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)

Politics, Philosophy & Economics, Volume 21, Issue 4, Page 347-371, November 2022. We sometimes have to choose between options that are seemingly incomparable insofar as they seem to be neither better than, worse than, nor equal to each other. This often happens when the available options are quite different from one another. For instance, consider a choice between prioritizing either criminal justice reform or healthcare reform as a public policy goal. Even after the relevant details of the goals and possible (...) reforms are filled in, it is plausible that neither goal is better than, worse than, nor equal to the other. Such seemingly incomparable options present a problem for rational choice since it is unclear how an agent might rationally choose between them. What we need are some principles to help govern rational choice when faced with seemingly incomparable options. I here present three such principles. While each principle is individually compelling, I show that they are jointly incompatible. I then argue that the correct response to this inconsistent triad is to reject the principle that rationally censures performing a sequence of choices one knows will result in a suboptimal outcome. The upshot is that when seeming incomparability is involved, an agent can money pump themselves without being less rational for it. (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)

Can quite different values be rationally weighed against one another? Can the value of one thing always be ranked as greater than, equal to, or less than the value of something else? If the answer to these questions is no, then in what areas do we find commensurability and comparability unavailable? And what are the implications for moral and legal decision making? This book struggles with these questions, and arrives at distinctly different answers.".

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)

Anselmian theism holds that there necessarily exists a being, God, who is essentially unsurpassable in power, knowledge, goodness, and wisdom. This being is also understood to be the creator and sustainer of all that is. In contemporary analytic philosophy of religion, this role is generally understood as follows: God surveys the array of possible worlds, and in his wisdom selects exactly one for actualization, based on its axiological properties. In this paper, I discuss an under-appreciated challenge for this account of (...) the Anselmian God’s selection of a world. In particular, I urge that there are failures of comparability between various possible worlds, and I argue that, given certain assumptions, these failures threaten the rationality of God’s choice of a world. To the extent that rationality is deemed necessary for unsurpassability, this result also challenges the core Anselmian notion that God is an unsurpassable being. (shrink)

John Broome has argued that incomparability and vagueness cannot coexist in a given betterness order. His argument essentially hinges on an assumption he calls the ‘collapsing principle’. In an earlier article I criticized this principle, but Broome has recently expressed doubts about the cogency of my criticism. Moreover, Cristian Constantinescu has defended Broome’s view from my objection. In this paper, I present further arguments against the collapsing principle, and try to show that Constantinescu’s defence of Broome’s position fails.

The incomparability of alternatives is thought to pose a problem for justified choice, particularly for proponents of comparativism better than,worse than,equally good,roughly equalon a par. namely, rejection of the transitivity of the relation In this paper, I argue that proponents of comparativism need not incur this cost. I defend the possibility of justified choice between incomparable alternatives on grounds that comparativists can accept. The possibility of incomparability has been met with resistance, in part because of the intuitive appeal (...) of comparativism. By defending the possibility of justified choice between incomparable alternatives on grounds that comparativists can accept, this paper supports further inquiry into the subject of incomparability. (shrink)

The incomparability of alternatives is thought to pose a problem for justified choice, particularly for proponents ofcomparativism– the view that comparative facts about alternatives determine what one rationally ought to choose. As a solution, it has been argued that alternatives judged incomparable by one of the three standard comparative relations, “better than,” “worse than,” and “equally good,” are comparable by some fourth relation, such as “roughly equal” or “on a par.” This solution, however, comes at what many would regard (...) as too high a cost – namely, rejection of the transitivity of the relation “at least as good as.” In this paper, I argue that proponents of comparativism need not incur this cost. I defend the possibility of justified choice between incomparable alternatives on grounds that comparativists can accept. The possibility of incomparability has been met with resistance, in part because of the intuitive appeal of comparativism. By defending the possibility of justified choice between incomparable alternatives on grounds that comparativists can accept, this paper supports further inquiry into the subject of incomparability. (shrink)

The _Principle of Indifference_ was once regarded as a linchpin of probabilistic reasoning, but has now fallen into disrepute as a result of the so-called _problem of multiple of partitions_. In ‘Evidential symmetry and mushy credence’ Roger White suggests that we have been too quick to jettison this principle and argues that the problem of multiple partitions rests on a mistake. In this paper I will criticise White’s attempt to revive POI. In so doing, I will argue that what underlies (...) the problem of multiple partitions is a fundamental tension between POI and the very idea of _evidential incomparability_. (shrink)

This chapter presents arguments for two slightly different versions of the thesis that the value of persons is incomparable. Both arguments allege an incompatibility between the demands of a certain kind of practical reasoning and the presuppositions of value comparisons. The significance of these claims is assessed in the context of the “Numbers problem”—the question of whether one morally ought to benefit one group of potential aid recipients rather than another simply because they are greater in number. It is argued (...) that many of the popular approaches to this problem—even ones that avoid the aggregation of personal value—are imperiled by the incomparability theses. -/- . (shrink)

People tend to rank values of all kinds linearly from good to bad, but there is little reason to think that this is reasonable or correct. This book argues, to the contrary, that values are often partially ordered and hence frequently incomparable. Proceeding logically from a small set of axioms, John Nolt examines the great variety of partially ordered value structures, exposing fallacies that arise from overlooking them. He reveals various ways in which incomparability is obscured: using linear indices (...) to summarize partially ordered data, relying on an inadequately defined concept of parity, or conflating incomparability with vagueness. Incomparability can enrich and clarify a range of topics including the paradoxes of Derek Parfit, rational decision theory, and the infinite values of theology. Finally, Nolt shows how to generalize many of the concepts introduced earlier, explores the intricate depths of certain noteworthy partially ordered value structures, and argues for the finitude of value. Incomparable Values will be of interest to scholars and advanced students working in ethics, value theory, rational decision theory, and logic. (shrink)

Open peer commentary on the article “Making Sense and Meaning: On the Role of Communication and Culture in the Reproduction of Social Systems” by Raivo Palmaru. Upshot: The author addresses implications arising from socializing observer-dependent heuristics. Above all, Palmaru’s terminology is called into question since its conceptual deficiencies with regard to the relation between an observing system and its environments cause naturalistic fallacy. The author’s reply espouses a concise reanalysis of the complementary relations of fundamentally incomparable domains, namely the observer (...) and the social system. (shrink)

We show that for every intermediate \ s-degree there exists an incomparable \ s-degree. As a consequence, for every intermediate \ Q-degree there exists an incomparable \ Q-degree. We also show how these results can be applied to provide proofs or new proofs of upper density results in local structures of s-degrees and Q-degrees.

In earlier papers (Lindström & Rabinowicz, 1989. 1990), we proposed a generalization of the AGM approach to belief revision. Our proposal was to view belief revision as a relation rather thanas a function on theories (or belief sets). The idea was to allow for there being several equally reasonable revisions of a theory with a given proposition. In the present paper, we show that the relational approach is the natural result of generalizing in a certain way an approach to belief (...) revision due to Adam Grove. In his (1988) paper, Grove presents two closely related modelings of functional belief revision, one in terms of a family of "spheres" around the agent's theory G and the other in terms of an epistemic entrenchment ordering of propositions. The "sphere"-terminology is natural when one looks upon theories and propositions as being represented by sets of possible worlds. Grove's spheres may be thought of as possible "fallback" theories relative to the agent's original theory: theories that he may reach by deleting propositions that are not "sufficiently" entrenched (according to standards of sufficient entrenchment of varying stringency). To put it differently, fallbacks are theories that are closed upwards under entrenchment The entrenchment ordering can be recovered from the family of fallbacks by the definition: A is at least as entrenched as B iff A belongs to every fallback to which B belongs. To revise a theory T with a proposition A, we go to the smallest sphere that contain A-worlds and intersect it with A. The relational notion of belief revision that we are interested in, results from weakening epistemic entrenchment by not assuming it to be connected. I.e., we want to allow that some propositions may be incomparable with respect to epistemic entrenchment. As a result, the family of fallbacks around a given theory will no longer have to be nested. This change opens up the possibility for several different ways of revising a theory with a given proposition. (shrink)

Anselmian theism holds that there necessarily exists a being, God, who is essentially unsurpassable in power, knowledge, goodness, and wisdom. This being is also understood to be the creator and sustainer of all that is. In contemporary analytic philosophy of religion, this role is generally understood as follows: God surveys the array of possible worlds, and in his wisdom selects exactly one for actualization, based on its axiological properties. In this paper, I discuss an under-appreciated challenge for this account of (...) the Anselmian God's selection of a world. In particular, I urge that there are failures of comparability between various possible worlds, and I argue that, given certain assumptions, these failures threaten the rationality of God's choice of a world. To the extent that rationality is deemed necessary for unsurpassability, this result also challenges the core Anselmian notion that God is an unsurpassable being. (shrink)

We show that the analogues of the embedding theorems of [3], proved for the countable models of set theory, do not hold when extended to the uncountable realm of ω1‐like models of set theory. Specifically, under the ⋄ hypothesis and suitable consistency assumptions, we show that there is a family of many ω1‐like models of, all with the same ordinals, that are pairwise incomparable under embeddability; there can be a transitive ω1‐like model of that does not embed into its own (...) constructible universe; and there can be an ω1‐like model of whose structure of hereditarily finite sets is not universal for the ω1‐like models of set theory. (shrink)

In support of a recent conjecture by Nielsen (1999), we prove that the phenomena of ‘incomparable entanglement’— whereby, neither member of a pair of pure entangled states can be transformed into the other via local operations and classical communication (LOCC)—is a generic feature when the states at issue live in an infinite-dimensional Hilbert space.  2002 Elsevier Science B.V. All rights reserved.

Klaas J. Kraay argues that the rational choice model for divine creation—according to which God chooses to actualize one world among possible alternatives based on its axiological properties—cannot succeed given failures of comparability across possible worlds. I argue that failure of comparability across worlds would not undermine the rationality of choosing one world to create among possible alternatives.

We establish Hjorth's theorem that there is a family of continuum-many pairwise strongly incomparable free actions of free groups, and therefore a family of continuum-many pairwise incomparable treeable equivalence relations.

This encyclopedia entry urges what it takes to be correctives to common (mis)understandings concerning the phenomenon of incommensurability and incomparability and briefly outlines some of their philosophical upshots.

In [3], Shi proved that there exist incomparable Zermelo degrees at γ if there exists an ω‐sequence of measurable cardinals, whose limit is γ. He asked whether there is a size antichain of Zermelo degrees. We consider this question for the ‐degree structure. We use a kind of Prikry‐type forcing to show that if there is an ω‐sequence of measurable cardinals, then there are ‐many pairwise incomparable ‐degrees, where γ is the limit of the ω‐sequence of measurable cardinals.

. Consistent valuation and societal prioritization of risks presupposes comparability among risks, that is, in order to rank risks in order of severity, and allocate risk preventative resources accordingly, we must be able to determine whether one risk is better or worse than another, and by how much. It is often claimed, however, that some risks are not amenable to this kind of comparison because they are incommensurable, which roughly means that they are not comparable with respect to a common (...) cardinal measure. The aim of this thesis is to i) consider what it means to say that two risks are incommensurable, ii) explore if incomparability - comparison failure with respect to a common ordinal scale - ever occurs, and how to model it if it does. Essay I is a critical examination of the most prominent argument for incomparability, the so-called small improvement argument. It is argued that the argument fails because it conflates incomparability and a kind of evaluative indeterminacy. Essay II outlines so-called margin of error principles for comparative value judgements. They are based on the idea that if a proposition concerning the value relation between two value-bearing options is true, but there are sufficiently similar cases in which it is false, it is not available to be known. The usefulness of these principles is demonstrated by utilizing them in an epistemological case against SIA. Essay III presents a novel account of incomplete preference orderings which acknowledges that incomparability can vary in degrees. This is achieved by means of a probabilistic analysis of preferences. (shrink)

This paper explores the possibility of incomparability. It first focuses on a challenge to the small-improvement argument for incomparability and then turns to some seemingly more promising, “huge-improvement” variations on the argument. After considering an important complication, it is argued that, whether or not options can be strictly incomparable, there is room for cases beyond not just classic cases of trichotomous comparability but even beyond cases involving options that, though not trichotomously comparable, are still comparable as both positive, (...) or both negative, or, even if indeterminate, both fairly neutral. In the relevant further cases, at least one option that combines a positive feature and a negative feature is not positive, negative, neutral, or even fairly neutral. To the extent that comparability is revealed as applicable in such cases, skepticism about incomparability can persist, but its significance is reduced by the revelation of how little comparability requires. (shrink)

Calhoun, W.C., Incomparable prime ideals of recursively enumerable degrees, Annals of Pure and Applied Logic 63 39–56. We show that there is a countably infinite antichain of prime ideals of recursively enumerable degrees. This solves a generalized form of Post's problem.

According to one prominent view of rationality, for the choice of alternative to be justified, it must be at least as good as other alternatives. Michael Jensen has recently invoked this view to argue that managers should act exclusively to maximize the long-run market value of economic enterprises. According to Jensen, alternative accounts of managerial responsibility, such as stakeholder theory, are to be rejected because they lack a single measure to compare alternatives as better or worse. Against Jensen’s account, this (...) paper argues that choosing the alternative that is at least as good as other alternatives need not preclude managers from respecting considerations in addition to long-run market value. The paper argues that such considerations may be incorporated into managerial decision-making by introducing constraints and priorities into the process of maximizing long-run market value and by allowing for “clumpy” values. (shrink)

In support of a recent conjecture by Nielsen (1999), we prove that the phenomena of ‘incomparable entanglement’— whereby, neither member of a pair of pure entangled states can be transformed into the other via local operations and classical communication (LOCC)—is a generic feature when the states at issue live in an infinite-dimensional Hilbert space.  2002 Elsevier Science B.V. All rights reserved.

The author defines moral dilemmas as situations where there is a moral requirement for an agent to adopt each of two alternatives, And the agent cannot adopt both, But neither moral requirement overrides the other. The author then argues that moral dilemmas are possible because conflicting moral requirements can be either symmetrical or incomparable in a way that is limited enough to be plausible but still strong enough to yield moral dilemmas.

The incomparability of two items is thought to pose a problem for making justified choices and for consequentialist theories that rely on comparing states of the world to judge the goodness of a particular course of action. In response, it has been argued that items thought incomparable by one of the three standard relations, ‘better than’, ‘worse than’ and ‘equally good’, are instead comparable by some fourth relation, such as ‘roughly equal’ or ‘on a par’. Against such accounts, this (...) article argues that values in virtue of which comparisons are made can be ‘clumpy’ and that in comparisons involving clumpy values, we have no reason to accept ‘roughly equal’ or ‘on a par’ as distinct from ‘equally good’. The article supports the possibility of incomparability by arguing for an interpretation of incomparability as an instance of incommensurability. (shrink)

Epistemologists are interested in what makes beliefs well justified. Even before considering competing theories of epistemic justification, however, we should ask what sort of valuational structure we are trying to explain. If, as far as epistemic justification is concerned, beliefs are like bank accounts, then all beliefs are comparable: just as in any bank account there must be more, less, or as much money as in any other, one belief must be better, worse, or as good as any other. Contemporary (...) epistemologists take for granted the assumption that beliefs are comparable in the same way that bank accounts are comparable: bank accounts have balances, sprinters have personal bests for the 100-meter dash, and beliefs have degrees of justifiedness. Alternatively, we could understand the justificatory status of a belief to be more like the expensiveness of a restaurant. Consider a concept of restaurant expensiveness on which the expensiveness of a restaurant is determined by the range of prices for meals at that restaurant. If meals at Restaurant A are $25 to $50 and meals at B are $10 to $20, then A is more expensive than B. Restaurant C, however , is neither more nor less expensive than A. Nor is it equally expensive, for there are restaurants that are more expensive than C that are not more expensive than A . Some evaluative concepts, like this concept of restaurant expensiveness, do not reduce values to single numbers. If epistemic justification is like this, then there may be pairs of beliefs such that neither is better justified than the other, but nor are they equally well justified. Such beliefs would be incomparable. While incomparability is familiar in the ethics literature, it has not previously been explored in epistemology. I discuss the implications of allowing for incomparability in epistemology, both for theories of epistemic justification and theories of knowledge. (shrink)

In his original semantics for counterfactuals, David Lewis presupposed that the ordering of worlds relevant to the evaluation of a counterfactual admitted no incomparability between worlds. He later came to abandon this assumption. But the approach to incomparability he endorsed makes counterintuitive predictions about a class of examples circumscribed in this paper. The same underlying problem is present in the theories of modals and conditionals developed by Bas van Fraassen, Frank Veltman, and Angelika Kratzer. I show how to (...) reformulate all these theories in terms of lower bounds on partial preorders, conceived of as maximal antichains, and I show that treating lower bounds as cutsets does strictly better at capturing our intuitions about the semantics of modals, counterfactuals, and deontic conditionals. (shrink)

Palaeontology developed as a field dependent upon comparison. Not only did reconstructing the fragmentary records of fossil organisms and placing them within taxonomic systems and evolutionary lineages require detailed anatomical comparisons with living and fossil animals, but the field also required thinking in terms of behavioural, biological and ecological analogies with modern organisms to understand how prehistoric animals lived and behaved. Yet palaeontological material often worked against making easy linkages, bringing a sense of mystery and doubt. This paper will look (...) at an animal whose study exemplified these problems: the Chalicothere. Increasingly recognized as a specific type from finds across North America and Eurasia from the early nineteenth century onwards, these prehistoric mammals showed short back legs terminating in pawed feet, long front limbs ending in sharp claws, a long flexible neck, and herbivorous grinding teeth. The Chalicothere became a significant organism within palaeontological studies, as the unexpected mix of characters made it a textbook example against the Cuvierian notion of “correlation of parts,” while explaining how the animal moved, fed and behaved became puzzling. However, rather than prevent comparisons, these actually led to comparative analogies becoming flexible and varied, with different forms of comparison being made with varying methods and degrees of confidence, and with the anatomy, movement and behaviour of giraffes, bears, horses, anteaters, primates and other organisms all serving at various points as potential models for different aspects of the animal. This paper will examine some of the attempts to reconstruct and define the Chalicotheres across a long timescale, using this to show how multiple comparisons and analogies could be deployed in a reconstructive and evolutionary science like palaeontology, and illustrate some of the limits and tensions in comparative methods, as they were used to reconstruct organisms which were thought to be incomparable to any modern animal. (shrink)

How do we know when what is happening between two people should be called psychoanalysis? What is a psychoanalytic process and how do we know when one is taking place? _Psychoanalysis Comparable and Incomparable_ describes the rationale and ongoing development of a six year programme of highly original meetings conducted by the European Psychoanalytic Federation Working Party on Comparative Clinical Methods. The project comprises over seventy cases discussed by more than five hundred experienced psychoanalysts over the course of sixty workshops. (...) Authored by a group of leading European psychoanalysts, this book explores ways for psychoanalysts using different approaches to learn from each other when they present their work to fellow psychoanalysts, and provides tools for the individual practitioner to examine and improve his or her own approach. As described in detail in its pages, sticking to the task led to some surprising experiences, raising fundamental questions about the way clinical discussion and supervision are conducted in psychoanalysis. Well known by many in the psychoanalytic community and the object of much interest and debate, this project is described by those who have had the closest contact with it and will satisfy a widely held curiosity in psychoanalysts and psychotherapists throughout the world. David Tuckett is winner of the 2007 Sigourney Award. (shrink)

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)

review of Ruth Chang's collection in which I argue that the apparent agreements between the authors disguise underlying important differences.

After briefly considering the ancient Greek and nineteenth-century history of incommensurables (magnitudes that do not have a common aliquot part) and incomparables (magnitudes such that the larger can never be surpassed by any finite number of additions of the smaller to itself), this paper undertakes two tasks. The first task is to consider whether the numerical accommodation of incommensurables by means of the extension of the ordered field of rational numbers to the field of reals is `similar' or analogous to (...) the numerical accommodation of incomparables by means of the extension of the ordered field of reals to the field of hyperreals. The second task is to evaluate several contemporary attempts to use concepts and techniques of the nonstandard mathematics of hyperreals to address classical, Zenonian puzzles concerning continuous magnitudes. The result of both these undertakings is, in a certain sense, `deflationary'. (shrink)

This paper presents an argument against the widespread view that ‘hard choices’ are hard because of the incomparability of the alternatives. The argument has two parts. First, I argue that any plausible theory of practical reason must be ‘comparativist’ in form, that is, it must hold that a comparative relation between the alternatives with respect to what matters in the choice determines a justified choice in that situation. If comparativist views of practical reason are correct, however, the incomparabilist view (...) of hard choices should be rejected. Incomparabilism about hard choices leads us to an implausible error theory about the phenomenology of hard choices, threatens an unattractive view of human agency, and leaves us in perplexity about what we are doing when we choose in hard choices. The second part of the argument explores the main competitor to comparativist views of practical reason, noncomparativist view, according tow which a choice is justified so long as it is not worse than any of the alternatives. This view is often assumed by rational choice theorists but has its best philosophical defense in work by Joseph Raz. On Raz’s view, incomparabilism about hard choices avoids the problems faced if comparativism is correct, but it faces different difficulties. I argue that Raz’s noncomparativist view mistakenly assimilates practical reason to more restricted normative domains such as the law. (shrink)

According to influential arguments from several branches of philosophy, there exist some gradable natural language expressions that violate the following principle: if x and y are both F to some degree, then either x is at least as F as y or y is at least as F as x. Dorr, Nebel and Zuehl (2022) (DNZ), who refer to this principle as ‘Comparability’, respond to these arguments and offer a systematic case in support of Comparability. In this paper, I respond (...) to DNZ and develop an opposing case against Comparability that (i) buttresses previous criticisms of the principle, (ii) identifies two new arguments against the principle, and (iii) rebuts DNZ’s prospective arguments in favour of the principle. (shrink)