The class forcing theorem, which asserts that every class forcing notion ${\mathbb {P}}$ admits a forcing relation $\Vdash _{\mathbb {P}}$, that is, a relation satisfying the forcing relation recursion—it follows that statements true in the corresponding forcing extensions are forced and forced statements are true—is equivalent over Gödel–Bernays set theory $\text {GBC}$ to the principle of elementary transfinite recursion $\text {ETR}_{\text {Ord}}$ for class recursions of length $\text {Ord}$. It is also equivalent to the existence of truth predicates for the (...) infinitary languages $\mathcal {L}_{\text {Ord},\omega }$, allowing any class parameter A; to the existence of truth predicates for the language $\mathcal {L}_{\text {Ord},\text {Ord}}$ ; to the existence of $\text {Ord}$ -iterated truth predicates for first-order set theory $\mathcal {L}_{\omega,\omega }$ ; to the assertion that every separative class partial order ${\mathbb {P}}$ has a set-complete class Boolean completion; to a class-join separation principle; and to the principle of determinacy for clopen class games of rank at most $\text {Ord}+1$. Unlike set forcing, if every class forcing notion ${\mathbb {P}}$ has a forcing relation merely for atomic formulas, then every such ${\mathbb {P}}$ has a uniform forcing relation applicable simultaneously to all formulas. Our results situate the class forcing theorem in the rich hierarchy of theories between $\text {GBC}$ and Kelley–Morse set theory $\text {KM}$. (shrink)

We show that the theory, consisting of the usual axioms of but with the power set axiom removed—specifically axiomatized by extensionality, foundation, pairing, union, infinity, separation, replacement and the assertion that every set can be well‐ordered—is weaker than commonly supposed and is inadequate to establish several basic facts often desired in its context. For example, there are models of in which ω1 is singular, in which every set of reals is countable, yet ω1 exists, in which there are sets of (...) reals of every size, but none of size, and therefore, in which the collection axiom fails; there are models of for which the Łoś theorem fails, even when the ultrapower is well‐founded and the measure exists inside the model; there are models of for which the Gaifman theorem fails, in that there is an embedding of models that is Σ1‐elementary and cofinal, but not elementary; there are elementary embeddings of models whose cofinal restriction is not elementary. Moreover, the collection of formulas that are provably equivalent in to a Σ1‐formula or a Π1‐formula is not closed under bounded quantification. Nevertheless, these deficits of are completely repaired by strengthening it to the theory, obtained by using collection rather than replacement in the axiomatization above. These results extend prior work of Zarach. (shrink)

The generic Vopěnka principle, we prove, is relatively consistent with the ordinals being non-Mahlo. Similarly, the generic Vopěnka scheme is relatively consistent with the ordinals being definably non-Mahlo. Indeed, the generic Vopěnka scheme is relatively consistent with the existence of a \-definable class containing no regular cardinals. In such a model, there can be no \-reflecting cardinals and hence also no remarkable cardinals. This latter fact answers negatively a question of Bagaria, Gitman and Schindler.

One of the numerous characterizations of a Ramsey cardinal κ involves the existence of certain types of elementary embeddings for transitive sets of size κ satisfying a large fragment of ZFC. We introduce new large cardinal axioms generalizing the Ramsey elementary embeddings characterization and show that they form a natural hierarchy between weakly compact cardinals and measurable cardinals. These new axioms serve to further our knowledge about the elementary embedding properties of smaller large cardinals, in particular those still consistent with (...) V = L. (shrink)

If ZFC is consistent, then the collection of countable computably saturated models of ZFC satisfies all of the Multiverse Axioms of Hamkins.

Some 40 years ago, Dana Scott proved that every countable Scott set is the standard system of a model of PA. Two decades later, Knight and Nadel extended his result to Scott sets of size ω₁. Here, I show that assuming the Proper Forcing Axiom (PFA), every A-proper Scott set is the standard system of a model of PA. I define that a Scott set X is proper if the quotient Boolean algebra X/Fin is a proper partial order and A-proper (...) if X is additionally arithmetically closed. I also investigate the question of the existence of proper Scott sets. (shrink)

We survey recent developments in the theory of class forcing for- malized in the second-order set-theoretic setting.

I introduced the notions of proper and piecewise proper families of reals to make progress on a long standing open question in the field of models of Peano Arithmetic [5]. A family of reals is proper if it is arithmetically closed and its quotient Boolean algebra modulo the ideal of finite sets is a proper poset. A family of reals is piecewise proper if it is the union of a chain of proper families each of whom has size ≤ ω1.Here, (...) I investigate the question of the existence of proper and piecewise proper families of reals of different cardinalities. I show that it is consistent relative to ZFC to have continuum many proper families of cardinality ω1 and continuum many piecewise proper families of cardinality ω2. (shrink)

We show that there is a [Formula: see text]-model of second-order arithmetic in which the choice scheme holds, but the dependent choice scheme fails for a [Formula: see text]-assertion, confirming a conjecture of Stephen Simpson. We obtain as a corollary that the Reflection Principle, stating that every formula reflects to a transitive set, can fail in models of [Formula: see text]. This work is a rediscovery by the first two authors of a result obtained by the third author in [V. (...) G. Kanovei, On descriptive forms of the countable axiom of choice, Investigations on nonclassical logics and set theory, Work Collect., Moscow, 3-136 ]. (shrink)

We construct a variety of inner models exhibiting features usually obtained by forcing over universes with large cardinals. For example, if there is a supercompact cardinal, then there is an inner model with a Laver indestructible supercompact cardinal. If there is a supercompact cardinal, then there is an inner model with a supercompact cardinal κ for which 2κ = κ+, another for which 2κ = κ++ and another in which the least strongly compact cardinal is supercompact. If there is a (...) strongly compact cardinal, then there is an inner model with a strongly compact cardinal, for which the measurable cardinals are bounded below it and another inner model W with a strongly compact cardinal κ, such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H^{V}_{\kappa^+} \subseteq {\rm HOD}^W}$$\end{document}. Similar facts hold for supercompact, measurable and strongly Ramsey cardinals. If a cardinal is supercompact up to a weakly iterable cardinal, then there is an inner model of the Proper Forcing Axiom and another inner model with a supercompact cardinal in which GCH + V = HOD holds. Under the same hypothesis, there is an inner model with level by level equivalence between strong compactness and supercompactness, and indeed, another in which there is level by level inequivalence between strong compactness and supercompactness. If a cardinal is strongly compact up to a weakly iterable cardinal, then there is an inner model in which the least measurable cardinal is strongly compact. If there is a weakly iterable limit δ of <δ-supercompact cardinals, then there is an inner model with a proper class of Laver-indestructible supercompact cardinals. We describe three general proof methods, which can be used to prove many similar results. (shrink)

Ehrenfeucht’s lemma asserts that whenever one element of a model of Peano arithmetic is definable from another, they satisfy different types. We consider here the analogue of Ehrenfeucht’s lemma for models of set theory. The original argument applies directly to the ordinal-definable elements of any model of set theory, and, in particular, Ehrenfeucht’s lemma holds fully for models of set theory satisfying V=HOD. We show that the lemma fails in the forcing extension of the universe by adding a Cohen real. (...) We go on to formulate a scheme of natural parametric generalizations of Ehrenfeucht’s lemma, namely, the principles of the form EL, which asserts that P-definability from A implies Q-discernibility. We also consider various analogues of Ehrenfeucht’s lemma obtained by using algebraicity in place of definability, where a set b is algebraic in a if it is a member of a finite set definable from a. Ehrenfeucht’s lemma holds for the ordinal-algebraic sets, we prove, if and only if the ordinal-algebraic and ordinal-definable sets coincide. Using a similar analysis, we answer two open questions posed earlier by the third author and C. Leahy, showing that algebraicity and definability need not coincide in models of set theory and the internal and external notions of being ordinal algebraic need not coincide. (shrink)

We show that the analogues of the embedding theorems of [3], proved for the countable models of set theory, do not hold when extended to the uncountable realm of ω1‐like models of set theory. Specifically, under the ⋄ hypothesis and suitable consistency assumptions, we show that there is a family of many ω1‐like models of, all with the same ordinals, that are pairwise incomparable under embeddability; there can be a transitive ω1‐like model of that does not embed into its own (...) constructible universe; and there can be an ω1‐like model of whose structure of hereditarily finite sets is not universal for the ω1‐like models of set theory. (shrink)

The stable core, an inner model of the form $\langle L[S],\in, S\rangle $ for a simply definable predicate S, was introduced by the first author in [8], where he showed that V is a class forcing extension of its stable core. We study the structural properties of the stable core and its interactions with large cardinals. We show that the $\operatorname {GCH} $ can fail at all regular cardinals in the stable core, that the stable core can have a discrete (...) proper class of measurable cardinals, but that measurable cardinals need not be downward absolute to the stable core. Moreover, we show that, if large cardinals exist in V, then the stable core has inner models with a proper class of measurable limits of measurables, with a proper class of measurable limits of measurable limits of measurables, and so forth. We show this by providing a characterization of natural inner models $L[C_1, \dots, C_n]$ for specially nested class clubs $C_1, \dots, C_n$, like those arising in the stable core, generalizing recent results of Welch [29]. (shrink)

Remarkable cardinals were introduced by Schindler, who showed that the existence of a remarkable cardinal is equiconsistent with the assertion that the theory of L\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${L}$$\end{document} is absolute for proper forcing :176–184, 2000). Here, we study the indestructibility properties of remarkable cardinals. We show that if κ is remarkable, then there is a forcing extension in which the remarkability of κ becomes indestructible by all <κ-closed ≤κ-distributive forcing and all two-step iterations of (...) the form Add∗R˙\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${Add*\dot{\mathbb R}}$$\end{document}, where R˙\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\dot{\mathbb R}}$$\end{document} is forced to be <κ-closed and ≤κ-distributive. In the process, we introduce the notion of a remarkable Laver function and show that every remarkable cardinal carries such a function. We also show that remarkability is preserved by the canonical forcing of the GCH. (shrink)

In this article, we explicate evidence-based nursing (EBN), critically appraise its framework and respond to nurses’ concern that EBN sidelines the caring elements of nursing practice. We use resources from care ethics, especially Vrinda Dalmiya’s work that considers care as crucial for both epistemology and ethics, to show how EBN is compatible with, and indeed can be enhanced by, the caring aspects of nursing practice. We demonstrate that caring can act as a bridge between ‘external’ evidence and the other pillars (...) of the EBN framework: clinical expertise; patient preferences and values. Drawing on an influential EBN handbook, section 1 presents the aims and features of EBN, including the normative principle that EBN should take place within a ‘context of caring’. We aim to understand this context and whether it can be neatly detached from the EBN framework, as the handbook seems to suggest. In section 2, we highlight the grounds for resistance to EBN from the nursing community, before mounting the argument that nursing practices can be understood fruitfully through feminist care ethics and/or virtue ethics lenses. In section 3, we deepen that analysis using Dalmiya’s concepts of care-knowing and care as a hybrid ethico-epistemic virtue, which are ideally suited to the complex practices of nursing. In section 4, we bring this rich understanding of care into conversation with EBN, showing that its framework cannot be adequately theorised without paying proper attention to care. Caring can be neither an innocuous background assumption of nor an afterthought to the EBN framework. (shrink)

Recent publications on Alfred Schutz suggest the importance of his musical thought for understanding his general viewpoint on intersubjectivity. Developing this proposition further, my article focuses on one aspect of Schutz’s writings on music: his attempts to amalgamate the aesthetic oppositions of the Dionysian/Apollonian by Friedrich Nietzsche and inner duration/spatialized time by Henri Bergson. Despite the seeming distortion of the initial meaning of the Dionysian impulse, I suggest that Schutz’s employment remains faithful to the aesthetic and cognitive theory of early (...) Nietzsche. To substantiate this, I draw a link between Nietzsche’s early theory of aesthetic cognition and the neurophysiology of the mid-nineteenth century. Furthermore, the way Schutz applied the Dionysian/Apollonian opposition to the problem of musical communication made his musical thought prefigure some neuro-scientifically inspired discussions of the present-day, like the one on joint attention. Schutz tackled the key paradox: why the experience of music can be shared but not directly communicated. (shrink)

The medicalisation of pregnancy and childbirth has been encouraged by the continuing growth of technology that can be applied to the reproductive journey. Technology now has the potential to fully separate reproduction from the human body with the prospect of ectogenesis—the gestation of a fetus outside of the human body. This paper considers the issues that have been caused by the general medicalisation of pregnancy and childbirth and the impact that ectogenesis may have on these existing issues. The medicalisation of (...) pregnancy and childbirth is criticised for its impact on the relationship between doctors and pregnant women and the way in which doctors treat fetuses. It is argued that ectogenesis may cause more imbalance in the doctor and intended parent relationship and may result in an increased lack of clarity regarding a doctor’s duty to the fetus. This paper finds that extensive guidance and revised legislation will be necessary to minimise the impact of ectogenesis on the existing issues caused by the medicalisation of reproduction. (shrink)

Numerous societal issues, from climate change to pandemics, require public engagement with scientific research. Such engagement reveals challenges that can arise when experts communicate with laypeople. One of the most common frameworks for framing these communicative interactions is the deficit model of science communication, which holds that laypeople lack scientific knowledge and/or positive attitudes towards science, and that imparting knowledge will fill knowledge gaps, lead to desirable attitude/behavior changes, and increase trust in science. §1 introduces the deficit model in more (...) detail and shows that adhering to this model often fails to achieve its aims, which motivates the main question of this chapter: how can Ludwik Fleck’s theory of thinking collectives address the persistent problem of deficit approaches in science communication? I suggest that it can do so by exposing the deficit model’s implicit assumption of an expert-lay divide. Accordingly, §2 lays out Fleck’s theory and §3 contrasts it with contemporary debates about science communication. Following this descriptive work, §4 draws on Fleck’s ideas to make four concrete suggestions for further questioning the expert-lay divide. (shrink)

This paper is divided into two parts. In Section 1, I explore and defend a “regulative view” of folk-psychology as against the “standard view”. On the regulative view, folk-psychology is conceptualized in fundamentally interpersonal terms as a “mind-making” practice through which we come to form and regulate our minds in accordance with a rich array of socially shared and socially maintained sense-making norms. It is not, as the standard view maintains, simply an epistemic capacity for coming to know about the (...) mental states and dispositions already there. Importantly, the regulative view can meet and beat the standard at its own epistemic game. But it also does more. In Section 2, I show how the regulative view makes progress on two other problems that remain puzzling on the standard view: the problem of “first-person authority” – accounting for the special features of self-knowledge; and the problem of “reactive.. (shrink)

This paper examines the methodological claim made famous by P.F. Strawson: that we understand what features are required for responsible agency by exploring our attitudes and practices of holding responsible. What is the presumed metaphysical connection between holding responsible and being fit to be held responsible that makes this claim credible? I propose a non-standard answer to this question, arguing for a view of responsible agency that is neither anti-realist (i.e. purely 'conventionalist') nor straightforwardly realist. It is instead ‘constructivist’. On (...) the ‘Scaffolding View’ I defend, reactive attitudes play an essential role in developing, supporting, and thereby maintaining the capacities that make for responsible agency. While this view has relatively novel implications for a metaphysical understanding of capacities, its chief virtue, in contrast with more standard views, is providing a plausibly defensible account of how so-called responsible agents genuinely deserve to be treated as such. (shrink)

Philosophers and social scientists have focussed a great deal of attention on our human capacity to trust, but relatively little on the capacity to hope. This is a significant oversight, as hope and trust are importantly interconnected. This paper argues that, even though trust can and does feed our hopes, it is our empowering capacity to hope that significantly underwrites—and makes rational—our capacity to trust.

The purpose of the study is to justify the feasibility and prospects of including case technology in the process of independent training of cadets in a military higher educational institution. Scientific novelty lies in justifying the increase in the effectiveness of independent training through the implementation of case study technology. The article analyzes the impact of case technology on the development of professionalism of future officers, determines the stages of work on cases, examines the activities of cadets and the tasks (...) that are solved during each stage. As a result, the approaches to the organization of independent training of cadets in the process of implementing the case-training technology are identified; the capability of case technology, as a means of forming competencies such as design, organizational and communicative, is indicated; the features of the cadets' activity aimed at the formation of the above-mentioned competencies are revealed. (shrink)

We call systematic disputes that are particularly hard to resolve deep disagreements. We can divide most theories of deep disagreements in analytic epistemology into two camps: the Wittgensteinian view and the fundamental epistemic principles view. This essay analyzes how both views deal with two of the most pressing issues a theory of deep disagreement must address: their source and their resolution. After concluding that the paradigmatic theory of each camp struggles on both fronts, the essay proceeds to show that, despite (...) their differences, these theories share some core assumptions and think of deep disagreements similarly: as grounded in fundamental epistemic differences between the parties. This underlying conception of deep disagreements the author calls “the fundamental model.” The essay provides reasons to be wary of this conception, and hence, mainstream epistemologists’ treatment of the issue. (shrink)

The commentaries in response to our feature paper1 are indicative of the varied perspectives that can be taken towards artificial amnion and placenta technology (AAPT) and more specifically its relationship with pregnancy (loss). Kennedy rightly argues that empirical research is essential for understanding the experiences of pregnancy loss and AAPT2 and our own advocacy of empirical research is evident in previous work.3–5 Kennedy also acknowledges the current impossibility of researching AAPT experiences since it has not yet been applied in clinical (...) settings. It is precisely for this reason that we draw on the reported experiences of neonatal intensive care (NICU) parents since they present the closest analogy which can be drawn with the technology.1 What we reject in Kennedy’s response, however, is the claim that the experiences of NICU parents have been ‘exploited’ by us.2 Studies documenting the experiences of NICU parents were drawn on to indicate that our considerations are more than mere speculation. As stated, the experiences of NICU parents, particularly the pregnant individual who experiences pregnancy loss, are the closest indication we have of how AAPT and what it entails may be experienced. To claim that this data is being exploited by us is to undermine the very purpose that such research is undertaken—to indicate how future care may be improved. It would be much more harmful to not use this information. While we believe that experiences of …. (shrink)

In Building Better Beings, Vargas develops and defends a naturalistic account of responsibility, whereby responsible agents must possess a feasibly situated capacity to detect and respond to moral considerations. As a preliminary step, he also offers a substantive account of how we might justify our practices of holding responsible—viz., by appeal to their efficacy in fostering a ‘valuable form of agency’ across the community at large, a form of agency that precisely encompasses sensitivity to moral considerations. But how do these (...) accounts relate to one another? Though I find much that is appealing in Vargas’s general approach, I challenge his insistence that these accounts should be treated as ‘conceptually independent’, arguing that this generates an objectionable “justification gap”: on his analysis, someone could remain an appropriate target of our responsibility practices and yet fail to be a morally responsible agent. In closing, I offer a potential solution to this problem, though it means re-visioning how the account of moral responsibility is conceptually tied to the justification of our responsibility practices. (shrink)

This paper describes a live ethics case project that can be used to teach ethics in a broad variety of business classes. The live case differs from regular cases in that it involves a current situation. Students select an on-going or current event that involves ethical violations and write a case about it. They then present their case and run a debate about the challenges and issues outlined in the case and the actions that could have or should have been (...) taken. The dynamic project fulfills the key criteria for effective ethics education since it increases awareness of the complexity of ethical challenges, allows application of concepts, creates a personal emotional engagement in the case, is relevant, holds students accountable for their position, and creates a setting that encourages students to think critically about ethics. (shrink)

En este capítulo, analizo a qué se refiere Kuhn cuando habla de metáfora en las Conferencias Notre Dame, pero sobre todo a explorar a qué no se refiere. En la primera sección, analizo por qué Kuhn usa el término “metáfora” para referirse al proceso de aprendizaje de lenguaje científico, en particular, los paralelismos que encuentra entre ambos fenómenos. En la segunda sección, se presentan algunos aspectos centrales de dos teorías influyentes sobre la metáfora: la teoría del mapeo estructural y la (...) teoría de las metáforas conceptuales. Luego, en la sección III, vuelvo a Kuhn para ver las diferencias cruciales que encontramos entre los procesos metafóricos y el proceso de ostensión del que habla el autor. Finalmente, presento una noción del segundo Wittgenstein, la idea de figura, para ofrecer una nueva forma de interpretar la idea de que el cambio revolucionario involucra un cambio en la metáfora central. (shrink)

Volume 3, Issue 1, March 2019, Page 48-59.

Is it possible to disagree with someone without considering them cognitively flawed? The answer seems to be a resounding yes: disagreeing with someone doesn't entail thinking less of them. You can disagree with someone and not think that they are unreasonable. Deep disagreements, however, may challenge this assumption. A disagreement is deep when it involves many interrelated issues, including the proper way to resolve the disagreement, resulting in its persistence. The parties to a deep disagreement can hold neutral or even (...) positive judgements of each other's epistemic character, as parties’ judging each other's epistemic character negatively (i.e., epistemic disdain) is not a defining feature of deep disagreements. When analysing real-life cases, however, we find that epistemic disdain is typical of deep disagreements. In this article, I analyse why this is the case. Given that epistemic disdain undermines cognitive peerhood, the prospects of deep disagreements between epistemic peers seem bleak. Finally, it is discussed how the phenomenon of epistemic disdain, as it relates to deep disagreements, may increase affective polarisation. (shrink)