This paper develops an account of pluralities based on the following simple claim: some things are nothing over and above the individual things they comprise. For some, this may seem like a mysterious statement, perhaps even meaningless; for others, like a truism, trivial and inferentially inert. I show that neither reaction is correct: the claim is both tractable and has important consequences for a number of debates in philosophy.

This article introduces a new reflection principle. It is based on the idea that whatever is true in all entities of some kind is also true in a set-sized collection of them. Unlike standard reflection principles, it does not re-interpret parameters or predicates. This allows it to be both consistent in all higher-order languages and remarkably strong. For example, I show that in the language of second-order set theory with predicates for a satisfaction relation, it is consistent relative to the (...) existence of a 2-extendible cardinal (Theorem 7.12) and implies the existence of a proper class of 1-extendible cardinals (Theorem 7.9). (shrink)

Modal structuralism promises an interpretation of set theory that avoids commitment to abstracta. This article investigates its underlying assumptions. In the first part, I start by highlighting some shortcomings of the standard axiomatisation of modal structuralism, and propose a new axiomatisation I call MSST (for Modal Structural Set Theory). The main theorem is that MSST interprets exactly Zermelo set theory plus the claim that every set is in some inaccessible rank of the cumulative hierarchy. In the second part of the (...) article, I look at the prospects for supplementing MSST with a modal structural reflection principle, as suggested in Hellman (2015). I show that Hellman’s principle is inconsistent (Theorem 5.32), and argue that modal structural reflection principles in general are either incompatible with modal structuralism or extremely weak. (shrink)

Why and under what conditions are individuals altruistic to family and friends in their social networks? Evolutionary psychology suggests that such behaviour is primarily the product of adaptations for kin- and reciprocal altruism, dependent on the degree of genetic relatedness and exchange of benefits, respectively. For this reason, individuals are expected to be more altruistic to family members than to friends: whereas family members can be the recipients of kin and reciprocal altruism, friends can be the recipients of reciprocal altruism (...) only. However, there is a question about how the effect of kinship is implemented at the proximate psychological level. One possibility is that kinship contributes to some general measure of relationship quality (such as ‘emotional closeness’), which in turn explains altruism. Another possibility is that the effect of kinship is independent of relationship quality. The present study tests between these two possibilities. Participants (N= 111) completed a self-report questionnaire about their willingness to be altruistic, and their emotional closeness, to 12 family members and friends at different positions in their extended social networks. As expected, altruism was greater for family than friends, and greater for more central layers of the network. Crucially, the results showed that kinship made a significant unique contribution to altruism, even when controlling for the effects of emotional closeness. Thus, participants were more altruistic towards kin than would be expected if altruism was dependent on emotional closeness alone – a phenomenon we label a ‘kinship premium’. These results have implications for the ongoing debate about the extent to which kin relations and friendships are distinct kinds of social relationships, and how to measure the ‘strength of ties’ in social networks. (shrink)

One prominent criticism of the abstractionist program is the so-called Bad Company objection. The complaint is that abstraction principles cannot in general be a legitimate way to introduce mathematical theories, since some of them are inconsistent. The most notorious example, of course, is Frege’s Basic Law V. A common response to the objection suggests that an abstraction principle can be used to legitimately introduce a mathematical theory precisely when it is stable: when it can be made true on all sufficiently (...) large domains. In this paper, we raise a worry for this response to the Bad Company objection. We argue, perhaps surprisingly, that it requires very strong assumptions about the range of the second-order quantifiers; assumptions that the abstractionist should reject. (shrink)

Potentialism is the view that the universe of sets is inherently potential. It comes in two main flavours: height-potentialism and width-potentialism. It is natural to think that height and width potentialism are just aspects of a broader phenomenon of potentialism, that they might both be true. The main result of this paper is that this is mistaken: height and width potentialism are jointly inconsistent. Indeed, I will argue that height potentialism is independently committed to an ultimate background universe of sets, (...) an ultimate V, up to its height. (shrink)

Bob Hale has defended a new conception of properties that is broadly Fregean in two key respects. First, like Frege, Hale insists that every property can be defined by an open formula. Second, like Frege, but unlike later definabilists, Hale seeks to justify full impredicative property comprehension. The most innovative part of his defense, we think, is a “definability constraint” that can serve as an implicit definition of the domain of properties. We make this constraint formally precise and prove that (...) it fails to characterize the domain uniquely. Thus, we conclude, there is no easy road to impredicative definabilism. (shrink)

Humans are strongly lateralized for manual gestures at both individual and population levels. In contrast, the laterality bias in primates is less strong, leading some to suggest that lateralization evolved after the Pan and Homo lineages diverged. However, laterality in humans is also context-dependent, suggesting that observed differences in lateralization between primates and humans may be related to external factors such as the complexity of the social environment. Here we address this question in wild chimpanzees and examine the extent to (...) which the laterality of manual gestures is associated with social complexity. Right-handed gestures were more strongly associated with goal-directed communication such as repair through elaboration in response to communication failure than left-handed gestures. Right-handed gestures occurred in evolutionarily urgent contexts such as in interactions with central individuals in the network, including grooming reciprocity and mating, whereas left-handed gestures occurred in less-urgent contexts, such as travel and play. Right-handed gestures occurred in smaller parties and in the absence of social competition relative to left-handed gestures. Right-handed gestures increased the rate of activities indicating high physiological arousal in the recipient, whereas left-handed gestures reduced it. This shows that right- and left-handed gestures differ in cognitive and social complexity, with right-handed gestures facilitating more complex interactions in simpler social settings, whereas left-handed gestures facilitate more rewarding interactions in complex social settings. Differences in laterality between other primates and humans are likely to be driven by differences in the complexity of both the cognitive skills underpinning social interactions and the social environment. (shrink)

Classes are a kind of collection. Typically, they are too large to be sets. For example, there are classes containing absolutely all sets even though there is no set of all sets. But what are classes, if not sets? When our theory of classes is relatively weak, this question can be avoided. In particular, it is well known that von Neuman–Bernays–Godel class theory is conservative over the standard axioms of set theory ): anything NGB can prove about the sets is (...) already provable in ZFC. In this paper I will prove a new conservativity result for a much broader range of class theories. It tells us that as long as our set theory T contains an independently well-motivated reflection principle, anything provable about the sets in any reasonable class theory extending T is already provable in T. (shrink)

Applied Christian Ethics addresses selected themes in Christian social ethics. Part one shows the roots of contributors in the realist school; part two focuses on different levels of the significance of economics for social justice; and part three deals with both existential experience and government policy in war and peace issues.

Ethical theories consistent with Christian moral sensibilities must assure each neighbor's dignity and recognize his or her unique needs. While utilitarian ethics and duty-based ethics may fail to some degree in these respects, virtue ethics offers perspectives that echo more faithfully the parable of the Good Samaritan.