Kevin T. Kelly. General Characteristics of Inductive Inference Over Arbitrary Sets of Data Representations.

Kevin T. Kelly. Version Spaces, Structural Descriptions and NP-Completeness.

In the aftermath of scandals such as those at Enron and WorldCom, there is a growing suspicion of the corporate world. For this reason it is more important than ever for firms to maintain a good reputation. In Building Reputational Capital, Kevin T. Jackson offers a practical guide to taking the high road--the only path that leads to lasting success. Based on extensive research and real-world experience, Building Reputational Capital reveals basic principles of integrity and fairness with which firms (...) can build an enduring reputation. More than image, a firm's reputation is a form of capital often neglected in the boardroom and overlooked in conventional analyses of financial statements. Speaking directly to the work experience of real people in practical business settings, Jackson couples each principle with straightforward actions that drive management systems, and he provides tested strategies--from downsizing techniques to e-commerce tips--that cultivate the hidden power of a good reputation. He outlines the advantages of a superior reputation, describes the vital role the firm's leader must play, offers ways to build and protect your reputation on the Internet, and shows how to rescue your reputation once disaster hits. Perhaps most important, he shows how to strike the right balance of virtues like authenticity, honesty, responsibility, and stewardship of the environment, employees, and the economy. Highlighted with real-life success stories--from giants like Hewlett-Packard to small firms like Thanksgiving Coffee Company, Building Reputational Capital offers a simple but effective guide for executives, managers, entrepreneurs, legal professionals, and corporate consultants. (shrink)

This article challenges an argument from Tom Donaldson''s recent bookThe Ethics of International Business with a claim that distributive justice, deemed in many circles to impose a duty of mutual aid on individuals and nations, establishes a basis for holding multinational corporations to such a duty as well. The root idea I advocate is that Rawls'' theory of justice can be deployed — beyond its original intent yet in line with its spirit — to underwrite aprima facie obligation of international (...) business to render aid to ameliorate suffering on behalf of the inhabitants of developing countries in which they operate. (shrink)

This book is devoted to a different proposal--that the logical structure of the scientist's method should guarantee eventual arrival at the truth given the scientist's background assumptions.

Written by a Roman Catholic theologian, these essays cover how pastoral ministry should be done in today's society, including how to handle church law and build a collaborative church as well as specific issues such as euthanasia and embryo research.

The journal NatureGenetics recently reported a statement made by Dr. Brigitte Boisselier declaring the rights of parents to clone themselves. Dr. Boisselier is the scientific director of ClonaidBeam me up, Scotty!”? How do we begin to respond to this offer and Dr. Boisselier's claim for its ethical justification?

This paper develops a framework in which to compare the discovery problems determined by a wide range of distinct hypothesis languages. Twelve theorems are presented which provide a comprehensive picture of the solvability of these problems according to four intuitively motivated criteria of scientific success.

Simplicity has long been recognized as an apparent mark of truth in science, but it is difficult to explain why simplicity should be accorded such weight. This chapter examines some standard, statistical explanations of the role of simplicity in scientific method and argues that none of them explains, without circularity, how a reliance on simplicity could be conducive to finding true models or theories. The discussion then turns to a less familiar approach that does explain, in a sense, the elusive (...) connection between simplicity and truth. The idea is that simplicity does not point at or reliably indicate the truth but, rather, keeps inquiry on the cognitively most direct path to the truth. (shrink)

Formal learning theory is an approach to the study of inductive inference that has been developed by computer scientists. In this paper, I discuss the relevance of formal learning theory to such standard topics in the philosophy of science as underdetermination, realism, scientific progress, methodology, bounded rationality, the problem of induction, the logic of discovery, the theory of knowledge, the philosophy of artificial intelligence, and the philosophy of psychology.

A priest's fifty years of stuggle to be faithful to the spirit of Vatican II, reminding us that the task of receiving Vatican II is far from complete.

The article aims to provide a standpoint from which to critically address two broad concerns. The first concern surrounds a naïve view of mindfulness, which takes it as a given that it is a good thing to cultivate mindfulness and attendant qualities like compassion because these virtues are key to improving the quality of life and bettering effective decisionmaking within business. Yet the virtue of mindfulness has roots in religious and spiritual traditions, and the virtue of compassion is complex and (...) contextual; neither of these virtues operate in a vacuum. Nor do they function independently from other virtues and values. Reasonable people of goodwill possessing the virtues of mindfulness and compassion in good measure, may nevertheless strongly disagree about what the compassionate, mindful thing to do is, particularly in a business setting. It is, moreover, conceivable that intensively cultivating mindfulness and compassion could lead one to reject altogether the “dog-eat dog” culture of competitive business that draws upon selective features of mindfulness meditation that lie in the corporate comfort zone yet which are not especially countercultural from a religious or spiritual vantage point. The second concern is that Western virtue-based business ethics is largely confined to academic philosophical theories. As such, virtue-driven business ethics is often more centered around developing theoretical wisdom than developing “hard core” practical wisdom earned through yoga asanas, meditation, chanting, and breathing, whereas for contemplative practices the reverse is the case, with practical wisdom emphasized over theoretical wisdom. Accordingly, the article examines prospects for cross-fertilization between, on the one hand, mindfulness and compassion interpreted as virtues in Eastern contemplative practices, and on the other hand, mindfulness and compassion as interpreted within Western virtue-oriented business ethics. Illuminating a pathway for such interpretative cross-pollination calls for an appropriate conceptual frame of reference that the article organizes around a set of interconnected themes. The first theme is that mindfulness and compassion represent key virtues within contemplative practices. This indicates a promising touchpoint between Eastern and Western traditions: their respective focus upon character, inner states, intrinsic motivation, and self-improvement toward ethicality in the world. The second theme is that such virtues in Eastern contemplative practices, as well as character traits integral to Western virtue-oriented approaches, denote contested “normative-interpretive” concepts that engage philosophical debate rather than indisputable empirical-criterial concepts that can be taken at face-value. The third theme advocates moving beyond behaviorist and neuropsychological accounts of virtue, approaching character traits of Eastern contemplative practice and Western virtue ethics through nonscientific inquiry into normative interpretive questions concerning such virtues. This supports debate over competing views of the nature, purpose, cultivation, and cultural context of mindfulness, compassion, and other virtues – issues arising as mindfulness enters the business management sphere -- to be conducted on normative grounds. With the background conceptual framework established, the article presents key points about the prospects for cross-fertilization between virtue ethics and contemplative practice, and why it matters, with reference to business ethics. (shrink)

I have applied a fairly general, learning theoretic perspective to some questions raised by Reichenbach's positions on induction and discovery. This is appropriate in an examination of the significance of Reichenbach's work, since the learning-theoretic perspective is to some degree part of Reichenbach's reliabilist legacy. I have argued that Reichenbach's positivism and his infatuation with probabilities are both irrelevant to his views on induction, which are principally grounded in the notion of limiting reliability. I have suggested that limiting reliability is (...) still a formidable basis for the formulation of methodological norms, particularly when reliability cannot possibly be had in the short run, so that refined judgments about evidential support must depend upon measure-theoretic choices having nothing to do in the short run with the truth of the hypothesis under investigation. To illustrate the generality of Reichenbach's program, I showed how it can be applied to methods that aim to solve arbitrary assessment and discovery problems in various senses. In this generalized Reichenbachian setting, we can characterize the intrinsic complexity of reliable inductive inference in terms of topological complexity. Finally, I let Reichenbach's theory of induction have the last say about hypothetico-deductive method. (shrink)

Spirituality, in the broad sense, provides a deeper foundation for principles of international business ethics than legalistic, command-based ethics programs. Spiritual-based principles and values are presupposed and endorsed by established legal and ethical principles for international business. Identifying such spiritual-based principles and values requires the exercise of moral imagination and an openness to values embraced by the world's religions. Once identified, a new realm of moral freedom is attained for multinational corporations which may help them move beyond an "ethics for (...) sale" orientation. (shrink)

In the absence of any institution for imposing legal liability on global business, the idea of instituting a cosmopolitan court for international corporate offenses is advocated. The proposal is then critically examined and defended in light of a number of key objections. Having both civil and criminal jurisdiction, such a tribunal could benefit domestic and international legal systems, multinational corporations, and victims of transnational and international corporate misdeeds. By laying down minimal global standards of corporate liability, resolving conflicts between the (...) regulatory activities of national legal systems, and affording redress for aggrieved parties in countries with inadequate legal infrastructures, the tribunal would promote the objectives of compensatory, distributive, procedural and retributive justice at the global level. Moreover, the court would assist corporations by translating compliance with minimum ethical standards to the bottom line. Ethical corporations will gain competitive advantage relative to unscrupulous firms, since the latter'ss activities will be sanctioned. The international community need not leave obedience to basic norms up to the voluntary good-will and discretion of multinational firms. (shrink)

This article asks whether a “law-as-integrity” approach to human rights adjudication provides a theoretical framework within which to make sense of authoritative regional interpretations of basic human rights for the global community. To focus analysis, I consider U.S. court interpretations of international human rights as an interpretive context. I argue that, with appropriate modification so as to include the world community as a “community of principle” for purposes of human rights adjudication, the law-as-integrity perspective permits disputes surrounding the legality of (...) human rights to revolve around competing interpretive claims backed up by justifying legal theories, rather than as ideological battles external to a juridical philosophy of rights. (shrink)

Ockham’s razor is the principle that, all other things being equal, scientists ought to prefer simpler theories. In recent years, philosophers have argued that simpler theories make better predictions, possess theoretical virtues like explanatory power, and have other pragmatic virtues like computational tractability. However, such arguments fail to explain how and why a preference for simplicity can help one find true theories in scientific inquiry, unless one already assumes that the truth is simple. One new solution to that problem is (...) the Ockham efficiency theorem, which states that scientists who heed Ockham’s razor retract their opinions less often and sooner than do their non-Ockham competitors. The theorem neglects, however, to consider competitors following random strategies and in many applications random strategies are known to achieve better worst-case loss than deterministic strategies. In this paper, we describe two ways to extend the result to a very general class of random, empirical strategies. The first extension concerns expected retractions, retraction times, and errors and the second extension concerns retractions in chance, times of retractions in chance, and chances of errors. (shrink)

Kevin T. Fitzgerald, S.J.; Knowledge Without Wisdom: Human Genetic Engineering Without Religious Insight, Christian bioethics: Non-Ecumenical Studies in Medical.

Ockham’s razor is the characteristic scientific penchant for simpler, more testable, and more unified theories. Glymour’s early work on confirmation theory eloquently stressed the rhetorical plausibility of Ockham’s razor in scientific arguments. His subsequent, seminal research on causal discovery still concerns methods with a strong bias toward simpler causal models, and it also comes with a story about reliability—the methods are guaranteed to converge to true causal structure in the limit. However, there is a familiar gap between convergent reliability and (...) scientific rhetoric: convergence in the long run is compatible with any conclusion in the short run. For that reason, Carnap suggested that the proper sense of reliability for scientific inference should lie somewhere between short-run reliability and mere convergence in the limit. One natural such concept is straightest possible convergence to the truth, where straightness is explicated in terms of minimizing reversals of opinion and cycles of opinion prior to convergence. We close the gap between scientific rhetoric and scientific reliability by showing that Ockham’s razor is necessary for cycle-optimal convergence to the truth, and that patiently waiting for information to resolve conflicts among simplest hypotheses is necessary for reversal-optimal convergence to the truth. (shrink)

Taking a Sartrean existentialist viewpoint towards business ethics, in particular, concerning the question of the nature of businesspersons’ moral character, provides for a dramatically distinct set of reflections from those afforded by the received view on character, namely that of Aristotelian-based virtue ethics. Insofar as Sartre’s philosophy places human freedom at center stage, I argue that the authenticity with which a businessperson approaches moral situations depends on the degree of consciousness he or she has of the various choices at stake. (...) Finally, I consider some practical changes in business ethics education, managerial decision-making, and business organizations that Sartrean reflections might prompt. (shrink)

Kevin T. Kelly, Cory Juhl and Clark Glymour. Reliability, Realism, and Relativism.

Kevin T. Kelly and Clark Glymour. Inductive Inference from Theory-Laden Data.

Sixty‐one million Americans and approximately a billion people worldwide live with some form of disability that limits one or more major life activities. The field of precision medicine continues to grapple with how to best serve disability communities. In this paper, we suggest that precision medicine faces an ethical tension between its goal to treat or cure disabling conditions and views that consider disability as a marginalized identity. We appeal to the concepts of recognition justice and distributive justice to argue (...) that the ELSI community should take a more proactive role in promoting disability inclusion in precision medicine's practice and research. We also highlight two priorities for the ELSI community moving forward: facilitating greater collaboration between genetics and genomic professionals and disability communities and advocating for inclusive research design and disability accommodations in the research process. (shrink)

This thesis examines the prospects for mechanical procedures that can identify true, complete, universal, first-order logical theories on the basis of a complete enumeration of true atomic sentences. A sense of identification is defined that is more general than those which are usually studied in the learning theoretic and inductive inference literature. Some identification algorithms based on confirmation relations familiar in the philosophy of science are presented. Each of these algorithms is shown to identify all purely universal theories without function (...) symbols. It is demonstrated that no procedure can solve this universal theory inference problem in the more usual senses of identification. The question of efficiency for theory inference systems is addressed, and some definitions of limiting complexity are examined. It is shown that several aspects of obvious strategies for solving the universal theory inference problem are NP-hard. Finally, some non-worst case heuristic search strategies are examined in light of these NP-completeness results. These strategies are based upon an isomorphism between clausal entailments of a certain class and partition lattices, and are applicable to the improvement of earlier work on language acquisition and logical inductive inference. (shrink)

Explaining the connection, if any, between simplicity and truth is among the deepest problems facing the philosophy of science, statistics, and machine learning. Say that an efficient truth finding method minimizes worst case costs en route to converging to the true answer to a theory choice problem. Let the costs considered include the number of times a false answer is selected, the number of times opinion is reversed, and the times at which the reversals occur. It is demonstrated that (1) (...) always choosing the simplest theory compatible with experience, and (2) hanging onto it while it remains simplest, is both necessary and sufficient for efficiency. †To contact the author, please write to: Department of Philosophy, Carnegie Mellon University, Baker Hall 135, Pittsburgh, PA 15213-3890; e-mail: [email protected]. (shrink)

This paper places formal learning theory in a broader philosophical context and provides a glimpse of what the philosophy of induction looks like from a learning-theoretic point of view. Formal learning theory is compared with other standard approaches to the philosophy of induction. Thereafter, we present some results and examples indicating its unique character and philosophical interest, with special attention to its unified perspective on inductive uncertainty and uncomputability.

The paper introduces a virtue-theoretic critique of recent “prosocial” revisions of shareholder primacy. The paper aims at widening the scope of virtue-based business ethics beyond its nearly exclusive focus on the character and virtue of managers, employees, and organizations. In contrast to MacIntyre-inspired research, the paper takes a “good intentions” approach that looks squarely at shareholders, regarding them as real people (not algorithms or institutions) occupying distinctive roles as principals of firms who are, ideally, virtuous moral agents. It is argued (...) that virtue ethics can play a part in questioning normative features of shareholder primacy and the objective function of the firm. Probing deeper than prudential and instrumental concerns, the paper poses normative questions about interpreting shareholder motivations: what _ought_ shareholders to prefer/want/desire?; _ought_ shareholders to desire profit maximization versus profit satisficing?; _ought_ shareholders to prefer other, prosocial goals, alongside of wealth?; should self-interested motivation in profit-seeking be considered virtuous? To lend focus, the paper taps Hart and Zingales’ recently revised “pro-social” version of shareholder primacy, as well as Munger and Russell’s analysis of “virtuous profit-seekers,” as normative frames of reference. Doing so enables a critical engagement with their respective accounts from the standpoint of virtue theory. Taking a “virtuous shareholders” perspective provides an aretaic standpoint, one that is appropriate, given that shareholders care about matters of justice and the common good. Shareholders are not simply concerned about making money. Hence it is the moral intentions and motivations of ideally virtuous shareholders which ought to modulate the profit-seeking activities of shareholders in accordance with maximizing their welfare. (shrink)

I propose that empirical procedures, like computational procedures, are justified in terms of truth-finding efficiency. I contrast the idea with more standard philosophies of science and illustrate it by deriving Ockham's razor from the aim of minimizing dramatic changes of opinion en route to the truth.

Belief revision theory concerns methods for reformulating an agent's epistemic state when the agent's beliefs are refuted by new information. The usual guiding principle in the design of such methods is to preserve as much of the agent's epistemic state as possible when the state is revised. Learning theoretic research focuses, instead, on a learning method's reliability or ability to converge to true, informative beliefs over a wide range of possible environments. This paper bridges the two perspectives by assessing the (...) reliability of several proposed belief revision operators. Stringent conceptions of minimal change are shown to occasion a limitation called inductive amnesia: they can predict the future only if they cannot remember the past. Avoidance of inductive amnesia can therefore function as a plausible and hitherto unrecognized constraint on the design of belief revision operators. (shrink)

There is a popular view that the alleged meaning shifts resulting from scientific revolutions are somehow incompatible with the formulation of general norms for scientific inquiry. We construct methods that can be shown to be maximally reliable at getting to the truth when the truth changes in response to the state of the scientist or his society.

According to a widely held view, people's commitments to laws are dependent on the existence in their community of a conventional practice of complying with certain fundamental laws. This conventionalism has significantly hampered our attempts to explain the normative practice of law. Ronald Dworkin has argued against conventionalism by bringing up the phenomenon of persistent fundamental legal controversies, but neither Dworkin nor his legal positivist respondents have correctly understood the real significance of such controversies. This article argues that such controversies (...) pose a deep challenge to any conception of our legal practice as a genuinely normative, rule-mediated, practice. The article also argues that what is needed to deflect this challenge is a new understanding – different from the widely held conventionalist understanding – of how people's commitments to laws are predicated on their fellows’ like commitments. (shrink)

Philosophical logicians proposing theories of rational belief revision have had little to say about whether their proposals assist or impede the agent's ability to reliably arrive at the truth as his beliefs change through time. On the other hand, reliability is the central concern of formal learning theory. In this paper we investigate the belief revision theory of Alchourron, Gardenfors and Makinson from a learning theoretic point of view.

Kevin T. Kelly and Clark Glymour. Why Bayesian Confirmation Does Not Capture the Logic of Scientific Justification.

One construal of convergent realism is that for each clear question, scientific inquiry eventually answers it. In this paper we adapt the techniques of formal learning theory to determine in a precise manner the circumstances under which this ideal is achievable. In particular, we define two criteria of convergence to the truth on the basis of evidence. The first, which we call EA convergence, demands that the theorist converge to the complete truth "all at once". The second, which we call (...) AE convergence, demands only that for every sentence in the theorist's language, there is a time at which the theorist settles the status of the sentence. The relative difficulties of these criteria are compared for effective and ineffective agents. We then examine in detail how the enrichment of an agent's hypothesis language makes the task of converging to the truth more difficult. In particular, we parametrize first-order languages by predicate and function symbol arity, presence or absence of identity, and quantifier prefix complexity. For nearly each choice of values of these parameters, we determine the senses in which effective and ineffective agents can converge to the complete truth on an arbitrary structure for the language. Finally, we sketch directions in which our learning theoretic setting can be generalized or made more realistic. (shrink)

The article posits the concept of economy of mutuality as an intellectual mediation space for shifts in emphasis between market and social structures within economic theory and practice. Economy of mutuality, it is contended, provides an alternative frame of reference to the dichotomy of market economy and social economy, for inquiry about what business is for and what values it presupposes and creates. The article centers around the objective of gaining a broadened understanding of business so as to include not (...) just market economy, but social enterprise and social economy. In pursuit of this objective, a range of various archetypes of business enterprise are considered in light of higher ends of economic life. The highest telos of business encompassing all such archetypes, it is argued, is founded on reciprocity and integral human development. The article concludes that, compared to market economy per se, economy of mutuality provides a better conceptual framework for business in undertaking the challenges of sustainability. (shrink)

There is renewed interest in the logic of discovery as well as in the position that there is no reason for philosophers to bother with it. This essay shows that the traditional, philosophical arguments for the latter position are bankrupt. Moreover, no interesting defense of the philosophical irrelevance or impossibility of the logic of discovery can be formulated or defended in isolation from computation-theoretic considerations.

Many ethical issues facing managers of multinational corporations are polycentric problems. That is, they involve a number of distinct centers -- each of which define rights and obligations of a multiplicity of affected parties -- and resolving matters around one center typically creates unpredictable repercussions around one or more of the other centers. Polycentricity is a normative phenomenon especially unsuited for adjudication, often requiring recourse to alternative processes of contract (or reciprocal adjustment) and managerial direction. This study explores how such (...) concerns about the limits of adjudication (and its various moral counterparts) apply to international decisionmaking connected to human rights obligations of multinational companies. The focus is on ethical problems such as controlling child labor in LDCs such as Bangladesh, India and Pakistan; setting wages in developing countries like Honduras; and conducting business transactions with rights-violating regimes, such as the governments of Nigeria and Myanmar. (shrink)

Trainees with disabilities in health-related professions are often subjected to structural ableism in medicine: the discriminatory manifestation of lowered expectations towards people with disabilities by medical professionals. In this case study, I reflect on my experiences as the first individual with significant disabilities to be offered a postdoctoral fellowship in clinical bioethics at the National Institutes of Health. I focus on the following question: What arrangements need to be in place in order for someone with my level of disability to (...) thrive as a clinical bioethicist? By telling my story, I show how the process of accommodating trainees with disabilities often requires creative problem-solving and a considerable amount of institutional resources. I also describe the team-based method that my mentors and I developed to enable me to complete rotations on the NIH’s bioethics consultation service. If more trainees with disabilities are to succeed in clinical bioethics trainee programs, the field will have to grapple with how to develop an infrastructure for providing disability related support across training programs. This article is the beginning of a dialogue about how to build such an infrastructure. (shrink)

Convergent realists desire scientific methods that converge reliably to informative, true theories over a wide range of theoretical possibilities. Much attention has been paid to the problem of induction from quantifier-free data. In this paper, we employ the techniques of formal learning theory and model theory to explore the reliable inference of theories from data containing alternating quantifiers. We obtain a hierarchy of inductive problems depending on the quantifier prefix complexity of the formulas that constitute the data, and we provide (...) bounds relating the quantifier prefix complexity of the data to the quantifier prefix complexity of the theories that can be reliably inferred from such data without background knowledge. We also examine the question whether there are theories with mixed quantifiers that can be reliably inferred with closed, universal formulas in the data, but not without. (shrink)

This chapter presents a new semantics for inductive empirical knowledge. The epistemic agent is represented concretely as a learner who processes new inputs through time and who forms new beliefs from those inputs by means of a concrete, computable learning program. The agent’s belief state is represented hyper-intensionally as a set of time-indexed sentences. Knowledge is interpreted as avoidance of error in the limit and as having converged to true belief from the present time onward. Familiar topics are re-examined within (...) the semantics, such as inductive skepticism, the logic of discovery, Duhem’s problem, the articulation of theories by auxiliary hypotheses, the role of serendipity in scientific knowledge, Fitch’s paradox, deductive closure of knowability, whether one can know inductively that one knows inductively, whether one can know inductively that one does not know inductively, and whether expert instruction can spread common inductive knowledge—as opposed to mere, true belief—through a community of gullible pupils. (shrink)

As moral injury is a still-emerging concept within the area of military mental health, prevalence estimates for moral injury and its precursor, potentially morally injurious events (PMIEs), remain unknown for many of the world’s militaries. The present study sought to estimate the prevalence of PMIEs in the Canadian Armed Forces (CAF), using data collected from CAF personnel deployed to Afghanistan, via logistic regressions controlling for relevant sociodemographic, military, and deployment characteristics. Analyses revealed that over 65% of CAF members reported exposure (...) to at least one event that would be considered a PMIE. The most commonly PMIEs individuals reported included seeing ill or injured women and children they were unable to help (48.4%), being unable to distinguish between combatants and non-combatants (43.6%), and finding themselves in a threatening situation where they were unable to respond due to the rules of engagement under which they were required to operate (35.4%). These findings provide support for both the presence of exposure to PMIEs in CAF members and the need for formal longitudinal data collection regarding PMIE exposure and moral injury development. (shrink)

Over the past two decades, several consistent procedures have been designed to infer causal conclusions from observational data. We prove that if the true causal network might be an arbitrary, linear Gaussian network or a discrete Bayes network, then every unambiguous causal conclusion produced by a consistent method from non-experimental data is subject to reversal as the sample size increases any finite number of times. That result, called the causal flipping theorem, extends prior results to the effect that causal discovery (...) cannot be reliable on a given sample size. We argue that since repeated flipping of causal conclusions is unavoidable in principle for consistent methods, the best possible discovery methods are consistent methods that retract their earlier conclusions no more than necessary. A series of simulations of various methods across a wide range of sample sizes illustrates concretely both the theorem and the principle of comparing methods in terms of retractions. (shrink)