The present work determines the arithmetic complexity of the index sets of u.c.e. families which are learnable according to various criteria of algorithmic learning. Specifically, we prove that the index set of codes for families that are TxtFex\-learnable is \-complete and that the index set of TxtFex\-learnable and the index set of TxtFext\-learnable families are both \-complete.

Jockusch and Lewis proved that every DNC function computes a bi-immune set. They asked whether every DNC function computes an effectively bi-immune set. We construct a DNC function that computes no effectively bi-immune set, thereby answering their question in the negative.

We exhibit a family of computably enumerable sets which can be learned within polynomial resource bounds given access only to a teacher but which requires exponential resources to be learned given access only to a membership oracle. In general, we compare the families that can be learned with and without teachers and oracles for four measures of efficient learning.

Given any oracle, A, we construct a basic sequence Q, computable in the jump of A, such that no A-computable real is Q-distribution-normal. A corollary to this is that there is a Δn+10 basic sequence with respect to which no Δn0 real is distribution-normal. As a special case, there is a limit computable sequence relative to which no computable real is distribution-normal.

Despite a decade of federal regulation and debate over the appropriateness of financial ties in research and their management, little is known about the actual decision-making processes of university conflict of interest (COI) committees. This paper analyzes in detail the discussions and decisions of three COI committees at three public universities in California. University committee members struggle to understand complex financial relationships and reconcile institutional, state, and federal policies and at the same time work to protect the integrity of the (...) scientific process, the autonomy and intellectual freedom of their faculty colleagues and students, and the financial interests of the university. (shrink)

The Monographs produced by the International Agency for Research on Cancer (IARC) apply rigorous procedures for the scientific review and evaluation of carcinogenic hazards by independent experts. The Preamble to the IARC Monographs, which outlines these procedures, was updated in 2019, following recommendations of a 2018 expert Advisory Group. This article presents the key features of the updated Preamble, a major milestone that will enable IARC to take advantage of recent scientific and procedural advances made during the 12 years since (...) the last Preamble amendments. The updated Preamble formalizes important developments already being pioneered in the Monographs Programme. These developments were taken forward in a clarified and strengthened process for identifying, reviewing, evaluating and integrating evidence to identify causes of human cancer. The advancements adopted include strengthening of systematic review methodologies; greater emphasis on mechanistic evidence, based on key characteristics of carcinogens; greater consideration of quality and informativeness in the critical evaluation of epidemiological studies, including their exposure assessment methods; improved harmonization of evaluation criteria for the different evidence streams; and a single-step process of integrating evidence on cancer in humans, cancer in experimental animals and mechanisms for reaching overall evaluations. In all, the updated Preamble underpins a stronger and more transparent method for the identification of carcinogenic hazards, the essential first step in cancer prevention. (shrink)

Successful psychopaths, defined as individuals in the general population who nevertheless possess some degree of psychopathic traits, are receiving increasing amounts of empirical attention. To date, little is known about such individuals, specifically with regard to how they respond to ethical dilemmas in business contexts. This study investigated this relationship, proposing a mediated model in which the positive relationship between psychopathy and unethical decision-making is explained through the process of moral disengagement, defined as a cognitive orientation that facilitates unethical choice. (...) The results of the study supported this model, and implications for theory and practice are discussed. (shrink)

Is a whole something more than the sum of its parts? Are there things composed of the same parts? If you divide an object into parts, and divide those parts into smaller parts, will this process ever come to an end? Can something lose parts or gain new ones without ceasing to be the thing it is? Does any multitude of things (including disparate things such as you, this book, and the tail of a cat) compose a whole of some (...) sort? Questions such as these have occupied us for at least as long as philosophy has existed. They define the field that has come to be known as mereology-the study of all relations of part to whole and of part to part within a whole-and have deep and far-reaching ramifications in metaphysics as well as in logic, the foundations of mathematics, the philosophy of language, the philosophy of science, and beyond. In Mereology, A. J. Cotnoir and Achille C. Varzi have compiled decades of advanced research into a comprehensive, up-to-date, and formally rigorous picture. The early chapters cover the more classical aspects of mereology; the rest of the book deals with variants and extensions. Whether you are an established professional philosopher, an interested student, or a newcomer, inside you will find all the tools you need to join this ever-evolving field of inquiry and theorize about all things mereological. (shrink)

In an earlier paper, we introduced the following pre-order on the subgroups of a given Polish group: if G is a Polish group and \ are subgroups, we say H is homomorphism reducible to L iff there is a continuous group homomorphism \ such that \\). We previously showed that there is a \ subgroup L of the countable power of any locally compact Polish group G such that every \ subgroup of \ is homomorphism reducible to L. In the (...) present work, we show that this fails in the countable power of the group of increasing homeomorphisms of the unit interval. (shrink)

Involvement of industry in academic research is widespread and associated with favorable outcomes for industry. The objective of this study was to review empirical data on the attitudes of researchers toward industry involvement and financial ties in research. A review of the literature for quantitative data from surveys on the attitudes of researchers to financial ties in research, reported in English, resulted in the 17 studies included. Review of these studies revealed that investigators are concerned about the impact of financial (...) ties on choice of research topic, research conduct and publication, but this concern is less among investigators already involved with industry. Researchers approve of industry collaboration and financial ties when the ties are indirectly related to the research, disclosure is upfront, and results and ideas are freely publicized. However, their trust in disclosure as a way to manage conflicts may reveal a lack of awareness of the actual impact of financial incentives on themselves and other researchers. (shrink)

In this paper, we study some new examples of ideals on $$\omega $$ with maximal Tukey type (that is, maximal among partial orders of size continuum). This discussion segues into an examination of a refinement of the Tukey order—known as the weak Rudin–Keisler order—and its structure when restricted to these ideals of maximal Tukey type. Mirroring a result of Fremlin (Note Mat 11:177–214, 1991) on the Tukey order, we also show that there is an analytic P-ideal above all other analytic (...) P-ideals in the weak Rudin–Keisler order. (shrink)

Given a class${\cal C}$of subgroups of a topological groupG, we say that a subgroup$H \in {\cal C}$is auniversal${\cal C}$subgroupofGif every subgroup$K \in {\cal C}$is a continuous homomorphic preimage ofH. Such subgroups may be regarded as complete members of${\cal C}$with respect to a natural preorder on the set of subgroups ofG. We show that for any locally compact Polish groupG, the countable powerGωhas a universalKσsubgroup and a universal compactly generated subgroup. We prove a weaker version of this in the nonlocally compact (...) case and provide an example showing that this result cannot readily be improved. Additionally, we show that many standard Banach spaces have universalKσand compactly generated subgroups. As an aside, we explore the relationship between the classes ofKσand compactly generated subgroups and give conditions under which the two coincide. (shrink)

BackgroundAustralian health and medical research funders support substantial research efforts, and incentives within grant funding schemes influence researcher behaviour. We aimed to determine to what extent Australian health and medical funders incentivise responsible research practices.MethodsWe conducted an audit of instructions from research grant and fellowship schemes. Eight national research grants and fellowships were purposively sampled to select schemes that awarded the largest amount of funds. The funding scheme instructions were assessed against 9 criteria to determine to what extent they incentivised (...) these responsible research and reporting practices: publicly register study protocols before starting data collection, register analysis protocols before starting data analysis, make study data openly available, make analysis code openly available, make research materials openly available, discourage use of publication metrics, conduct quality research, collaborate with a statistician, and adhere to other responsible research practices. Each criterion was answered using one of the following responses: “Instructed”, “Encouraged”, or “No mention”.ResultsAcross the 8 schemes from 5 funders, applicants were instructed or encouraged to address a median of 4 of the 9 criteria. Three criteria received no mention in any scheme. Importantly, most incentives did not seem strong as applicants were only instructed to register study protocols, discourage use of publication metrics and conduct quality research. Other criteria were encouraged but were not required.ConclusionsFunders could strengthen the incentives for responsible research practices by requiring grant and fellowship applicants to implement these practices in their proposals. Administering institutions could be required to implement these practices to be eligible for funding. Strongly rewarding researchers for implementing robust research practices could lead to sustained improvements in the quality of health and medical research. (shrink)

We consider some natural sets of real numbers arising in ergodic theory and show that they are, respectively, complete in the classes${\cal D}_2 \left( {{\bf{\Pi }}_3^0 } \right)$and${\cal D}_\omega \left( {{\bf{\Pi }}_3^0 } \right)$, that is, the class of sets which are 2-differences (respectively,ω-differences) of${\bf{\Pi }}_3^0 $sets.

Extended revised edition of "Logica" (2003).

In _Necropolitics_ Achille Mbembe—a leader in the new wave of Francophone critical theory—theorizes the genealogy of the contemporary world—a world plagued by ever-increasing inequality, militarization, enmity, and terror, as well as by a resurgence of racist, fascist, and nationalist forces determined to exclude and kill. He outlines how democracy has begun to embrace its dark side, or what he calls its “nocturnal body,” which is based on the desires, fears, affects, relations, and violence that drove colonialism. This shift has hollowed (...) out democracy, thereby eroding the very values, rights, and freedoms liberal democracy routinely celebrates. As a result, war has become the sacrament of our times, in a conception of sovereignty that operates by annihilating all those considered to be enemies of the state. Despite his dire diagnosis, Mbembe draws on post-Foucault debates on biopolitics, war, and race, as well as Fanon's notion of care as a shared vulnerability, to explore how new conceptions of the human that transcend humanism might come to pass. These new conceptions would allow us to encounter the Other not as a thing to exclude, but as a person with whom to build a more just world. (shrink)

Achille Varzi è uno dei maggiori metafisici viventi. Nel corso degli anni ha scritto testi fondamentali di logica, metafisica, mereologia, filosofia del linguaggio. Ha sconfinato nella topologia, nella geografia, nella matematica, ha ragionato di mostri e confini, percezione e buchi, viaggi nel tempo, nicchie, eventi e ciambelle; e non ha disdegnato di dialogare con gli abitanti di Flatlandia, con Neo e con Terminator. Tra le sue opere principali: Holes and Other Superficialities e Parts and Places. The Structures of Spatial Representation, (...) entrambi scritti insieme a R. Casati per MIT Press; Il mondo messo a fuoco, Laterza; e il suo libro più recente: Le tribolazioni del filosofare, con C. Calosi, per Laterza. -/- Da una giornata all’Università di Urbino nasce questa conversazione a molte voci sulla e con la filosofia di Achille C. Varzi. In un dialogo critico al quale l’Autore si presta con generosità e onestà intellettuale, Andrea Borghini, Francesco Calemi, Claudio Calosi, Elena Casetta, Valeria Giardino, Pierluigi Graziani, Patrizia Pedrini, Daniele Santoro e Giuliano Torrengo lo interrogano e mettono alla prova sui temi affrontati, nel corso degli anni, in campi diversi. Il risultato è un percorso che si snoda attraverso molti mondi, dalla logica alla metafisica, dalla filosofia del linguaggio alla filosofia della matematica, dalla mereologia alla filosofia del tempo, spingendosi in qualche caso oltre i confini del saggio filosofico. (shrink)

In _Critique of Black Reason_ eminent critic Achille Mbembe offers a capacious genealogy of the category of Blackness—from the Atlantic slave trade to the present—to critically reevaluate history, racism, and the future of humanity. Mbembe teases out the intellectual consequences of the reality that Europe is no longer the world's center of gravity while mapping the relations among colonialism, slavery, and contemporary financial and extractive capital. Tracing the conjunction of Blackness with the biological fiction of race, he theorizes Black reason (...) as the collection of discourses and practices that equated Blackness with the nonhuman in order to uphold forms of oppression. Mbembe powerfully argues that this equation of Blackness with the nonhuman will serve as the template for all new forms of exclusion. With _Critique of Black Reason_, Mbembe offers nothing less than a map of the world as it has been constituted through colonialism and racial thinking while providing the first glimpses of a more just future. (shrink)

In matters of responsibility, there are often two sides to the transaction: one party who holds another responsible, and the other who takes responsibility for her conduct. The first side has been closely scrutinized in discussions of the nature of responsibility, due to the influential Strawsonian conjecture that an agent is responsible if and only if it is appropriate to hold her responsible. This preoccupation with holding responsible – with its focus on the second-personal perspective and on responses like blame (...) – contrasts with a relative neglect of the perspective of the agent and the role that she has to play by taking responsibility. I aim to show that this neglect is undeserved – that taking responsibility is both distinct in character from holding responsible and fundamentally important in its own right. I develop a conception of taking responsibility that reveals an under-explored dimension of our responsibility practices. (shrink)

His argument rests on several premises: that achievement in sport has become a model for achievement in the workplace; that the two worlds share the same ...

Partendo dal problema della dignità umana e della sua libertà come fondamento del pensiero popperiano, questo saggio approfondisce, attraverso un confronto aperto con alcune significative linee di pensiero, il valore rilevante che la lezione antropoetica di Karl Popper, finalizzata a porre in luce il “carattere speciale” della mente umana, assume nella realtà contemporanea.

There are sometimes good reasons to define a criminal offense in a way that is over-inclusive, in the sense that the definition will encompass conduct that is not otherwise wrongful. But are these reasons ever sufficient? When, if ever, can such laws justifiably be made and enforced? When, if ever, can they permissibly be violated? In The Realm of Criminal Law, Antony Duff tackles this challenge head on. We find Duff’s strategy promising in many ways as an effort to reconcile (...) over-inclusive offenses with the wrongness constraint on criminalization. Nonetheless, we aim to move the discussion forward by raising questions about Duff’s solution and highlighting some limitations and costs. We begin in Part 2 by sketching the contours of Duff’s position; then in Part 3 we propose one refinement and offer two practical observations; and finally, in Part 4 we raise broader concerns. In particular, we question whether the problem of over-inclusive offenses is one that can or ought to be solved, or whether it is better conceived as a difficulty to be managed and mitigated. Of course, we should avoid undue harshness in the law where we can, and Duff’s approach is guided by this worthy ambition. But there may also be a limit to this. To the extent that the harshness cannot be avoided, perhaps this should be acknowledged and faced up to, rather than obscured or finessed. (shrink)

Bernard Williams’ Shame and Necessity (1993) was an influential early contribution to what has become a broader movement to rehabilitate shame as a moral emotion. But there is a tension in Williams’ discussion that presents an under-appreciated difficulty for efforts to rehabilitate shame. The tension arises between what Williams takes shame in its essence to be and what shame can do—the role that shame can be expected to play in ethical life. Williams can—and we argue, should—be read as avoiding the (...) difficulties stemming from this tension, but this requires a reevaluation of several of his central claims about shame’s role in ethical thought and experience. For instance, his broad claims that the “structures of shame” can “give a conception of one’s ethical identity” (93), and that shame “mediates … between ethical demands and the rest of life” (102), cannot be taken at face value. What emerges is a view that is in a sense less ambitious, but also more in tune with the spirit of Williams’ larger project. There may also, we suggest, be a more general lesson: We should be suspicious of the temptation to seek some special affinity between shame and ethical life, lest we distort our understanding of both. (shrink)

We tend to talk about parts in the same way in which we talk about whole objects. Yet a part is not something to be included in an inventory of the world over and above the whole to which it belongs, and a whole is not something to be included in an inventory over and above its own parts. This paper is an attempt to clarify a way of dealing with this tension which may be labeled the Minimalist View: an (...) element in the field of a part‐whole relation is to be included in an inventory of the world if, and only if, it does not overlap any other element that is itself included in the inventory. As it turns out, a clarification of this view involves both a defense of mereological extensionality and an account of the topological distinction between detached and undetached parts. (shrink)

Italian translation of "Schaum's Outline of Theory and Problems of Logic" (1988).

Many experiences of shame centrally involve exposure. This has suggested to a number of writers that shame is essentially a social emotion that involves being exposed to the view or appraisal of an audience—call this the Audience Thesis. Others reject the Audience Thesis on the basis of private experiences of shame that seem to involve no exposure. This disagreement marks a basic fault line in theorizing about shame. I develop and explore a simple but effective way to shield the Audience (...) Thesis from the private shame objection, by understanding the notion of an audience in a very minimal way. Rather than conceiving of the audience in terms of an other whose appraisal is an element in shame, we can conceive of shame generally as a response to appraisals of the subject—either by others or by the subject herself. On this view, shame requires an audience in the sense that it is not a first-order self-appraisal—like disappointment in or disapproval of oneself—but rather an appraisal of appraisals. This approach yields substantial benefits: it renders the private shame objection harmless; it explains why exposure cases strike us as particularly paradigmatic instances of shame; it clarifies what is happening when we feel shame before appraisals with which we do not agree; it helps to understand how it may be possible to feel shame in the face of neutral or even positive appraisals; and it captures a significant but neglected sense in which shame might be considered a social emotion. (shrink)

The so-called ‘argument from vagueness’ is among the most powerful and innovative arguments offered in support of the view that objects are four-dimensional perdurants. The argument is defective – I submit – and in a number of ways that are worth looking into. But each ‘defect’, each gap in the argument, corresponds to a model of change that is independently problematic and that can hardly be built into the common-sense picture of the world. So once all the gaps of the (...) argument are filled in, the three-dimensionalist is left with the burden of a response that cannot rely on a passive plea for common sense. The argument is not a threat to common sense as such; it is a threat to the three-dimensionalist faithfulness to common sense. (shrink)

Cities are mysteriously attractive. The more we get used to being citizens of the world, the more we feel the need to identify ourselves with a city. Moreover, this need seems in no way distressed by the fact that the urban landscape around us changes continuously: new buildings rise, new restaurants open, new stores, new parks, new infrastructures… Cities seem to vindicate Heraclitus’s dictum: you cannot step twice into the same river; you cannot walk twice through the same city. But, (...) as with the river, we want and need to say that it is the same city we are walking through every day. It is always different, but numerically self-identical. How is that possible? What sort of mysterious thing is a city? The answer, I submit, is that cities aren’t things. They are processes. Like rivers, cities unfold in time just as they extend in space, by having different temporal parts for each time at which they exist. And walking though one part and then again through another is, literally, walking through the same whole. (shrink)

That parthood is a transitive relation is among the most basic principles of classical mereology. Alas, it is also very controversial. In a recent paper, Ingvar Johansson has put forward a novel diagnosis of the problem, along with a corresponding solution. The diagnosis is on the right track, I argue, but the solution is misleading. And once the pieces are properly put together, we end up with a reinforcement of the standard defense of transitivity on behalf of classical mereology.

Standard lore has it that a proper name, or a definite description on its de re reading, is a temporally rigid designator. It picks out the same entity at every time at which it picks out an entity at all. If the entity in question is an enduring continuant then we know what this means, though we are also stuck with a host of metaphysical puzzles concerning endurance itself. If the entity in question is a perdurant then the rigidity claim (...) is trivial, though one is left wondering how it is that different speakers ever manage to pick out one and the same entity when a host of suitable, overlapping candidates are available. But what if the entity in question is neither a continuant nor a perdurant? What if the things we talk about in ordinary language are time‐bound entities that cannot truly be said to persist through time, or stage sequences whose unity resides exclusively in our minds ‐ like the “wave” at the stadium or the characters of a cartoon? In such cases the rigidi‐ ty claim can't be right and a counterpart‐theoretic semantics seems required. Is that bad? I say it isn't. And it had better not be, if that turns out to be the best metaphysical option we have. (shrink)

In order to fulfill ABET requirements, Northern Arizona University’s Civil and Environmental engineering programs incorporate professional ethics in several of its engineering courses. This paper discusses an ethics module in a 3rd year engineering design course that focuses on the design process and technical writing. Engineering students early in their student careers generally possess good black/white critical thinking skills on technical issues. Engineering design is the first time students are exposed to “grey” or multiple possible solution technical problems. To identify (...) and solve these problems, the engineering design process is used. Ethical problems are also “grey” problems and present similar challenges to students. Students need a practical tool for solving these ethical problems. The step-wise engineering design process was used as a model to demonstrate a similar process for ethical situations. The ethical decision making process of Martin and Schinzinger was adapted for parallelism to the design process and presented to students as a step-wise technique for identification of the pertinent ethical issues, relevant moral theories, possible outcomes and a final decision. Students had greatest difficulty identifying the broader, global issues presented in an ethical situation, but by the end of the module, were better able to not only identify the broader issues, but also to more comprehensively assess specific issues, generate solutions and a desired response to the issue. (shrink)

In experimental psychology and neuroscience, technological advances and multisensory research have contributed to gradually dismiss a version of reductionism. Empirical results no longer support a brain model in which distinct “modules” perform discrete functions, but rather, a brain of partially overlapping networks. A similarly changed brain model is extending to psychopathology and clinical psychology, and partly accounts for the problems of reductionism.

This book is a conceptualization of the literary aesthetic in relation to ethics, in particular, an ethics for a concern for the Self. Bringing Gilles Deleuze and Felix Guattari's constructivist thinking into a practical domain, Sabrina Achilles rethinks the ways in which literature is understood and taught. Through an interdisciplinary approach, literature is viewed from the position of a problem without any pre-given frame.

The paper proposes a notion of fairness which overcomes the conflict arising between efficiency and the absence of envy in economies with uncertainty and asymmetrically informed agents. We do it in general economies which include, as particular cases, the main models of differential information economies, providing in this framework a natural competitive equilibrium notion which satisfies the fair criterion. The analysis is conducted allowing the presence of large traders, which may cause the lack of perfect competition.

A critical survey of the main philosophical theories about events and event talk, organized in three main sections: (i) Events and Other Categories (Events vs. Objects; Events vs. Facts; Events vs. Properties; Events vs. Times); (ii) Types of Events (Activities, Accomplishments, Achievements, and States; Static and Dynamic Events; Actions and Bodily Movements; Mental and Physical Events; Negative Events); (iii) Existence, Identity, and Indeterminacy.

What sort of logic do we get if we adopt a supervaluational semantics for vagueness? As it turns out, the answer depends crucially on how the standard notion of validity as truth preservation is recasted. There are several ways of doing that within a supervaluational framework, the main alternative being between “global” construals (e.g., an argument is valid iff it preserves truth-under-all-precisifications) and “local” construals (an argument is valid iff, under all precisifications, it preserves truth). The former alternative is by (...) far more popular, but I argue in favor of the latter, for (i) it does not suffer from a number of serious objections, and (ii) it makes it possible to restore global validity as a defined notion. (shrink)

I focus on three mereological principles: the Extensionality of Parthood (EP), the Uniqueness of Composition (UC), and the Extensionality of Composition (EC). These principles are not equivalent. Nonetheless, they are closely related (and often equated) as they all reflect the basic nominalistic dictum, No difference without a difference maker. And each one of them—individually or collectively—has been challenged on philosophical grounds. In the first part I argue that such challenges do not quite threaten EP insofar as they are either self-defeating (...) or unsupported. In the second part I argue that they hardly undermine the tenability of EC and UC as well. (shrink)

'Her mouth was like the bloom of a rose, when the rose begins to part the lips of its petals. As soon as I saw, I was done for...All my dreams were of Leucippe.' Achilles Tatius' Leucippe and Clitophon is the most bizarre and risqué of the five 'Greek novels' of idealized love between boy and girl that survive from the period of the Roman empire. Stretching the capacity of the genre to its limits, Achilles' narrative covers adultery, (...) violence, evisceration, pederasty, virginity-testing, and an improbable happy ending. Ingenious and sophisticated in conception, Leucippe and Clitophon is in execution at once subtle, stylish, moving, brash, tasteless, and obscene. This new translation aims to capture Achilles' writing in all its exuberant variety. ABOUT THE SERIES: For over 100 years Oxford World's Classics has made available the widest range of literature from around the globe. Each affordable volume reflects Oxford's commitment to scholarship, providing the most accurate text plus a wealth of other valuable features, including expert introductions by leading authorities, helpful notes to clarify the text, up-to-date bibliographies for further study, and much more. (shrink)

In ‘Towards a Solution to the Sorites Paradox’, Graham Priest gives us a new account of the sorites based on fuzzy logic. The novelty lies in the suggestion that truth-value assignments should themselves be treated as fuzzy objects, i.e., objects about which we can make fuzzy identity statements. I argue that Priest’s solution does not have the explanatory force that Priest advocates. That is, it does not explain why we find the existence of a cut-off point counter-intuitive. I also argue (...) that this sort of explanation calls for a general theory that goes beyond the special case of linguistic vagueness, for the phenomenon is at bottom not linguistic. (shrink)

L’Afrique a participé jusqu’a présent, souvent à titre de laboratoire, au développement de la science occidentale. Les défis rencontrés par la théorie critique aujourd’hui sont l’instrumentalisation de la théorie par les gouvernements, la prolifération de pratiques critiques diverses, notamment écologique, et surtout l’émergence d’un capitalisme de l’image et de l’affect. C’est en relation avec la Chine que l’Afrique peut apporter de nouvelles réponses et relever ces défis.

I argue that Universalism (the thesis that mereological composition is unrestricted) entails Extensionalism (the thesis that sameness of composition is sufficient for identity) as long as the parthood relation is transitive and satisfies the Weak Supplementation principle (to the effect that whenever a thing has a proper part, it has another part disjoint from the first).