A common objection to Quine's set theory "New Foundations" is that it is inadequately motivated because the restriction on comprehension which appears to avert paradox is a syntactical trick. We present a semantic criterion for determining whether a class is a set which motivates NF.

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)

This paper presents an exposition of subsystems and of Quine's , originally defined and shown to be consistent by Crabbé, along with related systems and of type theory. A proof that (and so ) interpret the ramified theory of types is presented (this is a simplified exposition of a result of Crabbé). The new result that the consistency strength of is the same as that of is demonstrated. It will also be shown that cannot be finitely axiomatized (as can and (...) ). (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)

An ω-model (a model in which all natural numbers are standard) of the predicative fragment of Quine's set theory "New Foundations" (NF) is constructed. Marcel Crabbe has shown that a theory NFI extending predicative NF is consistent, and the model constructed is actually a model of NFI as well. The construction follows the construction of ω-models of NFU (NF with urelements) by R. B. Jensen, and, like the construction of Jensen for NFU, it can be used to construct α-models for (...) any ordinal α. The construction proceeds via a model of a type theory of a peculiar kind; we first discuss such "tangled type theories" in general, exhibiting a "tangled type theory" (and also an extension of Zermelo set theory with Δ 0 comprehension) which is equiconsistent with NF (for which the consistency problem seems no easier than the corresponding problem for NF (still open)), and pointing out that "tangled type theory with urelements" has a quite natural interpretation, which seems to provide an explanation for the more natural behaviour of NFU relative to the other set theories of this kind, and can be seen anachronistically as underlying Jensen's consistency proof for NFU. (shrink)

Related Works: Original Paper: M. Randall Holmes. Strong Axioms of Infinity in NFU. J. Symbolic Logic, Volume 66, Issue 1 , 87--116. Project Euclid: euclid.jsl/1183746361.

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)

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.

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)

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).

Three systems of double extension set theory have been proposed by Andrzej Kisielewicz in two papers. In this paper, it is shown that the two stronger systems are inconsistent, and that the third, weakest system does not admit extensionality for general sets or the use of general sets as parameters in its comprehension scheme. The parameter-free version of the comprehension principle of double extension set theory is also shown to be inconsistent with extensionality. The definitions of the systems and a (...) self-contained exposition of their properties is given, sufficient to develop the inconsistency proofs. (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.

This paper described a formal theory of type judgments for propositional logic notations of PM; I felt the need of my own automated type checker to check their examples. The type checker I wrote did indeed serve to help me referee the paper, but also took a rather different approach to notation and typing for propositional functions of PM, which proved worth writing up independently in our own paper: Holmes, M. Randall, “Polymorphic type– checking for the ramified theory (...) of types of Principia Mathematica”, in Fairouz Kamareddine, ed., Thirty–five Years of Automating Mathematics, Kluwer, 2003, pp. 173-215. (shrink)

This is a brief report on results reported at length in our paper [2], made for the purpose of a presentation at the workshop to be held in November 2011 in Cambridge on the Principia Mathematica of Russell and Whitehead ([?], hereinafter referred to briefly as PM ). That paper grew out of a reading of the paper [3] of Kamareddine, Nederpelt, and Laan. We refereed this paper and found it useful for checking their examples to write our own independent (...) computer type-checker for the type system of PM ([1]), which led us to think carefully about formalization of the language and the type system of PM A modern mathematical logician reading PM finds that it is not completely formalized in a modern sense. The type theory in particular is inarguably not formalized, as no notation for types is given at all! In PM itself, the only type annotations which appear are occasional numerical indices indicating order; the type notation we use here extends one introduced later by Ramsey. The authors of PM regard the absence of explicit indications of type as a virtue of their system: they call it “systematic ambiguity”; modern computer scientists refer to this as “polymorphism”. The language of PM is also not completely formalized, and it is typographically inconvenient for computer software to which ASCII input is to be given. The notation of PM for abstractions (propositional functions) does not use head binders; the order of the arguments of a complex expression is determined by the alphabetical order of the bound variables. For example ˆa < ˆb is the “less than” relation while ˆb < ˆa is the “greater than” relation (this is indicated by the alphabetical order of the variables). In PM , the fact that a variable is bound in a propositional function is indicated by circumflexing it. Variables bound by quantifiers are not circumflexed. A feature of the notation of [3], carried over into ours, is that no circumflexes are used: notations for propositions and the corresponding propositional functions are identical.. (shrink)

A common objection to Quine’s set theory “New Foundations” is that it is inadequately motivated because the restriction on comprehension which appears to avert paradox is a syntactical trick. We present a semantic criterion for determining whether a class is a set (a kind of symmetry) which motivates NF.

Andrzej Kisielewicz has proposed three systems of double extension set theory of which we have shown two to be inconsistent in an earlier paper. Kisielewicz presented an argument that the remaining system interprets ZF, which is defective: it actually shows that the surviving possibly consistent system of double extension set theory interprets ZF with Separation and Comprehension restricted to 0 formulas. We show that this system does interpret ZF, using an analysis of the structure of the ordinals.

Related Works: Original Paper: M. Randall Holmes. Strong Axioms of Infinity in NFU. J. Symbolic Logic, Volume 66, Issue 1, 87--116. Project Euclid: euclid.jsl/1183746361.

At the centenary of Frank H. Knight’s Risk, Uncertainty, and Profit (1921), we explore the continuing relevance of Knightian uncertainty to the theory and practice of entrepreneurship. There are three challenges facing such assessment. First, RUP is complex and difficult to interpret. The key but neglected element of RUP is that Knight’s account is not solely about risk and uncertainty as states of nature, but about how an agent’s beliefs about uncertain outcomes and confidence in those beliefs guide their choices. (...) Second, RUP is Knight’s only effort in this area. His subsequent career led elsewhere, so he did not engage with subsequent interpretations of this work. Third, much of the current literature emphasizes that decisions must be different under the two states of nature with a consequent misunderstanding of entrepreneurial agency. This paper addresses each challenge in sequence. First, we explicate Knight’s (1921) approach and explain why that approach is murky. Second, as a complement to Knight’s interpretation, we examine Frank P. Ramsey’s approach to subjective probabilities to help clarify Knight’s murky approach. What links Knight and Ramsey is a shared pragmatism about entrepreneurial agency under uncertainty that depends upon the beliefs about, and confidence in, their judgments of possible outcomes. This Knight-Ramsey approach does not require actor’s behaviors to be determined by the class of uncertain environment (whether risk, uncertainty, or ambiguity) they face. We focus on the response by the entrepreneur to the existence of uncertainty in all its forms. We argue that this reductive account provides a foundation to examine common problems in management, including managerial hubris, the interaction between entrepreneurs and venture capitalists, and the need for experimentation (such as prototyping and market research) in advance of new product and venture launches. Third, we critique current literature that favors epistemic purism about the ontology of risk and uncertainty and ignores Knight-Ramsey pragmatism in meeting uncertainty, such as using formal and informal institutions for uncertainty mitigation. Our account locates Frank Knight’s subtleties in entrepreneurial behavior firmly in the literature on entrepreneurial agency a century later. (shrink)

At the centenary of Frank H. Knight’s Risk, Uncertainty, and Profit, we explore the continuing relevance of Knightian uncertainty to the theory and practice of entrepreneurship. There are three challenges facing such assessment. First, RUP is complex and difficult to interpret. The key but neglected element of RUP is that Knight’s account is not solely about risk and uncertainty as states of nature, but about how an agent’s beliefs about uncertain outcomes and confidence in those beliefs guide their choices. Second, (...) RUP is Knight’s only effort in this area. His subsequent career led elsewhere, so he did not engage with subsequent interpretations of this work. Third, much of the current literature emphasizes that decisions must be different under the two states of nature with a consequent misunderstanding of entrepreneurial agency. This paper addresses each challenge in sequence. First, we explicate Knight’s approach and explain why that approach is murky. Second, as a complement to Knight’s interpretation, we examine Frank P. Ramsey’s approach to subjective probabilities to help clarify Knight’s murky approach. What links Knight and Ramsey is a shared pragmatism about entrepreneurial agency under uncertainty that depends upon the beliefs about, and confidence in, their judgments of possible outcomes. This Knight-Ramsey approach does not require actor’s behaviors to be determined by the class of uncertain environment they face. We focus on the response by the entrepreneur to the existence of uncertainty in all its forms. We argue that this reductive account provides a foundation to examine common problems in management, including managerial hubris, the interaction between entrepreneurs and venture capitalists, and the need for experimentation in advance of new product and venture launches. Third, we critique current literature that favors epistemic purism about the ontology of risk and uncertainty and ignores Knight-Ramsey pragmatism in meeting uncertainty, such as using formal and informal institutions for uncertainty mitigation. Our account locates Frank Knight’s subtleties in entrepreneurial behavior firmly in the literature on entrepreneurial agency a century later. (shrink)

The eponymous protagonist of Okja is an adorable “super-pig,” larger than an ordinary pig not only in size but also in heart and mind. The film explores and interrogates different ways of seeing Okja, different portraits of Okja’s moral status, as philosophers would put it. To the Mirando Corporation, Okja has no moral status. She is a mere product to be used as they see fit. To the Animal Liberation Front, Okja is a dramatic symbol of animals everywhere who are (...) mistreated and deprived of their rights. She represents a rare opportunity to further their political cause. To the young girl Mija, Okja is unquestionably a person, a member of her family, her lifelong companion and friend. The film subjects both the Mirando Corporation and the Animal Liberation Front to satirical critique and invites the audience to come to see Okja as Mija does. So what is the moral status of an animal like Okja? Philosophers like Peter Singer have argued on behalf of animals, super or not, and protested the ways they are treated, often focused on factory farming and the use of animals in scientific research. Like these philosophical animal advocates, this film pushes human beings to expand the boundaries of the moral community beyond their own species. Simply put, Okja argues that animals matter. (shrink)

This study examines the impact of a social desirability response bias as a personality characteristic (self-deception and impression management) and as an item characteristic (perceived desirability of the behavior) on self-reported ethical conduct. Findings from a sample of college students revealed that self-reported ethical conduct is associated with both personality and item characteristics, with perceived desirability of behavior having the greatest influence on self-reported conduct. Implications for research in business ethics are drawn, and suggestions are offered for reducing the effects (...) of a socially desirable response bias. (shrink)

Examines the work of Kierkegaard as an ironist, reevaluating the works he penned under pseudonyms to show both their ironic character and the serious purpose that informed the deception Kierkegaard carried out.

The study assesses how a social desirability (SD) bias influences the relationship between several independent and dependent variables commonly investigated in ethics research. The effect of a SD bias was observed when a questionnaire was administered under varying conditions of anonymity and with different measurement techniques for the SD construct. Findings reveal that a SD bias is present in the majority of relationships studied, and it most frequently plays a moderating role. While the measure of SD influences the strength and (...) type of relationship, condition of anonymity has relatively little effect on the level of SD. (shrink)

Background: The factors influencing parents’ willingness to enroll their children in biobanks are poorly understood. This study sought to assess parents’ willingness to enroll their children, and their perceived benefits, concerns, and information needs under different consent and data-sharing scenarios, and to identify factors associated with willingness. Methods: This large, experimental survey of patients at the 11 eMERGE Network sites used a disproportionate stratified sampling scheme to enrich the sample with historically underrepresented groups. Participants were randomized to receive one of (...) three consent and data-sharing scenarios. Results: In total, 90,000 surveys were mailed and 13,000 individuals responded (15.8% response rate). 5737 respondents were parents of minor children. Overall, 55% (95% confidence interval 50–59%) of parents were willing to enroll their youngest minor child in a hypothetical biobank; willingness did not differ between consent and data-sharing scenarios. Lower educational attainment, higher religiosity, lower trust, worries about privacy, and attitudes about benefits, concerns, and information needs were independently associated with less willingness to allow their child to participate. Of parents who were willing to participate themselves, 25% were not willing to allow their child to participate. Being willing to participate but not willing to allow one’s child to participate was independently associated with multiple factors, including race, lower educational attainment, lower annual household income, public health care insurance, and higher religiosity. Conclusions: Fifty-five percent of parents were willing to allow their youngest minor child to participate in a hypothetical biobank. Building trust, protecting privacy, and addressing attitudes may increase enrollment and diversity in pediatric biobanks. (shrink)

What constitutes a scientific discovery? What role do discoveries play in science, its dynamics and social practices? Must every discovery be attributed to an individual discoverer (or a small number of discoverers)? The paper explores these questions by first critically examining extant philosophical explications of scientific discovery—the models of scientific discovery, propounded by Kuhn, McArthur, Hudson, and Schindler. As a simple, natural and powerful alternative, we proffer the “change-driver model”: in a nutshell, it takes discoveries to be cognitive scientific results (...) that have epistemically advanced science. The model overcomes the shortcomings of its precursors, whilst preserving their insights. We demonstrate its intensional and extensional superiority, especially with respect to the link between scientific discoveries and the dynamics of science, as well as the award system of science. Both as an illustration, and as an application to a recent scientific and political controversy, we apply the considered models of discovery to one of the most momentous discoveries of science: the expansion of the universe. We oppose the 2018 proposal of the International Astronomers’ Union as too simplistic vis-`a-vis the historical complexity of the episode. The change- driver model yields a more nuanced and circumspect verdict: (i) The redshift-distance relation shouldn’t be named the “Hubble-Lemaître Law”, but “Slipher-Lundmark-Hubble-Humason Law”; (ii) Its interpretation in terms of an expanding universe, however, Lemaître ought to be given credit for; but (iii) The establishment of the expansion of the universe, as an evidentially sufficiently warranted result, is a communal achievement, emerging in the 1950s or 1960s. (shrink)

Quarterly publication discussing various topics in environmental ethics, including features, discussion papers, book reviews, editorial commentaries, and other text related to environmental philosophies. Some issues also include announcements and other news related to the environmental studies community.

This chapter contains section titled: Should We Stay or Should We Go Now? Frak the Numbers! Saving Starbuck? The Mark of Cain “Evil Men in the Gardens of Paradise?” Sacrifice Notes.

Aquinas talks about a topic with which fans of Sons of Anarchy are all too familiar: homicide and murder. For Aquinas, murder is the private, intentional homicide of an innocent human being and always wrong. This is called the Murder Principle. Murder is the killing of an innocent. This seems to leave the Sons—a private organization rather than an arm of the state—without the legitimate authority to take a life. Watching Sons of Anarchy makes us increasingly suspicious of the moral (...) authority of the state and its agents, who hardly seem like the competent “doctors” of Aquinas's amputation analogy. Contemporary philosophers have articulated the Doctrine of Double Effect (DDE), the gist of which states that it is sometimes morally permissible to bring about as a foreseen but unintended side‐effect what one would not be permitted to bring about intentionally. Aquinas's Murder Principle condemns much of what the Sons do. (shrink)

In the early days, before Superman's full array of superpowers “developed,” Clark Kent's reporter persona was necessary for gathering information. Although he was pretty tough and fast, Superman didn't yet have the flight, the super‐hearing, the super‐vision, or the super‐intelligence that he would later have. The strategies that explain why a mere mortal or even a Golden Age Superman might not be up to meeting the demands of the S‐principle full time won’t apply to today's Superman. We may face a (...) choice: either we say goodbye to Clark or we abandon the S‐Principle. Clark Kent and Superman: We've had glimpses of worlds in which one exists without the other. But they're both normally there, and occasionally at odds. Jor‐El wants Superman to embrace his destiny as Earth's savior, Martha wants Clark to find a life for himself, and Lex wants Superman to leave and let humanity find its own path. (shrink)

There is a paucity of research into the prevalence of academic dishonesty within Canada compared to other countries. Recently, there has been a call for a better understanding of the particular characteristics of educational integrity in Canada so that Canada can more meaningfully contribute to current discussions surrounding academic integrity. Here, we present findings from student and faculty surveys conducted within a medium-sized Canadian university. These surveys probed perceptions towards, and experiences with, academic dishonesty, in which we aimed to understand (...) how students and faculty regarded academically dishonest practices during their postsecondary careers. We also aimed to understand how often students engaged in, and faculty had witnessed, academic dishonesty, whether or not witnessing incidents of academic dishonesty corresponded with gender, year of experience, highest level of educational attainment, discipline, or their personal perceptions towards the importance of academic honesty, and whether students had been adequately taught what constitutes academic dishonesty. We found that an overwhelming majority of students viewed academic honesty as important, and that most students reported not engaging in academic dishonesty themselves despite 45.8% reporting that they had witnessed others engage in academic dishonesty. We also found that students were more likely to witness cheating as their postsecondary experience increased, that witnessing varied across disciplines and educational attainment, and that witnessing varied with student perceptions. However, we found no such patterns in faculty responses, but found that faculty are split on whether or not they believe incidents of academic honesty are increasing. (shrink)

Quarterly publication discussing various topics in environmental ethics, including features, discussion papers, book reviews, editorial commentaries, and other text related to environmental philosophies. Some issues also include announcements and other news related to the environmental studies community.

Quarterly publication discussing various topics in environmental ethics, including features, discussion papers, book reviews, editorial commentaries, and other text related to environmental philosophies. Some issues also include announcements and other news related to the environmental studies community.

Quarterly publication discussing various topics in environmental ethics, including features, discussion papers, book reviews, editorial commentaries, and other text related to environmental philosophies. Some issues also include announcements and other news related to the environmental studies community.