the deprivation account of the badness of death, which seeks to make sense of the badness of death given that there is no life after death. This paper focus on the other puzzle, and advances the argument that the deprivation account can also make sense of the badness of death given that there is life after death. (shrink)

Some religious requirements seem genuinely arbitrary in the sense that there seem to be no sufficient explanation of why those requirements with those contents should pertain. This paper aims to understand exactly what it might mean for a religious requirement to be genuinely arbitrary and to discern whether and how a religious practitioner could ever be rational in obeying such a requirement. We lay out four accounts of what such arbitrariness could consist in, and show how each account provides a (...) different sort of baseline for understanding how obedience to arbitrary requirements could, in principle, be rational. (shrink)

The endeavour to locate value in moral progress faces various substantive as well as more formal challenges. This paper focuses on challenges of the latter kind. After some preliminaries, Section 3 introduces two general kinds of “evaluative moral progress-claims”, and outlines a possible novel analysis of a descriptive notion of moral progress. While Section 4 discusses certain logical features of betterness in light of recent work in value theory which are pertinent to the notion of moral progress, Sections 5 and (...) 6 outline the ambiguous character of “making moral progress”. (shrink)

Most familiar approaches to social conflict moot reasonable ways of dealing with conflict, ways that aim to serve values such as legitimacy, justice, morality, fairness, fidelity to individual preferences, and so on. In this paper, I explore an alternative approach to social conflict that contrasts with the leading approaches of Rawlsians, perfectionists, and social choice theorists. The proposed approach takes intrinsic features of the conflict—what I call a conflict's evaluative ‘structure’—as grounds for a rational way of responding to that conflict. (...) Like conflict within a single person, social conflict can have a distinctive evaluative structure that supports certain rational responses over others. I suggest that one common structure in both intra- and interpersonal cases of conflict supports the rational response of ‘self-governance’. Self-governance in the case of social conflict involves a society's deliberating over the question, ‘What kind of society should we be?’ In liberal democracies, this rational response is also a reasonable one. (shrink)

The vagueness view holds that when evaluative comparisons are hard, there is indeterminacy about which comparative relation holds. It is sceptical about whether there are any incommensurate items (in some domain). The sceptical element of John Broome’s version of this view rests on a controversial principle. Robert Sugden advances a similar view which does not depend on this principle. Sugden’s argument fails as a vagueness view because it assumes rather than shows that there are no incommensurate items (in some domain). (...) Nonetheless, I argue that an interpretation of his argument constitutes a defensible vagueness view which is supported by intuition about examples. On this interpretation Sugden’s view can be mapped onto Broome’s application of a supervaluationist view of vagueness. It is also (on this reading) a close relative of James Griffin’s ‘rough equality’ view when this is interpreted in terms of vagueness. On an alternative interpretation Sugden’s view is not sceptical about incommensurateness. On this interpretation he does not defend a vagueness view and is ‘sceptical’ about the contribution of the philosophical literature to our understanding of rational choice. (shrink)

Gert (2004) has suggested that several different types of value relations, including parity, can be clearly distinguished from each other if one interprets value comparisons as normative assessments of preference, while allowing for two levels of normativity - requirement and permission. While this basic idea is attractive, the particular modeling Gert makes use of is flawed. This paper presents an alternative modeling, developed in Rabinowicz (2008), and a general taxonomy of binary value relations. Another version of value analysis is then (...) brought in, which appeals to appropriate emotions rather than preferences. It is also shown what the modeling of value relations would look like from such an emotion-centered perspective. The preference-based and the emotion-based approaches differ importantly from each other, but they give rise to isomorphic taxonomies. (shrink)

In Rabinowicz (2008), I considered how value relations can best be analysed in terms of fitting pro-attitudes. In the formal model of that paper, fitting pro-attitudes are represented by the class of permissible preference orderings on a domain of items that are being compared. As it turns out, this approach opens up for a multiplicity of different types of value relationships, along with the standard relations of ‘better’, ‘worse’, ‘equally as good as’ and ‘incomparable in value’. Unfortunately, the approach is (...) vulnerable to a number of objections. I believe these objections can be avoided if one re-interprets the underlying notion of preference: instead of treating preference as a ‘dyadic’ attitude directed towards a pair of items, we can think of it as a difference of degree between ‘monadic’ attitudes of favouring. Each such monadic attitude has just one item as its object. Given this re-interpretation, permissible preferences can be modelled by the class of permissible assignments of degrees of favouring to items in the domain. From this construction, we can then recover the old modelling in terms of the class of permissible preference orderings, but the previous objections to that model no longer apply. (shrink)

In hard cases of comparison, people are faced with two options neither of which is conceived of as better, worse, or equally good compared to the other. Most philosophers claim that hard cases can indeed be distinguished from cases in which two options are equally good, and can be characterized by a failure of transitive reasoning. It is a much more controversial matter and at the heart of an ongoing debate, whether the options in hard cases of comparison should be (...) interpreted as incomparable, on par, or roughly equal. So far, however, none of these claims and interpretations have been tested. This paper presents the first empirical investigation on hard cases, intransitive reasoning, and incomparability. Our results reveal that hard cases present real-world dilemmas in which a significant majority of people violate transitivity. After suggesting a way of operationalizing the notion of incomparability, we provide empirical evidence that the options in some hard cases are not considered to be incomparable. Theories of rough equality or parity seem to provide better interpretations of our results. (shrink)

According to Ruth Chang the three standard positive value relations: “better than”, “worse than” and “equally good” do not fully exhaust the conceptual space for positive value relations. According to her, there is room for a fourth positive value relation, which she calls “parity”. Her argument for parity comes in three parts. First, she argues that there are items that are not related by the standard three value relations. Second, that these items are not incomparable, and third, that the phenomena (...) she has focused on are not due to the vagueness of the comparative predicates. This paper focuses on the second part of the argument and an objection is presented. By assuming the Small Unidimensional Difference Principle, which is a key premise for the second part of the argument, Chang’s argument could be accused of begging the question. More so, by assuming this principle, the space for incomparability gets severely limited. If these worries are justified, then Chang’s argument for parity as a fourth form of comparability is unsuccessful. (shrink)

This thesis is the first step in a research project which aims to develop an accurate and robust theory of global justice. The thesis concerns the content of our duties of global justice, under strict compliance theory. It begins by discussing the basic framework of my theory of global justice, which consists in two aspects: duties of minimal wellbeing, which are universal, and duties of fairness and equality, which are associative and not universal. With that in place, it briefly discusses (...) the nature of duties of fairness and equality. I shall argue that they are associative, because they are derived from the form of cooperation at hand; and that there are three kinds of them in our contemporary world: states, local cooperation and trans-state cooperation. It is from their forms of cooperation that these duties are derived. After that, the thesis focuses exclusively on duties of minimal wellbeing. Against the usual account of these duties - the human-flourishing account - I argue for my human-life account. This account argues that the function of these duties is to secure a human life for individuals; and it begins with a Razian conception of wellbeing, which states that the wellbeing of an individual is fundamentally constituted by: (a) the satisfaction of his biological needs, and (b) his success in whole-heartedly pursuing socially defined and determined goals and activities which are in fact valuable. An account of what constitutes a human life is then derived from this conception of wellbeing – it is a life that consists in having a level of wellbeing that is higher than the satisfaction of biological needs, where this is constituted by the pursuit of goals and activities with a sense of what is worth doing; and this in turn consists in: (a) being able to forms ideas of what is worth doing, (b) being able to revise them in light of further reasons, and (c) being able to coordinate one's actions according to them. I then determine the specific objects of duties of minimal wellbeing (means for the satisfaction of biological needs, education, physical security, freedom of belief, association and expression, freedom of non-harmful conduct, and minimal resources), by determining what is involved in securing such a human life for individuals. (shrink)

This paper casts doubts on John Broome's view that vagueness in value comparisons crowds out incommensurability in value. It shows how vagueness can be imposed on a formal model of value relations that has room for different types of incommensurability. The model implements some basic insights of the ‘fitting attitudes’ analysis of value.

ABSTRACTConfucianism tends to play only a marginal role in current theorizing about justice, which is a global pursuit dominated by Western theory and its strong tendency to assume that justice refers to some substantive conception of distributive, socioeconomic justice. This article examines and compares reconstructions of Confucian justice by Joseph Chan, May Sim, and Fan Ruiping. Each reconstruction makes reference to both classical and modern Western justice theory and thus each involves a comparative approach; indeed, each reconstruction seeks ultimately, in (...) its own distinctive fashion, to present a version of Confucian justice that is comparable with modern Western justice theory. In this article we assess, critically and comparatively, the tertium comparationis and the arguments in each reconstruction. While our analysis does not wholly endorse any of the reconstructions, it shows that there is a richness and vitality to Confucian justice theory that merits proper consideration in justice theory co... (shrink)

Larry Temkin challenged what seems to be an analytic truth about comparatives: if A is Φ-er than B and B is Φ-er than C, then, A is Φ-er than C. Ruth Chang denies a related claim: if A is Φ-er than B and C is not Φ-er than B, but is Φ to a certain degree, then A is Φ-er than C. In this paper I advance a context-sensitive semantics of gradability according to which the data uncovered by Temkin and (...) Chang leave both statements intact. (shrink)

Ruth Chang has argued for the existence of a fourth positive value relation, distinct from betterness, worseness and equality, which she calls “parity.” In an earlier article I seemed to criticize Chang's suggestion by offering an interval model for the values of items that I claimed could accommodate all the phenomena characteristic of parity. Wlodek Rabinowicz, offering his own model of value relations, endorsed one central feature of my proposal: the need to distinguish permissible preferences from required ones. But he, (...) along with Chang, correctly pointed out that my own model suffered from serious formal problems. The present article offers an improved interval model, but the main point is not the model itself. Rather, it is the derivation of the model from a response-dependent account of practical rationality. This derivation helps explain why we have a concept for which Rabinowicz's model is appropriate, and therefore supplements Chang's existence arguments. My conclusion is that we can take Chang's view, Rabinowicz's and mine as providing a reinforcing set of views. (shrink)

The mythical1 hungry ass, facing two identical bundles of hay equidistant from him, has engendered two related questions. Can he choose one of the bundles, there seemingly being nothing to incline him one way or the other? If he can, the second puzzle — pertaining to rational choice — arises. It seems the ass cannot rationally choose one of the bundles, because there is no sufficient reason for any choice.2In what follows, I will argue that choice is possible even when (...) there is no option which is better than the others (section II), and that it is perfectly reasonable to choose an option even when there is no sufficient reason for it (section III). I will then (section IV) point to another puzzling feature of the .. (shrink)

Suppose it is possible to make approximate interpersonal comparisons of welfare gains and losses. Thus, if w\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w$$\end{document}, x\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x$$\end{document}, y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y$$\end{document} and z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z$$\end{document} are personal states, then it is sometimes possible to say “The welfare gain of the state change w⇝x\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} (...) \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w\leadsto x$$\end{document} is greater than the welfare gain of the state change y⇝z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y\leadsto z$$\end{document}”. We can represent this by the formula “⪰\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\succeq $$\end{document}”, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$$$\end{document} is a difference preorder: an incomplete preorder on the space of all possible personal state changes. A social state change is a bundle of personal state changes. A social difference preorder is an incomplete preorder on the space of social state changes, which satisfies Pareto and Anonymity axioms. We characterize a family of SDPs which are roughly utilitarian in nature. We also apply the model to redistributive taxation. (shrink)