Discussion of atomistic and monistic theses about abstract reality.

When an entity ontologically depends on another entity, the former ‘presupposes’ or ‘requires’ the latter in some metaphysical sense. This paper defends a novel view, Dependence Deflationism, according to which ontological dependence is what I call an aggregative cluster concept: a concept which can be understood, but not fully analysed, as a ‘weighted total’ of constructive and modal relations. The view has several benefits: it accounts for clear cases of ontological dependence as well as the source of disagreement in controversial (...) ones; it gives a nice story about the evidential relevance of modal, mereological and set-theoretic facts to ontological dependence; and it makes sense of debates over the relation's formal properties. One important upshot of the deflationary account is that questions of ontological dependence are generally less deep and less interesting than usually thought. (shrink)

In his influential article ‘Essence and Modality’, Fine proposes a definition of necessity in terms of the primitive essentialist notion ‘true in virtue of the nature of’. Fine’s proposal is suggestive, but it admits of different interpretations, leaving it unsettled what the precise formulation of an Essentialist definition of necessity should be. In this paper, four different versions of the definition are discussed: a singular, a plural reading, and an existential variant of Fine’s original suggestion and an alternative version proposed (...) by Correia which is not based on Fine’s primitive essentialist notion. The first main point of the paper is that the singular reading is untenable. The second that given plausible background assumptions, the remaining three definitions are extensionally equivalent. The third is that, this equivalence notwithstanding, Essentialists should adopt Correia’s version of the definition, since both the existential variant, which has de facto been adopted as the standard version of the definition in the literature, and the plural reading suffer from problems connected to Fine’s primitive essentialist notion. (shrink)

In this paper, we explore the idea that sets depend on, or are grounded in, their members. It is said that a set depends on each of its members, and not vice versa. Members do not depend on the sets that they belong to. We show that the intuitive modal truth conditions for dependence, given in terms of possible worlds, do not accurately capture asymmetric dependence relations between sets and their members. We extend the modal truth conditions to include impossible (...) worlds and give a more satisfactory account of the dependence of a set on its members. Focusing on the case of singletons, we articulate a logical framework in which to evaluate set-theoretic dependence claims, using a normal first-order modal logic. We show that on this framework the dependence of a singleton on its single members follows from logic alone. However, the converse does not hold. (shrink)

This paper raises the question under what circumstances a plurality forms a set, parallel to the Special Composition Question for mereology. The range of answers that have been proposed in the literature are surveyed and criticised. I argue that there is good reason to reject both the view that pluralities never form sets and the view that pluralities always form sets. Instead, we need to affirm restricted set formation. Casting doubt on the availability of any informative principle which will settle (...) which pluralities form sets, the paper concludes by affirming a naturalistic approach to the philosophy of set theory. (shrink)

A prominent version of mathematical structuralism holds that mathematical objects are at bottom nothing but "positions in structures," purely relational entities without any sort of nature independent of the structure to which they belong. Such an ontology is often presented as a response to Benacerraf's "multiple reductions" problem, or motivated on hermeneutic grounds, as a faithful representation of the discourse and practice of mathematics. In this paper I argue that there are serious difficulties with this kind of view: its proponents (...) rely on a distinction between "essential" and "nonessential" features of mathematical objects, and there's no good way to articulate this distinction which is compatible with basic structuralist commitments. But all is not lost. For I further argue that the insights motivating structuralism (or at least those worth preserving) can be preserved without formulating the view in ontologically committal terms. (shrink)

This article for the Stanford Encyclopedia for Philosophy provides a state of the art survey and assessment of the contemporary debate about truth-makers, covering both the case for and against truth-makers. It explores 4 interrelated questions about truth-makers, (1) What is it to be a truth-maker? (2) Which range, or ranges, of truths are eligible to be made true (if any are)? (3) What kinds of entities are truth-makers? (4) What is the motivation for adopting a theory of truth-makers? And (...) adds that there's another question to often put aside by metaphysicians but has critical consequences for truth-makers: (5) What are the truth-bearers? (shrink)

The non-well-founded set theories described by Aczel (1988) have received attention from category theorists and computer scientists, but have been largely ignored by philosophers. At the root of this neglect might lie the impression that these theories do not embody a conception of set, but are rather of mere technical interest. This paper attempts to dispel this impression. I present a conception of set which may be taken as lying behind a non-well-founded set theory. I argue that the axiom AFA (...) is justified on the conception, which provides, contra Rieger (Mind 109:241–253, 2000), a rationale for restricting attention to the system based on this axiom. By making use of formal and informal considerations, I then make a case that most of the other axioms of this system are also justified on the conception. I conclude by commenting on the significance of the conception for the debate about the justification of the Axiom of Foundation. (shrink)

I argue that absolutism, the view that absolutely unrestricted quantification is possible, is to blame for both the paradoxes that arise in naive set theory and variants of these paradoxes that arise in plural logic and in semantics. The solution is restrictivism, the view that absolutely unrestricted quantification is not possible. -/- It is generally thought that absolutism is true and that restrictivism is not only false, but inexpressible. As a result, the paradoxes are blamed, not on illicit quantification, but (...) on the logical conception of set which motivates naive set theory. The accepted solution is to replace this with the iterative conception of set. -/- I show that this picture is doubly mistaken. After a close examination of the paradoxes in chapters 2--3, I argue in chapters 4 and 5 that it is possible to rescue naive set theory by restricting quantification over sets and that the resulting restrictivist set theory is expressible. In chapters 6 and 7, I argue that it is the iterative conception of set and the thesis of absolutism that should be rejected. (shrink)

In a recent series of papers, I and others have advanced new logical approaches to familiar paradoxes. The key to these approaches is to accept full classical logic, and to accept the principles that cause paradox, while preventing trouble by allowing a certain sort ofnontransitivity. Earlier papers have treated paradoxes of truth and vagueness. The present paper will begin to extend the approach to deal with the familiar paradoxes arising in naive set theory, pointing out some of the promises and (...) pitfalls of such an approach. (shrink)

According to the iterative conception of sets, standardly formalized by ZFC, there is no set of all sets. But why is there no set of all sets? A simple-minded, though unpopular, “minimal” explanation for why there is no set of all sets is that the supposition that there is contradicts some axioms of ZFC. In this paper, I first explain the core complaint against the minimal explanation, and then argue against the two main alternative answers to the guiding question. I (...) conclude the paper by outlining a close alternative to the minimal explanation, the conception-based explanation, that avoids the core complaint against the minimal explanation. (shrink)

I develop a novel argument against the claim that ordinals are sets. In contrast to Benacerraf’s antireductionist argument, I make no use of covert epistemic assumptions. Instead, my argument uses considerations of ontological dependence. I draw on the datum that sets depend immediately and asymmetrically on their elements and argue that this datum is incompatible with reductionism, given plausible assumptions about the dependence profile of ordinals. In addition, I show that a structurally similar argument can be made against the claim (...) that cardinals are sets. (shrink)

The phrase ‘in virtue of’ is a mainstay of metaphysical discourse. In recent years, many philosophers have argued that we should understand this phrase, as metaphysicians use it, in terms of a concept of metaphysical dependence called ‘grounding’.This dissertation explores a range of central issues in the theory of grounding. Chapter 1 introduces the intuitive concept of grounding and discusses some compulsory questions in the theory of grounding. Chapter 2 focusses on scepticism on grounding, according to which the recent philosophical (...) interest in the topic is misguided. In chapter 3 I discuss grounding’s explanatory roles. Chapter 4 focusses on the claim that if an entity is grounded then it is an ontological free lunch. Chapter 5 discusses and rejects the claim that groundingis a relation between facts. This conclusion raises a problem: if grounding is not a relation between facts it becomes difficult to specify the connections between grounding and explanation and grounding and necessity. But not only is it desirable to specify these relations, it is essential for establishing that grounding is able to play the explanatory roles that are discussed in chapter 3. Chapter 6 responds to this problem by outlining an approach to grounding based on David Lewis’s (2003) theory of truthmaking. Against this backdrop I discuss, in chapter 7, some issues in the logic of grounding. (shrink)