O presente ensaio não se entende como uma pesquisa crítico-histórica das artes liberais e do seu papel nas universidades medievais, mas antes como uma reflexão filosófica sobre a lógica intrínseca destas artes e a sua interação com a filosofia aristotélica.

Resumo O fundamentalismo é um fenômeno relativamente recente, pelo menos como posição articulada e autoconsciente. O presente artigo apresenta, de início, uma reflexão sobre o surgimento do termo fundamentalismo, em breve resgate histórico. A seguir, identifica e analisa as características principais dessa atitude, tanto na sua configuração psicológica (subjetivismo fechado), como na sua teoria epistemológica implícita (fideísmo radical, fé ou submissão a uma autoridade religiosa como fonte exclusiva ou predominante de certeza epistemológica), na sua hermenêutica (liberalismo na interpretação de escrituras) (...) e na sua maneira de priorizar um ativismo radical, na perspectiva pragmática (tendência a medidas radicais, à militância e até ao terrorismo no prosseguimento efetivo dos seus fins). Aplicando essa análise, mostra que os debates atuais entre cristãos e "neoateus" parecem frequentemente fundamentalistas, mas não apenas do lado religioso da disputa. Talvez o nome de fundamentalista possa ser reivindicado por alguns que ficariam bem surpresos pela nova titulação. A última parte do artigo discute a religião "usada" e "abusada". Palavras-chave: Ateísmo; Fundamentalismo; Filosofia da religião.Fundamentalism is a relatively recent phenomenon, at least as an articulated and self-conscious position. The present article begins by reflecting on the origin of the term fundamentalism in a brief historical review. Then the principal characteristics of this attitude are identified and analyzed, both in its psychological configuration (a closed subjectivity), as well as in its implicit epistemological theory (radical fideism, faith or submission to a religiou authority as exclusive or predominant source of epistemological certitude), its hermeneutics (literalism in the interpretation of scriptures) and its manner of prioritizing radical activisim in its pragmatic perspective (tendency to radical means, to militancy and even terrorism in the effective persecution of its ends). Applying this analysis, current debates between Christians and "neo-atheists" appear frequently fundamentalist, but not only on the religious side of the dispute. Perhaps the name fundamentalist can be claimed even by some who would be quite surprised by the new christening. The last part of the article discusses "used" and "abused" religion. Key words : Atheism; fundamentalism; philosophy of religion. (shrink)

Many upper limb amputees experience an incessant, post-amputation “phantom limb pain” and report that their missing limbs feel paralyzed in an uncomfortable posture. One hypothesis is that efferent commands no longer generate expected afferent signals, such as proprioceptive feedback from changes in limb configuration, and that the mismatch of motor commands and visual feedback is interpreted as pain. Non-invasive therapeutic techniques for treating phantom limb pain, such as mirror visual feedback (MVF), rely on visualizations of postural changes. Advances in neural (...) interfaces for artificial sensory feedback now make it possible to combine MVF with a high-tech “rubber hand” illusion, in which subjects develop a sense of embodiment with a fake hand when subjected to congruent visual and somatosensory feedback. We discuss clinical benefits that could arise from the confluence of known concepts such as MVF and the rubber hand illusion, and new technologies such as neural interfaces for sensory feedback and highly sensorized robot hand testbeds, such as the “BairClaw” presented here. Our multi-articulating, anthropomorphic robot testbed can be used to study proprioceptive and tactile sensory stimuli during physical finger-object interactions. Conceived for artificial grasp, manipulation, and haptic exploration, the BairClaw could also be used for future studies on the neurorehabilitation of somatosensory disorders due to upper limb impairment or loss. A remote actuation system enables the modular control of tendon-driven hands. The artificial proprioception system enables direct measurement of joint angles and tendon tensions while temperature, vibration, and skin deformation are provided by a multimodal tactile sensor. The provision of multimodal sensory feedback that is spatiotemporally consistent with commanded actions could lead to benefits such as reduced phantom limb pain, and increased prosthesis use due to improved functionality and reduced cognitive burden. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Philosophy of Music Education Review 11.2 (2003) 157-169 [Access article in PDF] Praxis and the PossibleThoughts on the Writings of Maxine Greene and Paulo Freire Randall Everett Allsup Columbia University Authors in a recent edition of the Philosophy of Music Education Review have assayed various understandings of praxis within the domain of music learning and teaching. 1 Leadened (perhaps) by history, this six-letter word sustains a multiplicity of (...) meanings. Philosophers as disparate as Aristotle, Marx and Engels, Sartre, Marshall Sahlins, Pierre Bourdieu, and David Elliott have represented praxis as a form of cultural-doing—but in many instances, this is where the similarities end. 2 By stirring up the pot, editor Estelle Jorgensen has sought to lighten history's burden, lest the word praxis suffer ossification, territorial over-use, or (worse) over-simplification.My first encounters with the word praxis were not particularly weighty or didactic. In graduate work with Maxine Greene, I would often hear praxis associated with the arts, with possibilities, openings, or alternatives. Greene does not construct a philosophical stance around the word; rather the conditions for praxis exist within the imagination, within "the capacity to look at things as if they could be otherwise." 3 Recalling Emily Dickinson, Greene writes that "imagination lights the possible's slow fuse." 4 Praxis, I think, is not simply the capacity to imagine alternative scenarios, but is instead the slow burning fuse of possibility and action. [End Page 157] This definition connects nicely with Paulo Freire, the revolutionary pedagogue who is of late often paired with Greene. 5 "Human activity," writes Freire, "consists of action and reflection: it is praxis." 6 The result is action upon our world.I must admit that as a long time fan of Freire (and as a student and friend of Greene), I have often wondered what exactly it means to act upon one's world, to name's one's world—especially as a music educator. Can such action lead to the creation of a love song? Can we say that certain art forms like the blues, be-bop, and jam band are more praxial than others? Might an encounter with Brahms lead to potentially transformative action upon my world?Freirian praxis, for sure, is non-neutral; it occurs when we disavow the traditional separation between abstracted learning and real life. After September 11, for example, my freshman music education majors created a requiem for the missing and the dead. This was the most difficult teaching I have ever facilitated. Before beginning our composition, we listened to several movements from the Faure Requiem, particularly the "In Paradisum": "May the choir of angels receive thee, and with Lazarus, once a beggar, mayst thou have eternal rest." We tried to attain Faure's sense of peace and hope; we admired his assurance that those who were gone were at rest, at peace. And we shared stories, and talked about our feelings and our pain. 7During this time, even the ordinary became infused with meaning. Banal pop music, like Enrique Inglesia's Hero, became a tribute song for many, including myself. Suddenly full of significance and relevance, I attended more closely to the simplicity of his chords and the sotto voce color of his voice: Would you die for the one you love? Hold me in your arms tonight. I wanna be your hero... The fact that this was an ordinary love song written some time before the terrorist attacks does not negate my feelings for it. Like Faure's requiem, it brought me a degree of comfort. In addition, I found solace in the music of Aaron Copland, particularly Quiet City. The open intervals seemed to speak of piece, quietude, and space. The opening trumpet recitative was defiant, but introspective—the English horn felt dark and mournful. I wanted to be back in an earlier New York, like the one that Copland knew and loved.Of course, acting upon our world requires inaction: thinking, perceiving, reflecting, reconceptualizing, connecting. For Freire and Greene, transformational or liberatory practice cannot... (shrink)

The relationship between ethics and economics has never been easy. Opponents in a tug of war, friends in a warm embrace, ships passing in the night—the relationship has been highly variable. In recent years, the friendship model has been gaining credence, particularly among U.S. corporate executives. Increasingly, companies are launching ethics programs, values initiatives, and community involvement activities premised on management’s belief that “Ethics pays.”.

This work is a statement of the belief in humanity's potential to change the world for the better. It argues that people are born with a set of natural rights and that any society that violates those rights is flawed and should be changed.

This paper focuses on the possibility of conceiving a form of ontological nihilism, starting from D. M. Armstrong’s combinatorialism. This possibility has been suggested by Efird and Stoneham, by means of proposing an alternative strategy to the ‘subtraction argument’. They claim that it is possible to sustain such nihilism trough the concepts of construction and totality state of affairs. However, this hypothesis will require the acceptance of non-instanciated universals, that is, platonic universals. Yet this is opposite to requirements that are (...) basic for a combinatorial theory, which should uphold that universals exist only in their instantiations. Then, the assumption of platonic universals results in high costs for a combinatorial naturalism, according to which only exists what is located in space and time. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Protecting Human Research SubjectsThe Office for Protection from Research RisksJoan Paine Porter (bio)The office for Protection from Research Risks (OPRR), located within the National Institutes of Health, has two divisions: Human Subject Protections and Animal Welfare. This article will address the overall responsibilities and current projects relating to human subject protections.OPRR implements the Department of Health and Human Services' (HHS) regulations for the protection of human subjects (45 (...) CFR 46 (1991)). These regulations fulfill the responsibilities set forth in the Public Health Service (PHS) Act with regard to institutional review boards (IRB) and other protections for human subjects involved in biomedical and behavioral research conducted or supported by HHS. The PHS Act also mandates a program to clarify ethical issues in biomedical research involving human subjects and to address violations of the rights of human subjects of research (P.L. 99-158 (1985)).OPRR staff negotiate assurances of compliance with institutions conducting Federally-supported research involving human subjects. The assurances describe the policies and procedures each institution will follow to fulfill the requirements of the regulations, including specific IRB composition and functions.In addition, OPRR cosponsors with the Food and Drug Administration and awardee institutions a series of regional workshops that address human subject protections issues. These are attended by IRB members, institutional officials, research investigators, and others involved with human subject protections.Current ActivitiesNumerous activities are underway in OPRR. The following describes only a few of the initiatives and studies aimed at helping develop policies and procedures to protect human subjects of research."Points to Consider" DocumentsOPRR is currently working with the National Center for Human Genome Research (NCHGR) to develop a "points to consider" document for IRBs and others involved in large genetic pedigree studies. Many questions have emerged [End Page 279] on topics such as how subjects should be recruited, with whom information should be shared about others in the family, social risks of disclosure in employment and insurance, the circumstances and timing of informing subjects about study results, and involvement of children. OPRR and NCHGR hope to assemble a group this fall to review and refine the points that are now being developed jointly. A conference in South Carolina organized by the American Association for Advancement in the Sciences entitled "Project on the Ethical and Legal Implications of Genetic Testing," helped generate advice, and identified some of the more controversial issues in design and conduct of genetic research that involves large groups of family members.Another "points to consider" effort concerns enrollment of women and minorities in research. The PHS has established a policy that members of these groups must be included in research projects unless there are valid scientific justifications for their exclusion which can be made on a project by project basis (ADAMHA/NIH Policy 1990; NIH/ADAMHA Policy 1990). OPRR has commissioned several scholarly papers to help in developing "points to consider" for IRBs as they balance issues of equitable selection of subjects and protection of these selected individuals. Concerns about enrolling women of childbearing age in research that may be harmful to a fetus, and liability issues frequently dominate discussion in this area. The papers will address ethical, legal, and statistical aspects of enrollment of minorities and women as well as provide a description of various practices and roles of IRBs in addressing goals of fair selection of participants and appropriate protections.OPRR is also working with the PHS Panel on Women, Adolescents and Children to produce a document, also in the form of "points to consider" guidance, concerning enrollment of children who are wards of the state in HIV-related research. Subpart D of the regulations for the protection of human subjects (45 CFR 46) specifically addresses research involving children who are wards of the state, including the special protections that must be provided. State and local officials who have authority to enroll foster care children in research have indicated a desire to obtain advice in this area.The IRB GuidebookA major project that is now being completed under contract to the Poynter Center at Indiana University is the IRB Guidebook. It will revise and expand the current IRB Guidebook produced in the 1980s... (shrink)

Non-recent child sexual abuse and child sexual exploitation have received recent attention. Victims often do not report their ordeal at the time the incident occurred, and it is increas...

Different views of life documented for discussion. For 14-16 year-olds.

This article examines the inherently ethical nature of resource allocation in policing. Decision-makers must make trade-offs between values such as efficiency vs. equity, individual vs. collective benefit, and adopt principles of distribution which allocate limited resources fairly. While resource allocation in healthcare has been the subject of extensive discussion in both practitioner and academic literature, ethical resource allocation in policing has received almost no attention. We first consider whether approaches used in healthcare settings would be suitable for policing. Whilst there (...) are some high-level similarities in the relevant ethical considerations and processes—such as aiming for efficient and equitable resource deployment—there are also fundamental differences between the two settings, which we argue necessitate a different approach. These differences include: (1) greater diversity and incommensurability of the benefits of policing, (2) more frequent non-linearity in returns on policing investment, (3) greater “ethical divisibility” (permissible scaling down) of programs in policing, (4) a requirement to assess “need” for policing resources at the collective (rather than individual) level, and (5) clearer primacy of equity considerations in policing. Having drawn out the implications of these differences, we sketch some tentative proposals for ethical resource allocation in policing. (shrink)

In “Subordinating Speech,” Ishani Maitra defends the claim that ordinary instances of hate speech can sometimes constitute subordination. While she accepts that subordinating speech requires authority, she argues that ordinary speakers can acquire this authority via a process of “licensing.” I believe this account is interestingly mistaken, and in this paper I develop an alternative account. In particular, I take issue with what I see as the highly localized character of Maitra’s account, which effectively divorces the subordinating authority of ordinary (...) hate speech from the broader normative context, including social and pragmatic features that I claim play essential roles in subordination. (shrink)

Public health ethicists face two difficult questions. First, what makes something a matter of public health? While protecting citizens from outbreaks of communicable diseases is clearly a matter of public health, is the same true of policies that aim to reduce obesity, gun violence or political corruption? Second, what should the scope of the government’s authority be in promoting public health? May government enact public health policies some citizens reasonably object to or policies that are paternalistic? Recently, some theorists have (...) attempted to address these questions by arguing that something is a matter of public health if and only if it involves a health-related public good, such as clean water or herd immunity. Relatedly, they have argued that appeals to the promotion of public health should only be used to justify the provision of health-related public goods. This public goods conception of public health (PGC) is meant to enjoy advantages over its rivals in three respects: it provides a better definition of public health than rival views, it respects moral disagreement, and it avoids licensing objectionably paternalistic public health policies. We argue, however, that the PGC does just as poorly, or worse, than its rivals in all three respects. (shrink)

It was not until the nineteenth century that Western nations came to replace mutilation, corporal punishment, and banishment as the favored method of criminal punishment with the more humane concept of imprisonment. Even then, however, a convicted inmate was viewed as nothing more than a slave of the state, entitled only to the most basic of human rights and subject to the whim and peril of his jailor's desire. The shift to imprisonment gradually was accompanied by the additional humanitarian demand (...) that prisons avail their charges of some minimum standard of human decency and necessity, which, by the early part of the twentieth century, had come to encompass the provision of medical care. The courts later came to acknowledge this as a fundamental right grounded in the Eighth Amendment's proscription against cruel and unusual punishments.Although the scope of a prisoner's constitutional right to medical care has been the subject of much debate before the courts, the majority of cases have focused only on the circumstances under which the deprivation of care will amount to a violation of the rights guaranteed by the Eighth Amendment. (shrink)

This paper discusses a sequence of extensions ofNFU, Jensen's improvement of Quine's set theory “New Foundations” (NF) of [16].The original theoryNFof Quine continues to present difficulties. After 60 years of intermittent investigation, it is still not known to be consistent relative to any set theory in which we have confidence. Specker showed in [20] thatNFdisproves Choice (and so proves Infinity). Even if one assumes the consistency ofNF, one is hampered by the lack of powerful methods for proofs of consistency and (...) independence such as are available for use withZFC; very clever work has been done with permutation methods, starting with [18] and [5], and exemplified more recently by [14], but permutation methods can only be applied to show the consistency or independence of unstratified sentences (see the definition ofNFUbelow for a definition of stratification). For example, there is no method available to determine whether the assertion “the continuum can be well-ordered” is consistent with or independent ofNF. There is one substantial independence result for an assertion with nontrivial stratified consequences, using metamathematical methods: this is Orey's proof of the independence of the Axiom of Counting fromNF(see below for a statement of this axiom).We mention these difficulties only to reassure the reader of their irrelevance to the present work. Jensen's modification of “New Foundations” (in [13]), which was to restrict extensionality to sets, allowing many non-sets (urelements) with no elements, has almost magical effects. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Ideas 61.1 (2000) 153-156 [Access article in PDF] JHI 2000 Donald R. Kelley It was just sixty years ago that this Journal first made its appearance. Two hundred thirty-nine issues later it continues in a world transformed by war, overpopulation, cultural shocks, scientific and technological transformations, globalization, the avalanche of information produced by electronic exchange, and "the acceleration of just about everything." Yet despite (...) these factors and the strains of postmodernism and cultural alterity, 1 it has not entirely lost touch with its intellectual innocence, faith in humanistic learning, and reliance on enlightened reason. This continuity itself may seem something of a novelty in an age of distrust of history and "rage against reason," but for some of us it reflects the critical spirit and intellectual context out of which such ostensibly subversive attitudes emerge, and reemerge. To some extent, moreover, the diversity of opinions about intellectual traditions is the result of specializations, and special interests, which have changed the climate of opinion since the time of Arthur O. Lovejoy, principal founder of the JHI. Lovejoy's vision was super- as well as inter-disciplinary; that is, he intended to include in the agenda of "history of ideas" particular areas of history of philosophy, literature, art, science, social science, etc., as well as the larger intellectual and cultural areas into which these disciplinary histories extend. 2 These days, however, all these disciplinary traditions have their own more specialized journals and do not need to seek a vehicle in publications of more general interest. What remains then, for the most part, is the interdisciplinary arena in which larger questions of human experience should be posed--and for Lovejoy, as for so many epigones and critics, the main "larger question" is not only one of value but also of specifically historical inquiry. 3 [End Page 153]The first issue of the JHI appeared under the editorship of Lovejoy (with Philip Wiener, who did most of the editorial work, as "managing editor"), assisted by a committee that included two historians, three philosophers, four literary scholars, and one political theorist; namely, Crane Brinton, Gilbert Chinard, Morris Cohen, Francis Coker, Richard McKeon, Perry Miller, Marjory Nicolson, J. H. Randall, J. Salwyn Schapiro, and Louis Wright. The first volume featured contributions from each of these, including Wiener, except for Miller (vol. 2), Shapiro (vol. 3), McKeon (vol. 8), and Coker. 4 During the first five years there were also reviews of books by Dilthey (by Horace L. Friess), Croce (Schapiro), Mannheim (Randall), Perry Miller (Herbert W. Schneider), Van Wyck Brooks (Miller) and Alfred Kazin (F. O. Matthiessen). In this period the disciplinary distribution of articles was not markedly different from that of the past five years, nor was the ranking of fields by number of submissions. 5 Most important, the original interdisciplinary thrust of Lovejoy's agenda, if not his attachment to the spiritualist currency of "unit-ideas," has survived growing specialization.At first the history of ideas was pursued largely in the shadow of the history of philosophy; for in this field, according to Lovejoy, "is to be found the common seed-plot, the locus of initial manifestation in writing, of the greater number of the more fundamental and pervasive ideas, and especially of the ruling preconceptions, which manifest themselves in other regions of intellectual history." 6 Lovejoy's concern was always with concepts, especially "-isms," and yet in some ways he anticipated the "linguistic turn" of the later years of this century, pointing out in particular "the role of semantic shifts, ambiguities, and confusions, in the history of thought and taste," for "nearly all of the great catchwords have been equivocal--or rather, multivocal." 7 If ideas could be given stable definitions, they were nonetheless often, in the context of language, in conflict, even in the mind of a single thinker; for such was the "anomaly of knowledge." 8 So Lovejoy was at pains to distinguish the various meanings of [End Page 154] catch-words like "nature," "perfectibility," "romanticism," "progress," and "pragmatism," not to mention more inflammatory terms of wartime ideological debate.But... (shrink)

We analyze the representation of binary relations in general, and in particular of functions and of total antisymmetric relations, in monadic third order logic, that is, the simple typed theory of sets with three types. We show that there is no general representation of functions or of total antisymmetric relations in this theory. We present partial representations of functions and of total antisymmetric relations which work for large classes of these relations, and show that there is an adequate representation of (...) cardinality in this theory. The relation of our work to similar work by Henrard and Allen Hazen is discussed. This work can be understood as part of a program of assessing the capabilities of weak logical frameworks: our results are applicable for example, to the framework in David Lewis’s Parts of Classes. (shrink)

Sense of agency refers to the ability to influence one’s functioning and environment, relating to self-efficacy and wellbeing. In youth, agency may be challenged by external demands or redefinition of self-image. Music, having heightened relevance for the young, has been argued to provide feelings of self-agency for them. Yet, there is little empirical research on how music impacts adolescents’ daily sense of agency. The current study investigated whether music listening influences adolescents’ perceived agency in everyday life and which individual and (...) contextual determinants would explain such an influence. Participants were 44 adolescents (48% female, 36% with training in music, mean age 14), recruited through local schools. The mobile Experience Sampling app MuPsych was used to collect brief self-reports of personal music listening experiences during daily life. This method assessed the change in the listener’s perceived control over both their emotional states (Internal Agency), and their external environment (External Agency), over five minutes of music listening. Also measured were changes in mood states, and contextual variables (social situation, concurrent activity, and reason for listening). The impact of music on the sense of agency was analyzed using multilevel structural equation modelling. There was no general increase of agency across all music episodes, but agency fluctuations were determined by specific individual and contextual factors. Music training predicted increases in both internal and external agency during music listening, and external agency increased more for male participants. Certain context features such as listening alone, paying attention to music, and listening for enjoyment were related to greater increases in agency. Increases in agency significantly correlated with increases in mood valence. Our findings confirmed the plasticity and situational embeddedness of the sense of agency. Music indeed can support agency, but the impact is dependent on specific individual and contextual factors. Sense of agency can be seen as a health resource and significant part of youth development, and current findings provide new insight into when and by whom such affordance is likely to be employed. (shrink)

Systems TRC and TRCU of illative combinatory logic are introduced and shown to be equivalent in consistency strength and expressive power to Quine's set theory ‘New Foundations’ and the fragment NFU + Infinity of NF described by Jensen, respectively. Jensen demonstrated the consistency of NFU + Infinity relative to ZFC; the question of the consistency of NF remains open. TRC and TRCU are presented here as classical first-order theories, although they can be presented as equational theories; they are not constructive.

Two key research issues in the field of causal learning are how people acquire causal knowledge when observing data that are presented sequentially, and the level of abstraction at which learning takes place. Does sequential causal learning solely involve the acquisition of specific cause-effect links, or do learners also acquire knowledge about abstract causal constraints? Recent empirical studies have revealed that experience with one set of causal cues can dramatically alter subsequent learning and performance with entirely different cues, suggesting that (...) learning involves abstract transfer, and such transfer effects involve sequential presentation of distinct sets of causal cues. It has been demonstrated that pre-training can modulate classic causal learning phenomena such as forward and backward blocking. To account for these effects, we propose a Bayesian theory of sequential causal learning. The theory assumes that humans are able to consider and use several alternative causal generative models, each instantiating a different causal integration rule. Model selection is used to decide which integration rule to use in a given learning environment in order to infer causal knowledge from sequential data. Detailed computer simulations demonstrate that humans rely on the abstract characteristics of outcome variables to select a causal integration rule, which in turn alters causal learning in a variety of blocking and overshadowing paradigms. When the nature of the outcome variable is ambiguous, humans select the model that yields the best fit with the recent environment, and then apply it to subsequent learning tasks. Based on sequential patterns of cue-outcome co-occurrence, the theory can account for a range of phenomena in sequential causal learning, including various blocking effects, primacy effects in some experimental conditions, and apparently abstract transfer of causal knowledge. (shrink)

The usual construction of models of NFU (New Foundations with urelements, introduced by Jensen) is due to Maurice Boffa. A Boffa model is obtained from a model of (a fragment of) Zermelo–Fraenkel with Choice (ZFC) with an automorphism which moves a rank: the domain of the Boffa model is a rank that is moved. “Most” elements of the domain of the Boffa model are urelements in terms of the interpreted NFU. The main result of this paper is that the restriction (...) of the membership relation of the original model of set theory with automorphism to the domain of the Boffa model is first-order definable in the language of NFU. In particular, all information about the extensions in the original model of the urelements of the model of NFU is definable in terms of NFU. A corollary (answering a question of Thomas Forster) is that the urelements in a Boffa model are not homogeneous. (shrink)

This paper extends the results of an earlier paper by the author . New subsystems of the combinatory logic TRC shown in that paper to be equivalent to NF are introduced; these systems are analogous to subsystems of NF with predicativity restrictions on set comprehension introduced and shown to be consistent by Crabbé. For one of these systems, an exact equivalence in consistency strength and expressive power with the analogous subsystem of NF is established.

We show in (the usual set theory without Choice) that for any set X, the collection of sets Y such that each element of the transitive closure of is strictly smaller in size than X (the collection of sets hereditarily smaller than X) is a set. This result has been shown by Jech in the case (where the collection under consideration is the set of hereditarily countable sets).