Many recent writers in the philosophy of mathematics have put great weight on the relative categoricity of the traditional axiomatizations of our foundational theories of arithmetic and set theory. Another great enterprise in contemporary philosophy of mathematics has been Wright's and Hale's project of founding mathematics on abstraction principles. In earlier work, it was noted that one traditional abstraction principle, namely Hume's Principle, had a certain relative categoricity property, which here we term natural relative categoricity. In this paper, we show (...) that most other abstraction principles are not naturally relatively categorical, so that there is in fact a large amount of incompatibility between these two recent trends in contemporary philosophy of mathematics. To better understand the precise demands of relative categoricity in the context of abstraction principles, we compare and contrast these constraints to stability-like acceptability criteria on abstraction principles, the Tarski-Sher logicality requirements on abstraction principles studied by Antonelli and Fine, and supervaluational ideas coming out of Hodes' work. (shrink)

Benacerraf’s 1965 multiple-reductions argument depends on what I call ‘deferential logicism’: his necessary condition for number-set identity is most plausible against a background Quineanism that allows autonomy of the natural number concept. Steinhart’s ‘folkist’ sufficient condition on number-set identity, by contrast, puts that autonomy at the center — but fails for not taking the folk perspective seriously enough. Learning from both sides, we explore new conditions on number-set identity, elaborating a suggestion from Wright.

Neo-Fregeans have been troubled by the Nuisance Principle, an abstraction principle that is consistent but not jointly satisfiable with the favored abstraction principle HP. We show that logically this situation persists if one looks at joint consistency rather than satisfiability: under a modest assumption about infinite concepts, NP is also inconsistent with HP.

This article improves two existing theorems of interest to neologicist philosophers of mathematics. The first is a classification theorem due to Fine for equivalence relations between concepts definable in a well-behaved second-order logic. The improved theorem states that if an equivalence relation E is defined without nonlogical vocabulary, then the bicardinal slice of any equivalence class—those equinumerous elements of the equivalence class with equinumerous complements—can have one of only three profiles. The improvements to Fine’s theorem allow for an analysis of (...) the well-behaved models had by an abstraction principle, and this in turn leads to an improvement of Walsh and Ebels-Duggan’s relative categoricity theorem. (shrink)

The injective version of Cantor’s theorem appears in full second-order logic as the inconsistency of the abstraction principle, Frege’s Basic Law V (BLV), an inconsistency easily shown using Russell’s paradox. This incompatibility is akin to others—most notably that of a (Dedekind) infinite universe with the Nuisance Principle (NP) discussed by neo-Fregean philosophers of mathematics. This paper uses the Burali–Forti paradox to demonstrate this incompatibility, and another closely related, without appeal to principles related to the axiom of choice—a result hitherto unestablished. (...) It discusses both the general interest of this result, its interest to neo-Fregean philosophy of mathematics, and the potential significance of the Burali–Fortian method of proof. (shrink)

Objects appear to fall into different sorts, each with their own criteria for identity. This raises the question of whether sorts overlap. Abstractionists about numbers—those who think natural numbers are objects characterized by abstraction principles—face an acute version of this problem. Many abstraction principles appear to characterize the natural numbers. If each abstraction principle determines its own sort, then there is no single subject-matter of arithmetic—there are too many numbers. That is, unless objects can belong to more than one sort. (...) But if there are multi-sorted objects, there should be cross-sortal identity principles for identifying objects across sorts. The going cross-sortal identity principle, ECIA2 of Cook and Ebert (2005), solves the problem of too many numbers. But, I argue, it does so at a high cost. I therefore propose a novel cross-sortal identity principle, based on embeddings of the induced models of abstracts developed by Walsh (2012). The new criterion matches ECIA2’s success, but offers interestingly different answers to the more controversial identifications made by ECIA2. (shrink)

I argue that rather than aiming at the well-being of those whom we love, we should aim to share in their ends. The former stance runs the risk of being objectionably paternalistic and, as I explain, only the latter makes reciprocal relationships possible. I end by diagnosing our attraction to the idea that we should promote our loved-ones’ well-being.

Rawls' requirement that citizens of liberal democracies support only policies which they believe can be justified in 'public reason' depends on a certain ideal for the relationships between citizens. This is a valuable ideal, and thus citizens have reasons to try to achieve it. But it is not always possible to find the common ground that we would need in order to do so, and thus we should reject Rawls' strong claim that we have an obligation to defend our views (...) in public reason. Because I recognize that we have strong reasons to conduct our political enquiry within the guidelines of political liberalism, but deny that we always have an obligation to do so, one might call my view 'permissive political liberalism'. (shrink)

I attempt to vindicate our authority to create new practical reasons for others by making choices of own own. In The Doctrine of Right Kant argues that we have an obligation to leave the Juridical State of Nature and found the state. In a less familiar passage in Religion within the Bounds of Mere Reason he argues for an obligation to leave what he calls the Ethical State of Nature and join together in the Moral Community. I read both texts (...) as addressing and trying to resolve a tension between our individual freedom and our authority to make claims on one another. I explicate the political argument, and then develop the view that Kant sketches in the Religion, arguing that regarding others as capable of making choices that give you reasons to act is a condition of the full exercise of your autonomy. (shrink)

We often come to value someone or something through experience of that person or thing. You may thereby come to embrace a value that you did not grasp prior to the experience in question. Moreover, it seems that in a large and important subset of cases you could not have fully appreciated that value merely by considering a report of the reasons or arguments that purport to establish that it is valuable. Despite its ubiquity, this phenomenon goes missing in a (...) great deal of contemporary work in ethics and political philosophy. In this paper I further specify the phenomenon of interest by developing a series of examples. Then I support the claim that philosophers routinely overlook it by surveying several significant philosophical positions that do so. (shrink)

Christine M. Korsgaard has had a profound influence on moral philosophy over the past forty years. Through her writing and teaching she has developed a distinctive, rigorous, and historically informed way of thinking about ethics, agency, and the normative dimension of human life more generally. The twelve original essays in this volume are written in her honor on the occasion of her retirement from teaching. They engage questions that recur in her work: Why are we obligated to do what morality (...) demands? What features of our nature make us subject to moral obligation? What does it mean to be autonomous and responsible for what we do? What do we owe to nonhuman animals? Contributors include Stephen Darwall, Kyla Ebels-Duggan, Barbara Herman, Richard Moran, Japa Pallikkathayil, Faviola Rivera-Castro, T.M. Scanlon, Tamar Schapiro, Sharon Street, David Sussman, Sigrún Svavarsdóttir, and David Velleman. These essays shed light on Korsgaard's own views while staking out provocative new positions on the topics that feature centrally in her own work. (shrink)

Kant’s political theory stands in the social contract tradition, but departs significantly from earlier versions of social contract theory. Most importantly Kant holds, against Hobbes and Locke, that we have not merely a pragmatic reason but an obligation to exit the state of nature and found a state. Kant holds that each person has an innate right to freedom, but it is possible to simultaneously honor everyone’s right only under the rule of law. Since we are obligated to respect each (...) person’s right to freedom, and can do so only in a state, we are obligated to submit to the authority of the state if we have one, and to establish one if we do not. In the first half of the essay I reconstruct this argument in more detail. In the second half I survey four points of controversy: (i) What is the relationship between Kant’s political philosophy and his moral philosophy? (ii) How does the innate right to freedom support the postulate that we are permitted to acquire property and other private rights? (iii) How does the postulate support an obligation to found the state? (iv) How should we understand Kant’s views about political revolutions? (shrink)

I argue that we cannot adequately characterize the aims of education in terms of some formal conception of what it is to think well. Implementing any such aim requires reliance on and communication of further, substantive normative commitments. This reveals that a standard contrast between an old-fashioned approach to education that aims to communicate a particular normative outlook, and a progressive approach that aims to develop skills of critical reasoning and reflection is confused and misleading.

I argue that unfortunate formative circumstances do not undermine the warrant for either responsibility or blame. I then diagnose the tendency to think that formative circumstances do matter in this way, arguing that knowledge of these circumstances can play an essential epistemic role in our interpersonal interactions.

Our ordinary talk reflects a deep tension in the way that we think about love. On the one hand, we regard love as an especially important expression of our agency. Yet, on the other hand, we also think of love as something that happens to us, in the face of which we are passive and can be powerless. While it’s hard to see how to hold these two ways of thinking of love together, in this paper I argue that we (...) must find some way of doing so. I argue that we must think of love as a contentful attitude attributable to its agent, an expression of our selves. But familiar ways of understanding agency sort love into the category of things that happen to us, rather than that of things that we do: You cannot love at will, nor is love an attitude to which you could reason. I conclude that questions about the relationship of our agency to what we love are not superficial, but stem from deep tensions about the relationship between love and reasons. A resolution to these difficulties would provide important insight not only into the character of love, but also the nature of agency, and its relationship to values, reasoning and reasons. (shrink)

Berislav Marušić’s On the Temporality of Emotions is a lovely book. Marušić confronts a puzzle about grief and anger that many will find familiar from their own.

Many thinkers agree that facilitating the development of students’ autonomy is a proper aim of education generally and higher education in particular. I defend a version of the autonomy view, but not as I think its other advocates imagine it. I suggest that an important aim of education is the facilitation of intellectual virtues. What is right about the idea that education should facilitate students’ autonomy is best captured in virtue terms as intellectual charity and humility.

Kant rejects all of the standard accounts of the dependence of morality on religious claims or commitment. He nevertheless thinks that morality “leads to” religion. I defend an account of this “leading to” relationship, arguing that it is the result of Kant’s struggle to capture the practical import of the consequences of our actions within a moral theory that rejects the idea that we must maximize the good. On this view, the best way to acknowledge that the outcomes of our (...) actions matter, while maintaining uncompromising commitment to the moral law, is to hope in God. (shrink)

I argue that political liberals should not support the monopoly of a single educational approach in state sponsored schools. Instead, they should allow reasonable citizens latitude to choose the worldview in which their own children are educated. I begin by defending a particular conception of political liberalism, and its associated requirement of public reason, against the received interpretation. I argue that the values of respect and civic friendship that motivate the public reason requirement do not support the common demand that (...) citizens “bracket” their comprehensive commitments in politics. Rather, citizens should seek to enact policies the justification of which is compatible with the truth of their fellow reasonable citizens’ worldviews. Next I argue that no single educational approach can meet this standard of justification. Many believe that state sponsored education in a pluralist, liberal society ought to present multiple worldviews in a neutral way. I argue that this aspiration is unrealizable, and no other educational model will plausibly meet the justificatory demand. Finally, I address two objections to my favored alternative: that it may allow for the inculcation of disrespect, and that it violates children’s autonomy. Against the first, I claim that political liberals have no grounds for thinking that reasonable citizens will seek to inculcate disrespect. Finally, I argue that there is no conception of autonomy that can sustain the second. (shrink)

The principle that children’s freedom should be preserved in their upbringing is sometimes thought to provide an alternative to imposing a particular conception of the good on them. But to sustain the alternative we must distinguish between those desires and proclivities that are educated into a person and those that are his own. Several philosophers appeal to innate or presocial tendencies to ground this distinction, but that approach fails. The ability to exercise first person authority over a desire or commitment (...) provides a better conception of what it is for such a state to be one’s own. But such desires and commitments are not distinct from those educated into a person. While the ideal of autonomy, conceived in these terms, can still provide some guidance for upbringing, it will not substitute for teaching children a conception of the good. (shrink)

I articulate and defend the most central claims of contemporary Kantian moral theory. I also explain some of the most important internal disagreements in the field, contrasting two approaches to Kantian ethics: Kantian Constructivism and Kantian Realism. I connect the former to Kant’s Formula of Universal Law and the latter to his Formula of Humanity. I end by discussing applications of the Formula of Humanity in normative ethics.

Kant’s claim that virtue has nothing to do with the content of our desires, but depends only on the strength of will needed to manage our desires, depends on an unattractive conception of inclination that he inherits from Hume. Kantians can replace this with a better view of desire without giving up what is most attractive about the Kantian approach: the claim that reason can motivate, and the associated illuminating account of practical freedom.

In the Groundwork Kant dismisses theistic principles, along with all other competitors to his Categorical Imperative, claiming that they are heteronomous. By contrast, he asserts, the fundamental moral principle must be a principle of autonomy. I argue that the best case for this Kantian conclusion conflates our access to the reasons for our commitments with an ability to state these reasons such that they could figure in an argument. This conflation, in turn, results from a certain Kantian conception of inclination, (...) and its role in our moral psychology. These are views that we ought to reject. Having done so we will see that a theistic ethics based on desire or love for God would not face a distinctive problem of heteronomy. (shrink)

I consider how Christian philosophers should decide which questions are worth asking. I provide an interpretation and defense of Alvin Plantinga’s claim that Christian philosophers should strive for autonomy, and argue that this rules out some ways of settling on our questions. I then argue that the questions in which Christian philosophers should take an interest are those arising from or continuous with a distinctively Christian way of life.

Ripstein’s Kantian argument for the authority of the state purports to demonstrate that state authority is a necessary condition of each individual’s freedom. Ripstein regards an individual as free just in case her entitlement to control what is hers is not violated. After questioning whether his approach adequately distinguishes standards of legitimacy from standards of ideal justice, I argue for the superiority of an alternative conception of freedom. On the view that I defend a person is free just in case (...) she is able to move her body in space unimpeded by others. I argue that this conception allows for a more convincing version of the Kantian argument. (shrink)

This paper presents new constructions of models of Hume's Principle and Basic Law V with restricted amounts of comprehension. The techniques used in these constructions are drawn from hyperarithmetic theory and the model theory of fields, and formalizing these techniques within various subsystems of second-order Peano arithmetic allows one to put upper and lower bounds on the interpretability strength of these theories and hence to compare these theories to the canonical subsystems of second-order arithmetic. The main results of this paper (...) are: (i) there is a consistent extension of the hyperarithmetic fragment of Basic Law V which interprets the hyperarithmetic fragment of second-order Peano arithmetic, and (ii) the hyperarithmetic fragment of Hume's Principle does not interpret the hyperarithmetic fragment of second-order Peano arithmetic, so that in this specific sense there is no predicative version of Frege's Theorem. (shrink)

A common finding in the study of emotion and decision making is the tendency for fear and anxiety to decrease risk taking. The current meta-analysis summarises the strength and variability of this...

This paper sets out a predicative response to the Russell-Myhill paradox of propositions within the framework of Church’s intensional logic. A predicative response places restrictions on the full comprehension schema, which asserts that every formula determines a higher-order entity. In addition to motivating the restriction on the comprehension schema from intuitions about the stability of reference, this paper contains a consistency proof for the predicative response to the Russell-Myhill paradox. The models used to establish this consistency also model other axioms (...) of Church’s intensional logic that have been criticized by Parsons and Klement: this, it turns out, is due to resources which also permit an interpretation of a fragment of Gallin’s intensional logic. Finally, the relation between the predicative response to the Russell-Myhill paradox of propositions and the Russell paradox of sets is discussed, and it is shown that the predicative conception of set induced by this predicative intensional logic allows one to respond to the Wehmeier problem of many non-extensions. (shrink)

A crucial part of the contemporary interest in logicism in the philosophy of mathematics resides in its idea that arithmetical knowledge may be based on logical knowledge. Here an implementation of this idea is considered that holds that knowledge of arithmetical principles may be based on two things: (i) knowledge of logical principles and (ii) knowledge that the arithmetical principles are representable in the logical principles. The notions of representation considered here are related to theory-based and structure-based notions of representation (...) from contemporary mathematical logic. It is argued that the theory-based versions of such logicism are either too liberal (the plethora problem) or are committed to intuitively incorrect closure conditions (the consistency problem). Structure-based versions must on the other hand respond to a charge of begging the question (the circularity problem) or explain how one may have a knowledge of structure in advance of a knowledge of axioms (the signature problem). This discussion is significant because it gives us a better idea of what a notion of representation must look like if it is to aid in realizing some of the traditional epistemic aims of logicism in the philosophy of mathematics. (shrink)

In this paper I develop an account of member obligation: the obligations that fall on the members of an obligated collective in virtue of that collective obligation. I use this account to argue that unorganized collections of individuals can constitute obligated agents. I argue first that, to know when a collective obligation entails obligations on that collective’s members, we have to know not just what it would take for each member to do their part in satisfying the collective obligation, but (...) also what they should do if they cannot do their part because others will not do theirs. I go on to argue (contra recent proposals) that it is not good enough for members in this situation to reasonably believe that others will not do their part. Rather, for a member of an obligated collective to permissibly escape doing her part in a collective obligation, she must both reasonably doubt that others will do their part and stand ready to act in case others do as well. -/- This necessary condition for collective obligation points the way to plausible sufficient conditions – conditions that, I argue, allow unstructured collectives to bear obligations. For (a) if a collective’s members are individually obligated to be ready to do their part, in a given collective action, and (b) if that individual readiness makes it sufficiently likely that the collective will in fact act, then it is hard to see what could block an attribution of collective obligation. In particular, in that case there ought to be no additional objection that there is no existing, organized “agent” on which the obligation might fall. For agents are, simply, things that can act. To be able to act is just to be able to succeed by trying. Unstructured collectives try to do something, I argue, when each member acts on their willingness to do their part in that thing if others do theirs; sometimes they succeed, producing a collective action. Some unstructured collectives, therefore, can succeed by trying; therefore, they can act; therefore they are agents. (shrink)

Humans can attribute beliefs to others. However, it is unknown to what extent this ability results from an innate biological endowment or from experience accrued through child development, particularly exposure to language describing others' mental states. We test the viability of the language exposure hypothesis by assessing whether models exposed to large quantities of human language display sensitivity to the implied knowledge states of characters in written passages. In pre‐registered analyses, we present a linguistic version of the False Belief Task (...) to both human participants and a large language model, GPT‐3. Both are sensitive to others' beliefs, but while the language model significantly exceeds chance behavior, it does not perform as well as the humans nor does it explain the full extent of their behavior—despite being exposed to more language than a human would in a lifetime. This suggests that while statistical learning from language exposure may in part explain how humans develop the ability to reason about the mental states of others, other mechanisms are also responsible. (shrink)

Frege's Grundgesetze was one of the 19th century forerunners to contemporary set theory which was plagued by the Russell paradox. In recent years, it has been shown that subsystems of the Grundgesetze formed by restricting the comprehension schema are consistent. One aim of this paper is to ascertain how much set theory can be developed within these consistent fragments of the Grundgesetze, and our main theorem shows that there is a model of a fragment of the Grundgesetze which defines a (...) model of all the axioms of Zermelo-Fraenkel set theory with the exception of the power set axiom. The proof of this result appeals to G\"odel's constructible universe of sets, which G\"odel famously used to show the relative consistency of the continuum hypothesis. More specifically, our proofs appeal to Kripke and Platek's idea of the projectum within the constructible universe as well as to a weak version of uniformization (which does not involve knowledge of Jensen's fine structure theory). The axioms of the Grundgesetze are examples of abstraction principles, and the other primary aim of this paper is to articulate a sufficient condition for the consistency of abstraction principles with limited amounts of comprehension. As an application, we resolve an analogue of the joint consistency problem in the predicative setting. (shrink)

Some modern cosmological models predict the appearance of Boltzmann Brains: observers who randomly fluctuate out of a thermal bath rather than naturally evolving from a low-entropy Big Bang. A theory in which most observers are of the Boltzmann Brain type is generally thought to be unacceptable, although opinions differ. I argue that such theories are indeed unacceptable: the real problem is with fluctuations into observers who are locally identical to ordinary observers, and their existence cannot be swept under the rug (...) by a choice of probability distributions over observers. The issue is not that the existence of such observers is ruled out by data, but that the theories that predict them are cognitively unstable: they cannot simultaneously be true and justifiably believed. (shrink)

Fallacies are things people commit, and when they commit them they do something wrong. What kind of activities are people engaged in when they commit fallacies, and in what way are they doing something wrong? Many different things are called fallacies. The diversity of the use of the concept of a fallacy suggests that we are dealing with a family of cases not related by a common essence. However, we suggest a simple account of the nature of fallacies which encompasses (...) them all, viz., the term “fallacy” is our most general term for criticizing any general procedure used for the fixation of beliefs that has an unacceptably high tendency to generate false or unfounded beliefs, relative to that method of fixing beliefs. Very different sorts of things called fallacies are examined in the light of this account, e.g., denying the antecedent, circular arguments, so-called informal fallacies, and propositions said to be fallacies. We do not provide a theory of fallacies. Still, on our account pretty much all of those things that have been called fallacies are fallacies, and they have been called fallacies for pretty much the same reasons. (shrink)

A non-technical introduction to quantum mechanics, the Everett interpretation, and the emergence of spacetime.