The notion of comparative probability defined in Bayesian subjectivist theory stems from an intuitive idea that, for a given pair of events, one event may be considered “more probable” than the other. Yet it is conceivable that there are cases where it is indeterminate as to which event is more probable, due to, e.g., lack of robust statistical information. We take that these cases involve indeterminate comparative probabilities. This paper provides a Savage-style decision-theoretic foundation for indeterminate comparative probabilities.

This article draws on key models of monitoring and blockholding articulated in the incomplete contracts theory of the firm. Under incomplete contracts theory, different governance systems have incentive structures that entail different tradeoffs—tradeoffs between ownership concentration and liquidity, between monitoring and management initiative, and between private rent-seeking and activity benefiting shareholders as a group. The tradeoffs delimit opportunities for productive cross-reference. More specifically, blockholder systems, such as those in Europe, subsidize monitoring by permitting blockholders to reap private benefits (...) of control through self-dealing and insider trading. Market systems, such as those in the United States and Britain, regulate such private rent-seeking toward the end of maintaining an institutional framework that supports diffuse share ownership and liquid trading markets. It follows that a legal framework conducive to blockholding may be ill-equipped to foster dispersed equity ownership and thick trading markets and that a legal framework conducive to liquid trading markets may have properties that discourage blockholding. This gives rise to questions for law reform agendas on both sides of the Atlantic. In the United States, proponents ask for deregulation of controls on institutional investors, looking to encourage blockholding and more effective monitoring. In Europe, proponents ask for stronger securities regulation, looking to encourage deeper trading markets. This article suggests that each reform program may lead to disappointing results because neither assures conforming adjustments to the pertinent actors’ incentives. Alternatively, strict reforms that materially change prevailing incentive patterns could perversely destabilize workable arrangements without assuring the appearance of more effective alternatives. (shrink)

Peano arithmetic cannot serve as the ground of mathematics for it is inconsistent to infinity, and infinity is necessary for its foundation. Though Peano arithmetic cannot be complemented by any axiom of infinity, there exists at least one (logical) axiomatics consistent to infinity. That is nothing else than a new reading at issue and comparative interpretation of Gödel’s papers (1930; 1931) meant here. Peano arithmetic admits anyway generalizations consistent to infinity and thus to some addable axiom(s) of infinity. The most (...) utilized example of those generalizations is the complex Hilbert space. Any generalization of Peano arithmetic consistent to infinity, e.g. the complex Hilbert space, can serve as a foundation for mathematics to found itself and by itself. (shrink)

We prove that the generalized cancellation axiom for incomplete comparative probability relations introduced by Rios Insua and Alon and Lehrer is stronger than the standard cancellation axiom for complete comparative probability relations introduced by Scott, relative to their other axioms for comparative probability in both the finite and infinite cases. This result has been suggested but not proved in the previous literature.

Sober (1984) has considered the problem of determining the evidential support, in terms of likelihood, for a hypothesis that is incomplete in the sense of not providing a unique probability function over the event space in its domain. Causal hypotheses are typically like this because they do not specify the probability of their initial conditions. Sober's (1984) solution to this problem does not work, as will be shown by examining his own biological examples of common cause explanation. The proposed (...) solution will lead to the conclusion, contra Sober, that common cause hypotheses explain statistical correlations and not matchings between event tokens. (shrink)

Bayesian models typically assume that agents are rational, logically omniscient and opinionated. The last of these has little descriptive or normative appeal, however, and limits our ability to describe how agents make up their minds (as opposed to changing them) or how they can suspend or withdraw their opinions. To address these limitations this paper represents the attitudinal states of non-opinionated agents by sets of (permissible) probability and desirability functions. Several basic ways in which such states of mind can be (...) changed are then characterised and compared with those found in AGM style models of attitude revision. Finally these models are employed to describe how agents make up their mind when deliberating. (shrink)

This paper examines the incompleteness of collective preference. We provide a series of Arrovian impossibility theorems without completeness. First, we consider the notion of regularity introduced by Eliaz and Ok (2006, Games and Economic Behavior 56, 61–86); it is an appropriate richness property for strict preference when preference is allowed to be incomplete. We examine the implication of imposing regularity on collective preference. Second, we propose responsiveness, a variation of positive responsiveness. This axiom requires that some changes in individual (...) preferences make an alternative weakly better than another. Third, we consider coherency conditions for collective preferences; this conditionally requires the existence of comparable pairs in a certain manner. We prove an impossibility result for each condition using Arrovian axioms. (shrink)

I compare Frith and colleagues’ influential comparator account of how the sense of agency is elicited to the multifactorial weighting model advocated by Synofzik and colleagues. I defend the comparator model from the common objection that the actual sensory consequences of action are not needed to elicit the sense of agency. I examine the comparator model’s ability to explain the performance of healthy subjects and those suffering from delusions of alien control on various self-attribution tasks. It transpires that the comparator (...) model needs case-by-case adjustment to deal with problematic data. In response to this, the multifactorial weighting model of Synofzik and colleagues is introduced. Although this model is incomplete, it is more naturally constrained by the cases that are problematic for the comparator model. However, this model may be untestable. I conclude that currently the comparator model approach has stronger support than the multifactorial weighting model approach. (shrink)

The issue of causal comparability in the social sciences underlies matters of both generalization and extrapolation. After critiquing two existing interpretations of comparability, due to Hitchcock and Hausman, I propose a distinction between ontological and epistemic comparability. While the former refers to whether two cases are actually comparable, the latter respects that in cases of incomplete information, we need to rely on whatever evidence we have of comparability. I argue, using a political science case study, (...) that in those cases of imperfect information, an epistemic homogeneity criterion can be an adequate justification for generalization. (shrink)

RésuméLa théorie des expressions incomplétes dans Principia Mathematica, se fonde sur le principe déja appliqué par Russell dans “On Denoting”, selon lequel il est souhaitable dans certains cas, ?on;établir le statut syntaxique des expressions catégorématiques. Grâce à la théorie intensionnelle ramifyée des types, les expressions incomplétes réféientiellement pourront être logiquement caractérisées par un mode de dérivation principalement basé sur la quantification non‐objectuelle. Ľintroduction des classes cependant, n'est en aucune façon reliée à ce mode intensionnel de dérivation; il en résulte qu'elles (...) fonctionnent en réalité dans ce contexte comme des expressions pourvues de dénotation et ce n'est que par une analogie grammaticale que Russell peut les comparer, du point de vue de leur élimination, aux expressions incomplétes de Principia Mathematica.SummaryThe theory of incomplete symbols in Principia Mathematica is founded on the principle already applied by Russell in “On Denoting“ that it is desirable in certain cases, to establish the syntactical status of “categorematic” expressions. Thanks to the intensional ramified theory of types, the referentially incomplete expressions could be logically caracterised by a mode of derivation primarily based on non‐objectual quantification. The introduction of classes, however, is by no way linked to that intensional mode of derivation; as a result, they function, in that context, like denotative expressions and it is only by a “grammatical” analogy that Russell can compare them, as regards their elimination, to the incomplete symbols in Principia Mathematica.ZusammenfassungDie Theorie der unvollstandigen Symbole in den Principia Mathematica gründet auf dem von Russell schon in “On Denoting” angewendeten Prinzip, wonach es in gewissen Fällen wün‐schenswert ist, den syntaktischen Status kategorematischer Ausdrücke festzulegen. Aufgrund der intensionalen verzweigten Typentheorie könnte man die referentiell unvollständigen Ausdrücke logisch durch eine im wesentlichen auf die nicht objektuelle Quantifi‐kation abstellende Ableitungsart charakterisieren. Die Einführung der Klassen dagegen hüngt keineswegs von dieser intensionalen Ableitungsart ab; daraus geht hervor, dass sie in diesem Kontext wie denotative Ausdrücke funktionieren, und es gelingt Russell nur vermittels einer grammatische Analogie, sie hinsichtlich ihrer Elimination mit den unvollständigen Symbolen der Principia Mathematica zu vergleichen. (shrink)

In an alleged counter-example to the completeness of rational preferences, a career as a clarinettist is compared with a career in law. It seems reasonable to neither want to judge that the law career is at least as preferred as the clarinet career nor want to judge that the clarinet career is at least as preferred as the law career. The two standard interpretations of examples of this kind are, first, that the examples show that preferences are rationally permitted to (...) be incomplete and, second, that the examples show that preferences are rationally permitted to be indeterminate. In this paper, I shall argue that the difference between these interpretations is crucial for the money-pump argument for transitivity, which is the standard argument that rational preferences are transitive. I shall argue that the money-pump argument for transitivity fails if preferences are rationally permitted to be incomplete but that it works if preferences are rationally permitted to be indeterminate and rationally required to be complete. (shrink)

Much has been written about the fittingness, epistemic, and standing norms that govern blame. In this paper, we argue that there exists a norm of blame that has yet to receive philosophical discussion and without which an account of the ethics of blame will be incomplete: a norm proscribing comparatively arbitrary blame. By reflecting on the objectionableness of comparatively arbitrary blame, we stand to elucidate a substantive, and thus far overlooked, norm governing our attributions of responsibility. Accordingly, our aim (...) in this paper is to develop a comparative non-arbitrariness condition on blame that can enrich our understanding of the ethics of blame. (shrink)

The proofs of Gödel (1931), Rosser (1936), Kleene (first 1936 and second 1950), Chaitin (1970), and Boolos (1989) for the first incompleteness theorem are compared with each other, especially from the viewpoint of the second incompleteness theorem. It is shown that Gödel’s (first incompleteness theorem) and Kleene’s first theorems are equivalent with the second incompleteness theorem, Rosser’s and Kleene’s second theorems do deliver the second incompleteness theorem, and Boolos’ theorem is derived from the second incompleteness theorem in the standard way. (...) It is also shown that none of Rosser’s, Kleene’s second or Boolos’ theorems is equivalent with the second incompleteness theorem, and Chaitin’s incompleteness theorem neither delivers nor is derived from the second incompleteness theorem. We compare (the strength of) these six proofs with one another. (shrink)

Russell's notion of an incomplete symbol has become a standard against which philosophers compare their views on the relationship between language and the world. But Russell's exact characterization of incomplete symbols and the role they play in his philosophy are still disputed. In this paper, I trace the development of the notion of an incomplete symbol in Russell's philosophy. I suggest – against Kaplan, Evans, and others – that Russell's many characterizations of the notion of an (...) class='Hi'>incomplete symbol are compatible. To this end, I examine and reject arguments for the purported incompatibility between declaring an expression to be incomplete and incorporating that symbol into a compositional semantic theory. I then examine how Russell puts the notion of an incomplete symbol to use in metaphysics. (shrink)

This paper is concerned with social choice without completeness of social preference. Completeness requires that pairs of alternatives are perfectly comparable. We introduce the concept of minimal comparability, which requires that for any profile, there is some comparable pair of distinct alternatives. Complete silence should be avoided according to this condition. We show that there exists no normatively desirable aggregation rule satisfying minimal comparability.

The objective of this study is to analyze the measurability and interfirm comparability of sustainability performance through the qualitative content analysis of 12 sustainability reports of mining firms using the Global Reporting Initiative guidelines. The systematic comparison of information disclosed in 92 GRI indicators sheds light on the reasons underlying the impossibility of rigorously measuring and comparing the sustainability performance of firms from the same sector, which are supposed to be strictly following the same reporting guideline. These reasons include (...) qualitative aspects of sustainability, lack of compliance with GRI protocols, indicator contingency, ambiguous or incomplete information, data heterogeneity, and report opacity. The study makes it possible to return to the very notion of sustainability, its meaning, and flexible application by organizations. The results are discussed from three different theoretical perspectives, each of which proposes possible and complementary explanations of the main findings. (shrink)

ABSTRACT What characterizes human reasoning is the ability of dealing with incomplete information. Incomplete logic is developed for modeling incomplete knowledge. The most distinctive feature of incomplete logic is its semantics. This is an alternative presentation of partial semantics. In this paper, we will introduce the general notion of incomplete logic (ICL), compare it with partial logic, and give the resolution method for it. We will also show how ICL can be applied to nonmonotonic reasoning. (...) We define nonmonotonic derivation as monotonic derivation in ICL from the database and some consistent assumptions. The mechanism of ICL makes it easy to assert the consistency of an assumption without asserting the assumption itself. (shrink)

Reviews show that comprehensive studies of international media ethics are necessarily incomplete because not all countries have either media codes or comparable measurement instruments. This article reviews major studies of international and national approaches to media ethics and describes contexts for global studies and comparisons. The three likely universals of truth, responsibility, and the drive for free expression are hypothesized, and codes are explored to see which patterns endured.

The primary aim of this thesis is to examine some of the arguments that have been leveled against the idea that all value bearing entities are comparable. A secondary aim is to investigate some putative properties of the relation ‘better than', especially transitivity and vagueness. Also, some of the consequences of accepting incomparability are investigated, both with regards to other value theoretical issues, such as the implications for monadic value predicates, and with regards to more applied issues, such as the (...) comparison of risks. PAPER I is a critical examination of the so-called small-improvement argument for incomparability. It is demonstrated that the value structure this argument is able to distinguish is compatible not only with incomparability but also with a kind of evaluative indeterminacy that is distinct from incomparability. PAPER II argues that if the possibility of non-conventional value relations is granted it follows that some things that have value are neither good, bad, nor neutral. This counterintuitive conclusion is reached by combining two individually plausible analyses of value. PAPER III addresses the phenomenon of incomplete preferences. It is shown how it is possible to model incomplete preference orderings by means of probabilistic preferences, and how to reveal an agent's incomplete preference ordering within a behaviorist framework. PAPER IV examines another version of the small-improvement argument designed to establish the rationality of incomplete preferences. It is argued that while there might be reasons to believe each of the premises in this version, there is a conflict between these reasons. The conflict is such that we are not provided with a reason to believe the conjunction of the premises. And without support for the conjunction of the premises the small-improvement argument for incomparability fails. PAPER V defends the common sense claim that ‘better than' is transitive against the compelling counterexamples provided by Larry Temkin and Stuart Rachels. It is demonstrated that the contradiction that follows from accepting Temkin and Rachels' premises trades on the vagueness of ‘better than', and so does not warrant the rejection of transitivity, but rather the conclusion that ‘better than' is vague. PAPER VI applies the notions of incommensurability and incomparability to comparative risk analysis. It is argued that if risks are incommensurable, and thereby resistant to accurate comparisons in terms of severity, we cannot perform accurate and cost effective trade-offs between risks and their associated benefits. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Comparative Theology Is Not “Business-as-Usual Theology”:Personal Witness from a Buddhist ChristianPaul F. KnitterThe following reflections find their stimulus and start in a paper prepared for a doctoral seminar on comparative theology led by John Makransky at Boston College. I was asked whether I was a comparative theologian and, if so, what difference it had made in my professional work as a theologian and in my personal life as a (...) Christian. My basic answer was: “Yes, I think so” and “A lot!”In this reworking and expansion of that seminar paper, I hope to unpack that “a lot.” This will enable me not only to carry on the search for clarity (and integrity) that I began in my book Without Buddha I Could Not Be a Christian,1 but also to comment on the promise of comparative theology (CT) and why I fear that this promise all too often goes unfulfilled.What I’m getting at is capsulized in a blurb that John Thatamanil wrote for Michelle Voss Roberts’s Dualities: A Theology of Difference:2 “Comparative theology done well is a dangerous discipline precisely because it raises provocative questions and threatens to put an end to business-as-usual theology.” I fear that some comparative theologians are not taking their job seriously enough—or they are not following through with what they say they want to do. They are carrying on CT as business as usual.moving beyond business-as-usual theologyComparative theology, which two decades ago might have been considered a fade but has become a fixture in Christian theology,3 makes the bold, indeed thoroughly revolutionary, claim that in order to properly carry on the task of Christian theology, the traditional Christian sources—Scripture, tradition, the lex orandi or spiritual practice of the communities—are not enough. A Christian theologian, in order to be a Christian theologian, must also draw on the experience, teachings, scriptures, and insights of other spiritual communities and traditions. CT embodies Raimon Panikkar’s often [End Page 181] cited (at least by me!) dictum: In order to understand who I am, I have to ask who you are. And in order to know my God I have to ask about your God.4If what comparative theologians are here claiming is true, if my understanding of myself is incomplete without my understanding of you, then after I understand you, I can be pretty sure that my understanding of myself is going to be different—really different. Otherwise, I really didn’t need to talk with you in the first place! And this is why Thatamanil can rightly sound his unsettling warning that if CT is another form of business-as-usual theology, if it is not leading to some really new discoveries about oneself and one’s own tradition, then it is not living up to its job description. And that, as I tried to say elsewhere, is what I suspect might be the case.As I survey the output of comparative theologians over the past fifteen years, I sense that, with some promising exceptions among younger scholars, they are strong on comparing but soft on concluding. More pointedly, comparative theology, so far, has been more comparison than theology. There have been intricate comparisons of Christian texts with Hindu texts and of Christian theologians/mystics with Buddhist scholars/mystics. Striking similarities and stark differences have been noted and examined, sometime meticulously. But, where are the clear, creative, courageous conclusions as to what Christians can learn from these comparisons? What might need to be clarified or changed or discarded in traditional Christian doctrine? Especially on the neuralgic and politically dangerous (for Catholics) issue of the uniqueness of Christ, I am not aware of any comparativists who have learned anything new. I would be delighted to be proved wrong.5I suspect that the reason why some comparative theologians are caught in business-as-usual theology is that they are also still lingering under the influence of postmodernism’s and postliberal theology’s insistence on the dominance of diversity. In their concern to respect and defend differences, postmodernists and postliberals conclude that cultures and religions are more different than they... (shrink)

In Arrovian social choice theory assuming the independence of irrelevant alternatives, Murakami (1968) proved two theorems about complete and transitive collective choice rules that satisfy strict non-imposition (citizens’ sovereignty), one being a dichotomy theorem about Paretian or anti-Paretian rules and the other a dictator-or-inverse-dictator impossibility theorem without the Pareto principle. It has been claimed in the later literature that a theorem of Malawski and Zhou (1994) is a generalization of Murakami’s dichotomy theorem and that Wilson’s (1972) impossibility theorem is stronger (...) than Murakami’s impossibility theorem, both by virtue of replacing Murakami’s assumption of strict non-imposition with the assumptions of non-imposition and non-nullness. In this note, we first point out that these claims are incorrect: non-imposition and non-nullness are together equivalent to strict non-imposition for all transitive collective choice rules. We then generalize Murakami’s dichotomy and impossibility theorems to the setting of incomplete social preference. We prove that if one drops completeness from Murakami’s assumptions, his remaining assumptions imply (i) that a collective choice rule is either Paretian, anti-Paretian, or dis-Paretian (unanimous individual preference implies noncomparability) and (ii) that adding proposed constraints on noncomparability, such as the regularity axiom of Eliaz and Ok (2006), restores Murakami’s dictator-or-inverse-dictator result. (shrink)

We define and study the notions of stability and relevance for precedent-based reasoning, focusing on Horty’s result model of precedential constraint. According to this model, precedents constrain the possible outcomes for a focus case, which is a yet undecided case, where precedents and the focus case are compared on their characteristics (called dimensions). In this paper, we refer to the enforced outcome for the focus case as its justification status. In contrast to earlier work, we do not assume that all (...) dimension values of the focus case or the precedent cases have been established with certainty: rather, each dimension is assigned a set of possible values. We define a focus case as stable if its justification status is the same for every choice of the possible values. For focus cases that are not stable, we study the task of identifying relevance: which possible values should be excluded to make the focus case stable? In addition, we introduce the notion of possibility to verify if a user can assign an outcome to an unstable focus case without making the case base of precedents inconsistent. We show how the tasks of identifying justification, stability, relevance and possibility can be applied for human-in-the-loop decision support. Finally, we discuss the computational complexity of these tasks and provide efficient algorithms. (shrink)

In this paper, missing attribute values in incomplete data sets have three possible interpretations: lost values, attribute-concept values and ‘do not care’ conditions. For rule induction, we use characteristic sets and generalized maximal consistent blocks. Therefore, we apply six different approaches for data mining. As follows from our previous experiments, where we used an error rate evaluated by ten-fold cross validation as the main criterion of quality, no approach is universally the best. Thus, we decided to compare our six (...) approaches using complexity of rule sets induced from incomplete data sets. We show that the smallest rule sets are induced from incomplete data sets with attribute-concept values, while the most complicated rule sets are induced from data sets with lost values. The choice between interpretations of missing attribute values is more important than the choice between characteristic sets and generalized maximal consistent blocks. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Buddhists and Christians Through Comparative Theology and SolidarityPaul O. IngramBuddhists and Christians Through Comparative Theology and Solidarity. By James L. Fredericks. Maryknoll, NY: Orbis Books, 2004. xiv + 134 pp.This elegantly written book is not only a call to Christians to act in solidarity with persons of other faith traditions as well as persons professing no religious identity inmatters of social, economic, and ecological injustice. It is also (...) a challenge posed directly to Christians to create new forms of interreligious solidarity that can empower political and social solidarity with other religious communities. Of course, interreligious solidarity requires interreligious dialogue, but Fredericks seeks to [End Page 223] deepen the Christian theological underpinnings of interreligious dialogue by delineating a model of theology of religions that moves beyond the usual exclusivist, inclusivist, and pluralist models. He calls this model "comparative theology of religions."Given the fact of religious pluralism that necessitates living out one's own particular faith tradition surrounded by faithful persons who religiously dwell in other faith traditions, and given the emerging realities of globalization, no religious tradition can remain in isolation from any other religious tradition nor from the growing secularization that challenges persons of faith wherever persons of faith are found on this planet. The issue is not the emergence of a new global religion that will supercede Buddhism, Christianity, Islam, Judaism, or Hinduism. Fredericks argues that the contrary is the case: as globalization and secularization erodes national, civic, and religious identity everywhere, interreligious solidarity that does not wash over the diversity of religious traditions becomes more important as a source of personal and communal identity, meaning, and purpose.Since the meaning of "solidarity" depends on how one understands the practice of interreligious dialogue and the theological models through which interreligious dialogue is practiced, Fredericks expends much effort describing and critiquing the exclusivist, inclusivist, and pluralist models of theology of religions. Strictly speaking, as Fredericks notes, "theology of religions" is not an issue for other religious traditions. Appropriating the words of Martin Luther, "this is most certainly true." Given the universal claims of Christian theology about the significance of the Incarnation and the notion that Christ is the only way to salvation, non-Christian traditions do not face the same exclusivist or inclusivist issues in their dialogues with other religious persons. This traditional Christian claim makes theology of religions a distinctively Christian problem.Fredericks's main criticism of exclusivist and inclusivist theologies are that both models are "replacement theologies" because (1) they either argue for the "replacement" of all non-Christian traditions with a form of Christian tradition, either Catholic or some form of Protestant tradition, or (2) they replace non-Christian traditions by assuming that what is good and true in these traditions is found most completely in Christian tradition. Much Protestant theology is exclusivist, and the current theological stance of Roman Catholicism is inclusivist. Both approaches are thoroughly nondialogical and thereby incapable of promoting Christian solidarity with non-Christians, and Fredericks's arguments in this regard are exceptionally cogent.But the trouble with pluralist theologies of religion, Fredericks argues, is that they "require Christians to recognize that Jesus is not the full revelation of God to the World. God is always more. Christians need to pay attention to other religious traditions in order to learn more of God" (p. 9). The idea of the incompleteness of God's revelation in Jesus doesn't make this revelation less true. But theological pluralism does make it difficult for Christians to be faithful to the depth and fullness of Christian faith and experience. In so doing, it fosters a debilitating theological relativism that often falsifies both Christian traditions and the non-Christian traditions with which one is engaged in dialogue. Given the relativist theology inherent in pluralist [End Page 224] theologies, it is difficult to see, Fredericks concludes, how theological pluralism can promote interreligious solidarity.So how should Christians understand the nature of religious pluralism and thereby enter into dialogue with faithful non-Christians in order to create religious solidarity, a sort of unity in diversity that aids all religious persons in confronting issues of injustice that are not religion-specific and haunt all religious human... (shrink)

Smith et al.'s article provides a convincing argument for devoting increased research attention to comparative metacognition. However, this increased attention should be complemented with establishing links with comparative theory of mind (ToM) research, which are currently missing. I present a task in which pairs of subjects are presented with incomplete information in an object-choice situation that could be used to establish that link.

Richard Bradley’s Bayesian model of preference kinematics is compared with Sven Ove Hansson’s AGM-style model of preference revision. Both seek to model the revision of preference orders as a consequence of retaining consistency when some preferences change. Both models are often interpreted normatively, as giving advice on how an agent should revise her preferences. I raise four criticisms of the Bayesian model: it is unrealistic; it neglects an important change mechanism; it disregards endogenous information relevant to preference change, in particular (...) about similarity and incompleteness; and its representational framework, when expanded with similarity comparisons, may give misleading advice. These criticisms are based on a principle of conservatism, and on two proposals of similarity metrics for the Bayesian model. The performance of the Bayesian model, with and without the similarity metrics, is then tested in three different cases of preference change, and compared to the performance of the AGM model. (shrink)

This essay discusses implications of incommensurability of values for justified decision-making, ethics and justice. Under particular conditions incommensurability of values causes what might be called ‘incomplete comparability’ of options. Some leading theorists interpret this in terms of ‘imprecise equality’ and ‘imprecise comparability.’ This interpretation is mistaken and conceals the implications of incommensurability for practical and ethical reasoning. The aim of this essay is to show that, in many cases, incommensurability prevents the assignment of determinate weights to competing (...) values. This may have problematic consequences for a complete and impartial justification of decisions concerning conflicting values to the extent that they depend on the need of weighing them. (shrink)

The overall aim of this paper is to compare the human development and social quality approaches in the context of quality of life in general and in relation to development in particular. It commences with a broad overview of several perspectives including: prudential values; Sen's capability approach; Berger-Schmitt and Noll's overarching quality of life construct; Phillips' quality of life construct; and Doyal and Gough's theory of Human Needs. en HD and SQ are introduced. HD emphasises well-being, enlarging people's choices, living (...) a long and healthy life, being educated and enjoying a decent standard of living. All this is predicated on the UNDP's insistence that it is people who comprise the real wealth of nations: HD emphasises the well-being of individuals. Two sets of tensions are then discussed: first between the ability to exercise individual freedom and the constraints upon freedom imposed by the provision of compulsory education of children which facilitates that freedom ; and secondly, the relationship between personhood, social relationships and collective capabilities. This is followed by an exploration of whether HD's individualistic orientation is a weakness or whether its explicit incompleteness is a strength. The paper concludes with a discussion of possible ways forward in developing the HD construct, either by incorporating the notion of 'the social' within its framework or else via strategies of using it in partnership with the social quality theory that can both extend it and provide it with a richer theoretical justification. (shrink)

This short book has two main purposes. The first is to explain Kurt Gödel's first and second incompleteness theorems in informal terms accessible to a layperson, or at least a non-logician. The author claims that, to follow this part of the book, a reader need only be familiar with the mathematics taught in secondary school. I am not sure if this is sufficient. A grasp of the incompleteness theorems, even at the level of ‘the big picture’, might require some experience (...) with the rigor of mathematical proof. Moreover, since the incompleteness theorems concern formal deductive systems, it would help for a potential reader of this book to have some familiarity with at least elementary logic.There are, of course, a number of informal expositions of the incompleteness theorems. I do not see the need to engage in a comparative study. The second, and more important, goal of the present book is to discuss some alleged consequences of the incompleteness theorems and, in particular, to debunk thoroughly claims in philosophy, religion, and literary criticism that are supposed to be established, or at least bolstered, by incompleteness. The execution of this part of the book is masterly. The arguments are clear and compelling.The book has eight chapters, each between eight and twenty-eight pages. The opening chapter gives a very brief account of the incompleteness theorems and a sketch of Gödel's life and work. Chapter 2 gives the main overview of the incompleteness theorems, plus …. (shrink)

The sense of agency over bodily actions is the feeling that one is the agent of one's actions. In this paper I examine the prospects of Frith and colleagues' influential comparator account of how the sense of agency over one's bodily actions is elicited, in comparison to the multifactorial weighting model advocated by Synofzik and colleagues in response to some problems with this account. I examine two problems for the comparator model. I consider the common objection that the actual sensory (...) consequences of action aren't needed to elicit the sense of agency with a look at the data which most strongly suggests this, namely the preserved sense of agency over phantom limb movements. I consider a problem for the comparator model in the behaviour of normal subjects placed in unusual circumstances, in particular I consider the 'wheel of fortune' studies which some take to be problematic for the comparator model. I argue that neither of these objections are devastating for the comparator model and that the comparator model plus some plausible assumptions can explain these data. However, these assumptions are not part of the original comparator model. In the end we get a version of the comparator model modified to deal with problematic cases in a manner that could be seen as somewhat ad hoc. To deal with this the multifactorial weighting model of Synofzik and colleagues is introduced. Although this model is incomplete a single version can be offered which is naturally constrained by the cases which are problematic for the comparator model. However, it is not clear what, if anything, could count as evidence against the multifactorial weighting model. Despite being generated with the data in mind it may be untestable. I conclude that currently the comparator model has stronger support than the multifactorial weighting model. (shrink)

We generalize a previous result about dynamically consistent menu preferences to the case where preferences are not necessarily complete. We show that, as it is the case when preferences are complete, a subjective state space version of dynamic consistency is linked to a comparative theory of preference for flexibility. In words, an objective signal is interpreted as an event in the agent’s subjective state space and the agent acts in a dynamically consistent way after that if and only if we (...) can attribute all the differences between the agent’s preferences before and after the signal to the fact that the agent values flexibility more before the signal than after. (shrink)

Scientific models share one central characteristic with fiction: their relation to the physical world is ambiguous. It is often unclear whether an element in a model represents something in the world or presents an artifact of model building. Fiction, too, can resemble our world to varying degrees. However, we assign a different epistemic function to scientific representations. As artifacts of human activity, how are scientific representations allowing us to make inferences about real phenomena? In reply to this concern, philosophers of (...) science have started analyzing scientific representations in terms of fictionalization strategies. Many arguments center on a dyadic relation between the model and its target system, focusing on structural resemblances and “as if” scenarios. This chapter provides a different approach. It looks more closely at model building to analyze the interpretative strategies dealing with the representational limits of models. How do we interpret ambiguous elements in models? Moreover, how do we determine the validity of model-based inferences to information that is not an explicit part of a representational structure? I argue that the problem of ambiguous inference emerges from two features of representations, namely their hybridity and incompleteness. To distinguish between fictional and non-fictional elements in scientific models my suggestion is to look at the integrative strategies that link a particular model to other methods in an ongoing research context. To exemplify this idea, I examine protein modeling through X-ray crystallography as a pivotal method in biochemistry. (shrink)

This article leverages the theme of friendship in Oliver O’Donovan's Entering into Rest as a locus of comparison between his earlier Resurrection and Moral Order and the Ethics as Theology trilogy. It does so by using demonstrable methodological differences between the two moral-theological projects to illumine a fundamental theological coherence. The article pursues this task in five sections. The first expounds O’Donovan's reflection on friendship in Entering into Rest. The second articulates the triadic approach adopted in these reflections. The third (...) examines a similar triad articulated by O’Donovan in Resurrection and Moral Order. The fourth uses this triadic connection to demonstrate theological coherence between Resurrection and Moral Order and the Ethics as Theology trilogy. The fifth concludes by raising a critical query about the arbitrariness of the triadic methodology employed by O’Donovan. The article’s conclusion considers more generally—in light of O’Donovan's pursuit of ethics as theology—the inherent contingency and incompleteness of theological ethics. (shrink)

ABSTRACT In this paper, we show that Kant’s complex concept of an “intuitive understanding”, which operates in his work as a tool for defining the peculiar character of our understanding, is critically absorbed by Hegel’s concept of “objective thought.” By means of this concept, Hegel first rejects the representational conception of thought that is implied by the Kantian concept of an intuitive understanding and, second, he proposes a way of comprehending thought that allows a new conception of the relationship between (...) the subject and reality. (shrink)

The focus of this essay is on conflicts of values and rival options in public decision-making, ethics and justice that seem to require us to balance the values or options against each other. The aim is to investigate implications of the so-called fourth value relation between competing valuable options for the possibility to weigh and balance them. The fourth value relation applies to many alternatives that represent important but conflicting or incompletely compatible human values. In this essay I will try (...) to show that in those cases it is implausible that relative weights and right balances can be determined or that they exist at all. The fourth value relation has significant implications for the resolution of the relevant value conflicts because it renders the answer to the question ‘what’s the right thing to do?’ indeterminate as far as this answer depends on comparative weights of the alternatives. (shrink)

The incompleteness theorems constitute the mathematical core of Gödel’s philosophical challenge. They are given in their “most satisfactory form”, as Gödel saw it, when the formality of theories to which they apply is characterized via Turing machines. These machines codify human mechanical procedures that can be carried out without appealing to higher cognitive capacities. The question naturally arises, whether the theorems justify the claim that the human mind has mathematical abilities that are not shared by any machine. Turing admits that (...) non-mechanical steps of intuition are needed to transcend particular formal theories. Thus, there is a substantive point in comparing Turing’s views with Gödel’s that is expressed by the assertion, “The human mind infinitely surpasses any finite machine”. The parallelisms and tensions between their views are taken as an inspiration for beginning to explore, computationally, the capacities of the human mathematical mind. (shrink)

A defective word is defined by paradigm as incomplete compared with the major class it belongs to. Defectiveness signifies the unwanted intrusion of morphological idiosyncrasy into syntax. Although this phenomenon has been a constant subject of studies, it has been ill incorporated into the theories of language. This present volume brings together scholars from various theoretical schools for an overdue typological view of defectiveness. It concentrates on some samples of idiosyncratic gaps which are assumed as indicative of the phenomenon (...) of defectiveness. Before delving into the specified topics of each chapter, this introductory chapter presents a typology of defective paradigms. It discusses terms used to describe defectiveness in synchronotic terms, and the possible diachrony of defective paradigms. (shrink)

Compared to standard liberal approaches to public reason and justification, the asymmetric convergence model of public justification allows for the public justification of laws and policies based on a convergence of quite different and even publicly inaccessible reasons. The model is asymmetrical in the sense of identifying a broader range of reasons that may function as decisive defeaters of proposed laws and policies. This paper raises several critical questions about the asymmetric convergence model and its central but ambiguous presumption against (...) coercion. By drawing on the theory of structural coercion, a main conclusion of the paper is that the asymmetric convergence model ultimately encounters the very incompleteness problems that its proponents often associate with more familiar consensus models of public justification. The paper also develops an alternative, Rawlsian-inspired account of public justification that includes elements of both convergence and consensus but not asymmetry. The Rawlsian model enables us to understand how democratic decisions may possess a degree of procedural, but still morally significant, liberal legitimacy under conditions of pluralism even when citizens fail to agree fully about either the premises of or conclusions to their political arguments. (shrink)

This article presents and compares various algebraic structures that arise in axiomatic unsharp quantum physics. We begin by stating some basic principles that such an algebraic structure should encompass. Following G. Mackey and G. Ludwig, we first consider a minimal state-effect-probability (minimal SEFP) structure. In order to include partial operations of sum and difference, an additional axiom is postulated and a SEFP structure is obtained. It is then shown that a SEFP structure is equivalent to an effect algebra with an (...) order determining set of states. We also consider σ-SEFP structures and show that these structures distinguish Hilbert space from incomplete inner product spaces. Various types of sharpness are discussed and under what conditions a Brouwer complementation can be defined to obtain a BZ-poset is investigated. In this case it is shown that every effect has a best lower and upper sharp approximation and that the set of all Brouwer sharp effects form an orthoalgebra. (shrink)

In this article, I compare Gottlob Frege's and Bernard Bolzano's rationalist conceptions of arithmetic. Each philosopher worked out a complicated system of propositions, all of which were set forth as true. The axioms, or basic truths, make up the foundations of the subject of arithmetic. Each member of the system which is not an axiom is related (objectively) to the axioms at the base. Even though this relation to the base may not yet be scientifically proven, the propositions of the (...) system include all of the truths of the science of arithmetic. I conclude the article by analyzing the respective views of Frege and Bolzano in the light of Gödel's first incompleteness theorem. (shrink)

Until comparatively recently the complete text of the Opus Postutmum has not been available to students of the Kantian philosophy.Prior to the publication of Adickes’ commentary on this material in 1920, students of Kant were almost wholly dependent upon Reicke's incomplete and markedly inadequate edition of 1882–84. 2 Adickes’ commentary, with its abundance of quoted passages, provided an access to a great deal of material hitherto unavailable. But it was not until the publication of the Academy Edition in 1936 (...) that the complete text became available for independent study. Because of the recent publication of the full text, most accounts of Kant's position in the Opus Postumum have depended upon Reicke's inadequate edition or, as has been more frequently the case, upon Adickes’ commentary. 3 In view of the considerable influence it has had upon Kant interpretation, Adickes’ analysis of the Opus Postumum warrants a careful and critical examination. (shrink)

This article considers an asymmetric contest with incomplete information. There are two types of players: informed and uninformed. Each player has a different ability to translate effort into performance in terms of the contest success function. While one player’s type is known to both players, the other is private information and known only to the player himself. We compare the Bayesian Nash equilibrium outcome of a one-sided private information contest to the Nash equilibrium with no private information, in which (...) both players know the type of the other player. We show conditions under which uncertainty increases the investment of the uninformed player and the rent dissipation of the contest, while decreasing the expected net payoff of the informed player. In addition, we consider conditions under which the informed player—before knowing his own type—prefers that the uninformed player knows his type. Moreover, we show conditions for the existence/non-existence of equilibrium in a two-stage contest in which the informed player declares his type (or does not declare) in the first stage and in the second stage the two players play according to the information available to them. (shrink)

Gentzen’s approach by transfinite induction and that of intuitionist Heyting arithmetic to completeness and the self-foundation of mathematics are compared and opposed to the Gödel incompleteness results as to Peano arithmetic. Quantum mechanics involves infinity by Hilbert space, but it is finitist as any experimental science. The absence of hidden variables in it interpretable as its completeness should resurrect Hilbert’s finitism at the cost of relevant modification of the latter already hinted by intuitionism and Gentzen’s approaches for completeness. This paper (...) investigates both conditions and philosophical background necessary for that modification. The main conclusion is that the concept of infinity as underlying contemporary mathematics cannot be reduced to a single Peano arithmetic, but to at least two ones independent of each other. Intuitionism, quantum mechanics, and Gentzen’s approaches to completeness an even Hilbert’s finitism can be unified from that viewpoint. Mathematics may found itself by a way of finitism complemented by choice. The concept of information as the quantity of choices underlies that viewpoint. Quantum mechanics interpretable in terms of information and quantum information is inseparable from mathematics and its foundation. (shrink)

We study the effective versions of several notions related to incompleteness, undecidability, and inseparability along the lines of Pour-El’s insights. Firstly, we strengthen Pour-El’s theorem on the equivalence between effective essential incompleteness and effective inseparability. Secondly, we compare the notions obtained by restricting that of effective essential incompleteness to intensional finite extensions and extensional finite extensions. Finally, we study the combination of effectiveness and hereditariness, and prove an adapted version of Pour-El’s result for this combination.

We often describe lives (or parts of lives) as meaningful or as not meaningful. It is also common to characterize them as more or less meaningful. Some lives, we tend to think, are more meaningful than others. But how then can one compare lives with respect to how much meaning they contain? Can one? This paper argues that (i) only a notion of rough equality can be used when comparing different lives with respect to their meaning, and that (ii) the (...) relation of being more meaningful is not transitive. It follows that all attempts to rank different lives in terms of meaning can at best lead to partially indeterminate and incomplete rankings. One should also give up on the idea of “maximizing” meaning. I will use Thaddeus Metz’s important recent book “Meaning in Life. An Analytic Study” as a foil for my discussion. (shrink)

In this paper I compare parametric and nonparametric regression models with the help of a simulated data set. Doing so, I have two main objectives. The first one is to differentiate five concepts of simplicity and assess their respective importance. The second one is to show that the scope of the existing philosophical literature on simplicity and model selection is too narrow because it does not take the nonparametric approach into account, S112–S123, 2002; Forster and Sober in The British Journal (...) for the Philosophy of Science 45, 1–35, 1994; Forster, 2001, in Philosophy of Science 74, 588–600, 2007; Hitchcock and Sober in The British Journal for the Philosophy of Science 55, 1–34, 2004; Mikkelson in Philosophy of Science 73, 440–447, 2006; Baker 2013). More precisely, I point out that a measure of simplicity in terms of the number of adjustable parameters is inadequate to characterise nonparametric models and to compare them with parametric models. This allows me to weed out false claims about what makes a model simpler than another. Furthermore, I show that the importance of simplicity in model selection cannot be captured by the notion of parametric simplicity. ‘Simplicity’ is an umbrella term. While parametric simplicity can be ignored, there are other notions of simplicity that need to be taken into consideration when we choose a model. Such notions are not discussed in the previously mentioned literature. The latter therefore portrays an incomplete picture of why simplicity matters when we choose a model. Overall I support a pluralist view according to which we cannot give a general and interesting justification for the importance of simplicity in science. (shrink)

This article compares the theory of Meinongian objects proposed by Edward Zalta with a theory of fiction formulated within an early Russellian framework. The Russellian framework is the second-order intensional logic proposed by Nino B. Cocchiarelly as a reconstruction of the form of Logicism Russell was examining shortly after writing The Principles of Mathematics. A Russellian theory of denoting concepts is developed in this intensional logic and applied as a theory of the "objects' of fiction. The framework retains the Orthodox (...) early Russellian ontology of existents, possible non-existents, and properties and relations in intension. This avoids the assumption, found in Meinongian theories, of impossible and incomplete objects. It also obviates the need to preserve consistency by distinguishing a new "mode of predication", or a "distinction in kinds of predicates". Thus, it is argued that an early Russellian theory forms a powerful rival to a Meinongian theory of objects. (shrink)

OBJECTIVES: To compare 2005 and 1995 ethics guidelines from journal editors to authors regarding requirements for institutional review board (IRB) approval and conflict-of-interest (COI) disclosure. DESIGN: A descriptive study of the ethics guidelines published in 103 English-language biomedical journals listed in the Abridged Index Medicus in 1995 and 2005. Each journal was reviewed by the principal author and one of four independent reviewers. RESULTS: During the period, the proportion of journals requiring IRB approval increased from 42% (95% CI 32.2% to (...) 51.2%, p<0.001) to 76% (95% CI 66.4% to 83.1%, p<0.001). In 2005, an additional 9% referred to the Declaration of Helsinki or the International Committee of Medical Journal Editors' Uniform requirements for ethical guidelines; 15% (95% CI 8.5% to 22.5%, p<0.01) provided ambiguous or no requirements. The proportion of journals requiring COI disclosure increased from 75% (95% CI 66.6% to 83.3%, p<0.05) to 94% (95% CI 89.4% to 98.6%, p<0.05); 41% had comprehensive requirements, while some addressed only funding source (6%), were vague (10%) or both (14%). Criteria for authorship rose from 40% (95% CI 30.5% to 49.5%, p<0.05) to 72% (95% CI 63.3% to 80.7%, p<0.05). Journals with higher impact factors were more likely to require IRB approval (p<0.01). Journals in anaesthesia and radiology all required IRB approval; requirements in other disciplines varied. CONCLUSIONS: Instructions to authors regarding ethical standards have improved. Some remain incomplete, especially regarding the scope of disclosure of COI. The ethical guidelines presented to authors need further clarification and standardisation. (shrink)

This article compares the theory of Meinongian objects proposed by Edward Zalta with a theory of fiction formulated within an early Russellian framework. The Russellian framework is the second-order intensional logic proposed by Nino B. Cocchiarelly as a reconstruction of the form of Logicism Russell was examining shortly after writing The Principles of Mathematics. A Russellian theory of denoting concepts is developed in this intensional logic and applied as a theory of the "objects' of fiction. The framework retains the Orthodox (...) early Russellian ontology of existents, possible non-existents, and properties and relations in intension. This avoids the assumption, found in Meinongian theories, of impossible and incomplete objects. It also obviates the need to preserve consistency by distinguishing a new "mode of predication", or a "distinction in kinds of predicates". Thus, it is argued that an early Russellian theory forms a powerful rival to a Meinongian theory of objects. (shrink)

Amartya Sen argues that for the advancement of justice identification of ‘perfect’ justice is neither necessary nor sufficient. He replaces ‘perfect’ justice with comparative justice. Comparative justice limits itself to comparing social states with respect to degrees of justice. Sen’s central thesis is that identifying ‘perfect’ justice and comparing imperfect social states are ‘analytically disjoined’. This essay refutes Sen’s thesis by demonstrating that to be able to make adequate comparisons we need to identify and integrate criteria of comparison. This is (...) precisely the aim of a theory of justice (such as John Rawls’s theory): identifying, integrating and ordering relevant principles of justice. The same integrated criteria that determine ‘perfect’ justice are needed to be able to adequately compare imperfect social states. Sen’s alternative approach, which is based on social choice theory, is incapable of avoiding contrary, indeterminate or incoherent directives where plural principles of justice conflict. (shrink)