We explain and explore class-theoretic potentialism—the view that one can always individuate more classes over a set-theoretic universe. We examine some motivations for class-theoretic potentialism, before proving some results concerning the relevant potentialist systems (in particular exhibiting failures of the $\mathsf {.2}$ and $\mathsf {.3}$ axioms). We then discuss the significance of these results for the different kinds of class-theoretic potentialists.

A central area of current philosophical debate in the foundations of mathematics concerns whether or not there is a single, maximal, universe of set theory. Universists maintain that there is such a universe, while Multiversists argue that there are many universes, no one of which is ontologically privileged. Often forcing constructions that add subsets to models are cited as evidence in favour of the latter. This paper informs this debate by analysing ways the Universist might interpret this discourse that seems (...) to necessitate the addition of subsets to V. We argue that despite the prima facie incoherence of such talk for the Universist, she nonetheless has reason to try and provide interpretation of this discourse. We analyse extant interpretations of such talk, and analyse various tradeoffs in naturality that might be made. We conclude that the Universist has promising options for interpreting different forcing constructions. (shrink)

It is standard in set theory to assume that Cantor's Theorem establishes that the continuum is an uncountable set. A challenge for this position comes from the observation that through forcing one can collapse any cardinal to the countable and that the continuum can be made arbitrarily large. In this paper, we present a different take on the relationship between Cantor's Theorem and extensions of universes, arguing that they can be seen as showing that every set is countable and that (...) the continuum is a proper class. We examine several principles based on maximality considerations in this framework, and show how some (namely Ordinal Inner Model Hypotheses) enable us to incorporate standard set theories (including ZFC with large cardinals added). We conclude that the systems considered raise questions concerning the foundational purposes of set theory. (shrink)

In this paper, I consider the problem of omission for the counterfactual comparative account of harm. A given event harms a person, on this account, when it makes her worse off than she would have been if it had not occurred. The problem arises because cases in which one person merely fails to benefit another intuitively seem harmless. The account, however, seems to imply that when one person fails to benefit another, the first thereby harms the second, since the second (...) person would have been better off if the first had benefited her. I argue that the cases of failing to benefit at issue are in fact cases of harming. They are cases of preventive harm. I also argue that we can explain away the intuition that no harm occurs in these cases, and that the relevant implication of the counterfactual comparative account is consistent with a variety of plausible views about the moral significance of harm. (shrink)

Let's suppose you think that there are no uncountable sets. Have you adopted a restrictive position? It is certainly tempting to say yes---you've prohibited the existence of certain kinds of large set. This paper argues that this intuition can be challenged. Instead, I argue that there are some considerations based on a formal notion of restrictiveness which suggest that it is restrictive to hold that there are uncountable sets.

The Continuum Hypothesis featured top of Hilbert's list of 23 problems in 1900. Today, we still consider the question, with various programmes pulling in different directions. This conceptual diversity raises a puzzle: In what sense do we disagree when we talk about it? A standard assumption takes it that the content of our thought about classes and the Continuum Hypothesis has not changed. Assuming a moderate view of how content is determined, I reject this assumption but also argue that whilst (...) the Continuum Hypothesis can have different content for different agents, it can also be determinate for certain programmes. In particular, I suggest that there is a fault line between those who think there are uncountable sets and recent countabilist programmes. (shrink)

Multiverse Views in set theory advocate the claim that there are many universes of sets, no-one of which is canonical, and have risen to prominence over the last few years. One motivating factor is that such positions are often argued to account very elegantly for technical practice. While there is much discussion of the technical aspects of these views, in this paper I analyse a radical form of Multiversism on largely philosophical grounds. Of particular importance will be an account of (...) reference on the Multiversist conception, and the relativism that it implies. I argue that analysis of this central issue in the Philosophy of Mathematics indicates that Radical Multiversism must be algebraic, and cannot be viewed as an attempt to provide an account of reference without a softening of the position. (shrink)

In the contemporary philosophy of set theory, discussion of new axioms that purport to resolve independence necessitates an explanation of how they come to be justified. Ordinarily, justification is divided into two broad kinds: intrinsic justification relates to how `intuitively plausible' an axiom is, whereas extrinsic justification supports an axiom by identifying certain `desirable' consequences. This paper puts pressure on how this distinction is formulated and construed. In particular, we argue that the distinction as often presented is neither well-demarcated nor (...) sufficiently precise. Instead, we suggest that the process of justification in set theory should not be thought of as neatly divisible in this way, but should rather be understood as a conceptually indivisible notion linked to the goal of explanation. (shrink)

This book offers a defense of the Property Theory of Content, according to which properties rather than propositions are the contents of our beliefs, desires, and other cognitive attitudes. New arguments for the theory are offered, objections are answered, and applications to problems in the philosophy of mind are discussed.

In recent years there has been an outpouring of work at the intersection of social movement studies and organizational theory. While we are generally in sympathy with this work, we think it implies a far more radical rethinking of structure and agency in modern society than has been realized to date. In this article, we offer a brief sketch of a general theory of strategic action fields (SAFs). We begin with a discussion of the main elements of the theory, describe (...) the broader environment in which any SAF is embedded, consider the dynamics of stability and change in SAFs, and end with a respectful critique of other contemporary perspectives on social structure and agency. (shrink)

Zero provides a challenge for philosophers of mathematics with realist inclinations. On the one hand it is a bona fide cardinal number, yet on the other it is linked to ideas of nothingness and non-being. This paper provides an analysis of the epistemology and metaphysics of zero. We develop several constraints and then argue that a satisfactory account of zero can be obtained by integrating an account of numbers as properties of collections, work on the philosophy of absences, and recent (...) work in numerical cognition and ontogenetic studies. (shrink)

Often philosophers, logicians, and mathematicians employ a notion of intended structure when talking about a branch of mathematics. In addition, we know that there are foundational mathematical theories that can find representatives for the objects of informal mathematics. In this paper, we examine how faithfully foundational theories can represent intended structures, and show that this question is closely linked to the decidability of the theory of the intended structure. We argue that this sheds light on the trade-off between expressive power (...) and meta-theoretic properties when comparing first-order and second-order logic. (shrink)

The independence phenomenon in set theory, while pervasive, can be partially addressed through the use of large cardinal axioms. A commonly assumed idea is that large cardinal axioms are species of maximality principles. In this paper I question this claim. I show that there is a kind of maximality (namely absoluteness) on which large cardinal axioms come out as restrictive relative to a formal notion of restrictiveness. Within this framework, I argue that large cardinal axioms can still play many of (...) their usual foundational roles. (shrink)

Informal rigour is the process by which we come to understand particular mathematical structures and then manifest this rigour through axiomatisations. Structural relativity is the idea that the kinds of structures we isolate are dependent upon the logic we employ. We bring together these ideas by considering the level of informal rigour exhibited by our set-theoretic discourse, and argue that different foundational programmes should countenance different underlying logics (intermediate between first- and second-order) for formulating set theory. By bringing considerations of (...) perturbations in modal space to bear on the debate, we will suggest that a promising option for representing current set-theoretic thought is given by formulating set theory using quasi-weak second-order logic. These observations indicate that the usual division of structures into \particular (e.g. the natural number structure) and general (e.g. the group structure) is perhaps too coarse grained; we should also make a distinction between intentionally and unintentionally general structures. (shrink)

This paper assessed whether belief in conspiracy theories was associated with a particularly cognitive style (worldview). The sample comprised 223 volunteers recruited via convenience sampling and included undergraduates, postgraduates, university employees and alumni. Respondents completed measures assessing a range of cognitive-perceptual factors (schizotypy, delusional ideation and hallucination proneness) and conspiratorial beliefs (general attitudes towards conspiracist thinking and endorsement of individual conspiracies). Positive symptoms of schizotypy, particularly the cognitive-perceptual factor, correlated positively with conspiracist beliefs. The best predictor of belief in conspiracies (...) was delusional ideation. Consistent with the notion of a coherent conspiratorial mindset, scores across conspiracy measures correlated strongly. Whilst findings supported the view that belief in conspiracies, within the sub-clinical population, was associated with a delusional thinking style, cognitive-perceptual factors in combination accounted for only 32% of the variance. (shrink)

According to Jerome Wakefield’s harmful dysfunction analysis of medical disorder, the inability of some internal part or mechanism to perform its natural function is necessary, but not sufficient, for disorder. HDA also requires that the part dysfunction be harmful to the individual. I consider several problems for HDA’s harm criterion in this article. Other accounts on which harm is necessary for disorder will suffer from all or almost all of these problems. Comparative accounts of harm imply that one is harmed (...) when one is made worse off, that is, worse off than one otherwise would have been. Non-comparative accounts imply that one is harmed when one is put into some kind of condition or state that is, in some way, bad in itself. I argue that whether harm is construed comparatively or non-comparatively, HDA’s harm criterion is problematic. I tentatively conclude that an analysis of medical disorder should not make use of the concept of harm. (shrink)

This book focuses on the nature and importance of harm by providing a sustained defense of the counterfactual comparative account, in particular by extending the account to allow for a certain kind of plural or collective harm. According to the counterfactual comparative account, an event harms a person provided that she would have been better off had it not occurred. On the account defended in this book, there are cases in which some events harm a given individual even though none (...) of them by itself harms the individual. This account aims to solve the most serious problem facing the counterfactual comparative account, a problem involving cases of preemption. In such cases, an intuitively harmful event does not make the victim worse off, since if it had not occurred, another event would have had a similar adverse influence on the victim’s well-being. The book also contains discussion of various competing accounts of harm, and it replies to other challenges to the counterfactual comparative account. It argues, moreover, that an action that would harm a person does not thereby have a strong moral reason against it; if there is a strong moral reason against a harmful act, the reason is due primarily to some of the act’s other features. (shrink)

We introduce and consider the inner-model reflection principle, which asserts that whenever a statement \varphi(a) in the first-order language of set theory is true in the set-theoretic universe V, then it is also true in a proper inner model W \subset A. A stronger principle, the ground-model reflection principle, asserts that any such \varphi(a) true in V is also true in some non-trivial ground model of the universe with respect to set forcing. These principles each express a form of width (...) reflection in contrast to the usual height reflection of the Lévy–Montague reflection theorem. They are each equiconsistent with ZFC and indeed \Pi_2-conservative over ZFC, being forceable by class forcing while preserving any desired rank-initial segment of the universe. Furthermore, the inner-model reflection principle is a consequence of the existence of sufficient large cardinals, and lightface formulations of the reflection principles follow from the maximality principle MP and from the inner-model hypothesis IMH. We also consider some questions concerning the expressibility of the principles. (shrink)

This article articulates and assesses an imperatival approach to the foundations of mathematics. The core idea for the program is that mathematical domains of interest can fruitfully be viewed as the outputs of construction procedures. We apply this idea to provide a novel formalisation of arithmetic and set theory in terms of such procedures, and discuss the significance of this perspective for the philosophy of mathematics.

Set-theoretic and category-theoretic foundations represent different perspectives on mathematical subject matter. In particular, category-theoretic language focusses on properties that can be determined up to isomorphism within a category, whereas set theory admits of properties determined by the internal structure of the membership relation. Various objections have been raised against this aspect of set theory in the category-theoretic literature. In this article, we advocate a methodological pluralism concerning the two foundational languages, and provide a theory that fruitfully interrelates a `structural' perspective (...) to a set-theoretic one. We present a set-theoretic system that is able to talk about structures more naturally, and argue that it provides an important perspective on plausibly structural properties such as cardinality. We conclude the language of set theory can provide useful information about the notion of mathematical structure. (shrink)

I provide an examination and comparison of modal theories for underwriting different non-modal theories of sets. I argue that there is a respect in which the `standard' modal theory for set construction---on which sets are formed via the successive individuation of powersets---raises a significant challenge for some recently proposed `countabilist' modal theories (i.e. ones that imply that every set is countable). I examine how the countabilist can respond to this issue via the use of regularity axioms and raise some questions (...) about this approach. I argue that by comparing them with the `standard' uncountabilist theory, a new approach that brings in arbitrariness rather than the strict controls of forcing is desirable. (shrink)

Let mathematical justification be the kind of justification obtained when a mathematician provides a proof of a theorem. Are Gettier cases possible for this kind of justification? At first sight we might think not: The standard for mathematical justification is proof and, since proof is bound at the hip with truth, there is no possibility of having an epistemically lucky justification of a true mathematical proposition. In this paper, I argue that Gettier cases are possible (and indeed actual) in mathematical (...) reasoning. By analysing these cases, I suggest that the Gettier phenomenon indicates some upshots for actual mathematical practice. (shrink)

To date, philosophical discussions of animal ethics and Critical Animal Studies have been dominated by Western perspectives and Western thinkers. This book makes a novel contribution to animal ethics in showing the range and richness of ideas offered to these fields by diverse Asian traditions. Asian Perspectives on Animal Ethics is the first of its kind to include the intersection of Asian and European traditions with respect to human and nonhuman relations. Presenting a series of studies focusing on specific Asian (...) traditions, as well as studies that put those traditions in dialogue with Western thinkers, this book looks at Asian philosophical doctrines concerning compassion and nonviolence as these apply to nonhuman animals, as well as the moral rights and status of nonhuman animals in Asian traditions. Using Asian perspectives to explore ontological, ethical and political questions, contributors analyze humanism and post-humanism in Asian and comparative traditions and offer insight into the special ethical relations between humans and other particular species of animals. This book will be of interest to students and scholars of Asian religion and philosophy, as well as to those interested in animal ethics and Critical Animal Studies. (shrink)

Much of the discussion of set-theoretic independence, and whether or not we could legitimately expand our foundational theory, concerns how we could possibly come to know the truth value of independent sentences. This paper pursues a slightly different tack, examining how we are ignorant of issues surrounding their truth. We argue that a study of how we are ignorant reveals a need for an understanding of set-theoretic explanation and motivates a pluralism concerning the adoption of foundational theory.

In this article, I survey some philosophical attitudes to talk concerning `the' universe of sets. I separate out four different strands of the debate, namely: (i) Universism, (ii) Multiversism, (iii) Potentialism, and (iv) Pluralism. I discuss standard arguments and counterarguments concerning the positions and some of the natural mathematical programmes that are suggested by the various views.

How does our language relate to reality? This is a question that is especially pertinent in set theory, where we seem to talk of large infinite entities. Based on an analogy with the use of models in the natural sciences, we argue for a threefold correspondence between our language, models, and reality. We argue that so conceived, the existence of models can be underwritten by a weak notion of existence, where weak existence is to be understood as existing in virtue (...) of language. (shrink)

Alexander Anievas and Kerem Nişancıoğlu’sHow the West Came to Ruleis an important intervention within Marxist historical debates which seeks to use the theory of uneven and combined development to explain the origin and rise to dominance of capitalism. The argument is shaped by a critique of Political Marxist ‘internalist’ explanations of the process, to which the authors counterpose an account which emphasises its inescapably ‘inter-societal’ nature. While recognising the many contributions that the book makes to our historical understanding, this article (...) argues that these insights do not depend onUCD, and could have been arrived at without reference to it. In particular, it will try to show thatUCDis inapplicable in periods before the consolidation of capitalism, but might be more usefully extended spatially rather than chronologically. (shrink)

The prevailing view of Advaita Vedānta as world negating and disengaged with worldly activity provides little space for an ethic of environmental care, or a psychology for eco-resilience beyond passive indifference. However, many sources for environmentalism within the Advaita Vedānta tradition and its canon of texts remain untapped. In this paper, I explore the ritual ecology found in chapter three of the Bhagavadgītā as the ground to construct an Advaitin ecotheology and ecopsychology. This all-encompassing ritual ecology, described as a sacrificial (...) wheel, provides meaning to the world and views individuals as essential causative participants within it. Individuals have the religious and moral responsibility to perpetuate the wheel through a relationship of mutual reciprocity with the gods based on an economy of food. I extend this ritual ecology to include moral responsibility to the environment and explain how it provides a religious psychology and practice for handling our anxiety of imminent ecological catastrophe. (shrink)

Should we be Roschians about the concept of disease, rather than taking a classical approach? A classical concept of disease defines disease in terms of necessary and sufficient conditions; any things and only things which meet this definition are members of the class. In Roschian concepts of disease, it is supposed that degree of similarity to a prototype determines membership in the class of diseases. In this paper, the two approaches are pitched against one another in a series of tests (...) which appear on first sight to favor Roschian accounts.These tests are 1) the capacity to accommodate the variety of the class of disease, 2) the capacity to explain controversies about disease attribution, and 3) the capacity to... (shrink)

The opening of the fourth session of The Neutral — the course given by Roland Barthes at the Collège de France in 1977–8 — is marked by a dramatic spillage of ink. Rather than take this as an incidental, trivial moment, I read it as one of the many ‘ink blots’ that colour the work of Barthes. Tracing his ‘almost obsessive relation to writing instruments’ and the material act of inscription, this essay relates the ‘ink blots’ to the development of (...) a ‘non-arrogant’, non-expressive, non-idealistic theory of language in The Neutral and other texts by Barthes. (shrink)

Aux États-Unis, une structure appelée Love Makes a Family [C’est l’amour qui fait une famille] fut créée en 1999 pour soutenir le droit à l’adoption des couples homosexuels. Elle a également joué un rôle central dans le développement des unions civiles. Dix ans plus tard, sa directrice, Ann Stanback, annonçait qu’ayant atteint ses objectifs Love Makes a Family arrêterait son activité à la fin de l’année, et qu’elle-même en profiterait pour passer plus de temps avec sa femme Charlotte: «L’esse..

In the light of the publication of Roland Barthes's Mourning Diary, this essay examines how the influential theory of the photographic punctum has cinematic roots which are repressed in Barthes's Camera Lucida. My aim is not to repeat familiar arguments about how Barthes's ‘The Third Meaning’ anticipates the photographic punctum in a cinematic context; it is, rather, to attend specifically to Mourning Diary as a much closer, more precise precursor which has been visible only since 2009, and which casts new (...) light upon the work of Roland Barthes. (shrink)

Advances in medical capability should be accompanied by discussion of their ethical implications. In the military medical context there is a growing interest in developing prophylactic interventions that will mitigate the effects of trauma and improve survival. The ethics of this novel capability are currently unexplored. This paper describes the concept of trauma prophylaxis and outlines some of the ethical issues that need to be considered, including within concept development, research and implementation. Trauma prophylaxis can be divided into interventions that (...) do not and those that do have medical enhancement as an unintended side effect of their prophylactic action. We conclude that type 1 interventions have much in common with established military medical prophylaxis, and the potentially enhancing qualities of type 2 interventions raise different issues. We welcome further debate on both interventions. (shrink)