Some hold that the lesson of Russell’s paradox and its relatives is that mathematical reality does not form a ‘definite totality’ but rather is ‘indefinitely extensible’. There can always be more sets than there ever are. I argue that certain contact puzzles are analogous to Russell’s paradox this way: they similarly motivate a vision of physical reality as iteratively generated. In this picture, the divisions of the continuum into smaller parts are ‘potential’ rather than ‘actual’. Besides the intrinsic interest of (...) this metaphysical picture, it has important consequences for the debate over absolute generality. It is often thought that ‘indefinite extensibility’ arguments at best make trouble for mathematical platonists; but the contact arguments show that nominalists face the same kind of difficulty, if they recognize even the metaphysical possibility of the picture I sketch. (shrink)

Languages involving modalities and languages involving vagueness have each been thoroughly studied. On the other hand, virtually nothing has been said about the interaction of modality and vagueness. This paper aims to start filling that gap. Section 1 is a discussion of various possible sources of vague modality. Section 2 puts forward a model theory for a quantified language with operators for modality and vagueness. The model theory is followed by a discussion of the resulting logic. In Section 3, the (...) framework will permit us to address a puzzle raised by Elizabeth Barnes and Robert Williams. (shrink)

Semantic interpretations of both natural and formal languages are usually taken to involve the specification of a domain of entities with respect to which the sentences of the language are to be evaluated. A question that has received much attention of late is whether there is unrestricted quantification, quantification over a domain comprising absolutely everything there is. Is there a discourse or inquiry that has absolute generality? After framing the debate, this article provides an overview of the main arguments for (...) and against the possibility of unrestricted quantification, highlighting some of the broader implications of the debate. (shrink)

There are certain metaphysically interesting arguments ‘from vagueness’, for unrestricted mereological composition and for four-dimensionalism, which involve a claim to the effect that idioms for unrestricted quantification are precise. An elaboration of Lewis’ argument for this claim, which assumes the view of vagueness as semantic indecision, is presented. It is argued that the argument also works according to other views on the nature of vagueness, which also require for an expression to be vague that there are different admissible alternatives of (...) the relevant sort, such as epistemicism, as defended by Williamson. Recent attempts to resist the argument are discussed and rejected. (shrink)

Jan Westerhoff defends an account of thoroughgoing non-foundationalism that he calls “irrealism,” which is implicitly modeled on a Madhyamaka Buddhist view. In this paper, I begin by raising worries about the irrealist’s account of human cognition as taking place in a brain-based representational interface. Next, I pose first-order and higher-order challenges to how the irrealist—who defends a kind of global error theory—can sensibly accommodate an unlocalized appearance-reality distinction, both metaphysically and epistemologically. Finally, although Westerhoff insists that irrealism itself is not (...) an ontological theory and that the irrealist’s rejection of absolutely general quantification precludes his commitment to any ultimately true theories, I propose strategies inspired by the Svātantrika commentarial tradition of Madhyamaka for how the irrealist might develop a lightweight account of unrestricted quantification that could be used to advance a lightweight ultimately true theory. This, I suggest, may allow the irrealist to (i) preserve a commitment to an unlocalized appearance-reality distinction, (ii) underwrite a distinction between ordinary veridical states and metaphysically accurate epistemic states, and (iii) provide an explanation for the massive error that he claims characterizes ordinary cognition. (shrink)

Call a quantifier ‘unrestricted’ if it ranges over absolutely all objects. Arguably, unrestricted quantification is often presupposed in philosophical inquiry. However, developing a semantic theory that vindicates unrestricted quantification proves rather difficult, at least as long as we formulate our semantic theory within a classical first-order language. It has been argued that using a type theory as framework for our semantic theory provides a resolution of this problem, at least if a broadly Fregean interpretation of type theory is assumed. However, (...) the intelligibility of this interpretation has been questioned. In this paper I introduce a type-free theory of properties that can also be used to vindicate unrestricted quantification. This alternative emerges very naturally by reflecting on the features on which the type-theoretic solution of the problem of unrestricted quantification relies. Although this alternative theory is formulated in a non-classical logic, it preserves the deductive strength of classical strict type theory in a natural way. The ideas developed in this paper make crucial use of Russell’s notion of range of significance. (shrink)

Discussion of new axioms for set theory has often focused on conceptions of maximality, and how these might relate to the iterative conception of set. This paper provides critical appraisal of how certain maximality axioms behave on different conceptions of ontology concerning the iterative conception. In particular, we argue that forms of multiversism (the view that any universe of a certain kind can be extended) and actualism (the view that there are universes that cannot be extended in particular ways) face (...) complementary problems. The latter view is unable to use maximality axioms that make use of extensions, where the former has to contend with the existence of extensions violating maximality axioms. An analysis of two kinds of multiversism, a Zermelian form and Skolemite form, leads to the conclusion that the kind of maximality captured by an axiom differs substantially according to background ontology. (shrink)

Penelope Maddy has recently addressed the set-theoretic multiverse, and expressed reservations on its status and merits ([Maddy, 2017]). The purpose of the paper is to examine her concerns, by using the interpretative framework of set-theoretic naturalism. I first distinguish three main forms of 'multiversism', and then I proceed to analyse Maddy's concerns. Among other things, I take into account salient aspects of multiverse-related mathematics , in particular, research programmes in set theory for which the use of the multiverse seems to (...) be crucial, and show how one may provide responses to Maddy's concerns based on a careful analysis of 'multiverse practice'. (shrink)

This thesis comprises three main chapters—each comprising one relatively standalone paper. The unifying theme is fragmentalism about truth, which is the view that the predicate “true” either expresses distinct concepts or expresses distinct properties. -/- In Chapter 1, I provide a formal development of alethic pluralism. Pluralism is the view that there are distinct truth properties associated with distinct domains of subject matter, where a truth property satisfies certain truth-characterizing principles. On behalf of pluralists, I propose an account of logic (...) and semantics that shows how they can answer central conceptual and logical challenges for their view. -/- In Chapter 2, I motivate and develop a modal account of propositions on the basis of an iterative conception of propositions, where the modality is logico-mathematical. The modal account of propositions takes the conception to motivate an inherently potential hierarchy of propositions. I show that the account helps provide satisfying solutions to the intensional paradoxes of Russell-Myhill, Kaplan, and Prior. -/- In Chapter 3, I propose that “true” is polysemous. I suggest that “true” is initially polysemous between correspondence truth and disquotational truth, and further polysemous between the meanings corresponding to the subconcepts of the concept truth generated by the indefinite extensibility of that concept. I show that the proposal provides satisfying solutions to the semantic paradoxes. (shrink)

Abstract: This paper explores the relevance of themes from Wittgenstein's Tractatus Logico-Philosophicus to the ongoing discussion of metaphysical nihilism. I set out by showing how metaphysical nihilism is of paramount importance for cosmological arguments. Metaphysical nihilism is the position that there might have been nothing. Two conflicting intuitions emerge from a survey of discussions of metaphysical nihilism: Firstly, that metaphysical nihilism is true, and secondly, that formulations of the position are somehow unclear or nonsensical. By considering formalizations of philosophical language, (...) the second intuition is sharpened, while the first intuition is given expression through the Tractarian distinction between what is said and what is shown by our symbolism. I conclude by exploring and rejecting objections to making metaphysical nihilism a scientific, rather than a philosophical question. (shrink)

This paper explores the relevance of Wittgenstein’s early work for treatments of religion. The first section briefly outlines some different interpretive possibilities with respect to early Wittgenstein’s thinking. The following section explores the idea that what is important about early Wittgenstein’s work resides not in the meaning of the text as such, but in elements of his Jewish heritage. The third section outlines the immediate task that is undertaken in the body of the TLP. Central notions are those of mysticism (...) and logical form and the two concluding sections explore how these notions play a role in treatments of religion. This they do either by exhibiting parallels between themes in theology and themes in logic, or they do so by playing a role in assessing cosmological arguments in the philosophy of religion. (shrink)

The Russellian argument against the possibility of absolutely unrestricted quantification can be answered by the partisan of that quantification in an apparently easy way, namely, arguing that the objects used in the argument do not exist because they are defined in a viciously circular fashion. We show that taking this contention along as a premise and relying on an extremely intuitive Principle of Determinacy, it is possible to devise a reductio of the possibility of absolutely unrestricted quantification. Therefore, there are (...) intuitive reasons to believe that the counter-argument fails to support the possibility of absolutely unrestricted quantification. (shrink)