Cantor argued that absolute infinity is beyond mathematical comprehension. His arguments imply that the domain of mathematics cannot be grasped by mathematical means. We argue that this inability constitutes a foundational problem. For Cantor, however, the domain of mathematics does not belong to mathematics, but to theology. We thus discuss the theological significance of Cantor’s treatment of absolute infinity and show that it can be interpreted in terms of negative theology. Proceeding from this interpretation, we refer to the recent debate (...) on absolute generality and argue that the method of diagonalization constitutes a modern version of the vianegativa. On our reading, negative theology can evoke an attitude of humility with respect to the boundedness of the human condition. Along these lines, we think that the foundational problem of mathematics concerning its domain can be addressed through a methodological attitude of humility. (shrink)

An increasing amount of contemporary philosophy of mathematics posits, and theorizes in terms of special kinds of mathematical modality. The goal of this paper is to bring recent work on higher-order metaphysics to bear on the investigation of these modalities. The main focus of the paper will be views that posit mathematical contingency or indeterminacy about statements that concern the ‘width’ of the set theoretic universe, such as Cantor’s continuum hypothesis. Within a higher-order framework I show that contingency about the (...) width of the set-theoretic universe refutes two orthodoxies concerning the structure of modal reality: the view that the broadest necessity has a logic of S5, and the ‘Leibniz biconditionals’ stating that what is possible, in the broadest sense of _possible_, is what is true in some possible world. Nonetheless, I suggest that the underlying picture of modal set-theory is coherent and has attractions. (shrink)

Realists about possible worlds typically identify possible worlds with abstract objects, such as propositions or properties. However, they face a significant objection due to Lewis (1986), to the effect that there is no way to explain how possible worlds-as-abstract objects represent possibilities. In this paper, I describe a response to this objection on behalf of realists. The response is to identify possible worlds with propositions, but to deny that propositions are abstract objects, or indeed objects at all. Instead, I argue (...) that realists should follow Prior (1971) and others in treating higher-order quantification (e.g. quantification into predicate or sentence position) as a genuine form of quantification in its own right, and so in particular, not to be analysed in terms of first-order quantification over abstract objects. I argue that from this ‘higher-orderist’ perspective, there is a relatively straightforward answer to the question of how possible worlds-as-propositions succeed in representing possibilities. (shrink)

The Geach‐Kaplan sentence is alleged to be an example of a non‐first‐orderizable sentence, and the proof of the alleged non‐first‐orderizability is credited to David Kaplan. However, there is also a widely shared intuition that the Geach‐Kaplan sentence is still first‐orderizable by invoking sets or other extra non‐logical resources. The plausibility of this intuition is particularly crucial for first‐orderism, namely, the thesis that all our scientific discourse and reasoning can be adequately formalized by first‐order logic. I first argue that the Geach‐Kaplan (...) sentence is, in fact, not first‐orderizable even by invoking extra non‐logical resources, in any sense that is acceptable for first‐orderism and adequately corresponds to the sense in which the Geach‐Kaplan sentence is deemed to be non‐first‐orderizable simpliciter via Kaplan's proof. To address this problem on behalf of first‐orderism, I then propose an alternative conception of first‐orderizability in the sense of which the Geach‐Kaplan sentence and any other second‐order sentences become first‐orderizable by invoking extra non‐logical resources; furthermore, in certain circumstances, they are first‐orderizable without incurring any extra ontological commitment. My analysis also turns out to yield (as a biproduct) a significant enhancement of the so‐called paradox of plurality. (shrink)

Higher-order logic, with its type-theoretic apparatus known as the simple theory of types (STT), has increasingly come to be employed in theorizing about properties, relations, and states of affairs—or ‘intensional entities’ for short. This paper argues against this employment of STT and offers an alternative: ordinal type theory (OTT). Very roughly, STT and OTT can be regarded as complementary simplifications of the ‘ramified theory of types’ outlined in the Introduction to Principia Mathematica (on a realist reading). While STT, understood as (...) a theory of intensional entities, retains the Fregean division of properties and relations into a multiplicity of categories according to their adicities and ‘input types’ and discards the division of intensional entities into different ‘orders’, OTT takes the opposite approach: it retains the hierarchy of orders (though with some modifications) and discards the categorisation of properties and relations according to their adicities and input types. In contrast to STT, this latter approach avoids intensional counterparts of the Epimenides and related paradoxes. Fundamental intensional entities lie at the base of the proposed hierarchy and are also given a prominent part to play in the individuation of non-fundamental intensional entities. (shrink)

According to ontological nihilism nothing exists. A recent argument purports to show that this view is indefensible, since its most plausible formulations are tacitly committed to quantificational claims that are inconsistent with the nihilist's view that there aren't any existents. I show that this objection begs the question against the nihilist. The objector's argument relies on an equivalence principle implying that claims which nihilists regard as non-quantificational should nonetheless be interpreted as equivalent to quantified claims, given that both kinds of (...) assertion are assertible in the same contexts. This style of reasoning is supported by considerations of charity, which suggest that similarities in inter-personal patterns of language use provide evidence of similarity in the semantic content of speakers' utterances. However, related considerations also suggest that intra-personal patterns of language use constitute significant semantic evidence of the same kind. When we appreciate this and further recognize that nihilists are disposed to assert the negation of any positive, quantified claim, we see that it is too much to expect principles of semantic interpretation based on considerations of charity to decisively refute nihilism. This result has interesting metametaphysical implications concerning the role and scope of analogous arguments based on principles of interpretive charity. (shrink)

Quine maintained that philosophical and scientific theorizing should be conducted in an untyped language, which has just one style of variables and quantifiers. By contrast, typed languages, such as those advocated by Frege and Russell, include multiple styles of variables and matching kinds of quantification. Which form should our theories take? In this article, I argue that expressivity does not favour typed languages over untyped ones.

This paper revisits a challenge for contextualist approaches to paradoxes such as the Liar paradox and Russell’s paradox. Contextualists argue that these paradoxes are to be resolved by appeal to context dependence. This can offer some nice and effective ways to avoid paradox. But there is a problem. Context dependence is, at least to begin with, a phenomenon in natural language. Is there really such context dependence as the solutions to paradoxes require, and is it really just a familiar linguistic (...) phenomenon at work? Not so clearly. In earlier work, I argued that the required form of context dependence does not look like our most familiar instances of context dependence in natural language. I called this extraordinary context dependence. In this paper, I shall explore, somewhat tentatively, a way that we can see the context dependence needed to address paradoxes as not so extraordinary. Doing so will also allow us to connect thinking about the context dependence of quantifier domains with some interesting ideas about the distinctive semantic properties of certain quantifiers. (shrink)

Nominalists need some account of correctness for sentences committed to the existence of abstract objects. This paper proposes a new statement of such conditions specifically for properties. The account builds on an earlier proposal of mine, but avoids the counter-examples against the latter pointed out by Thomas Schindler, particularly, the sentence ‘There are inexpressible properties’. I argue that the new proposal is independently motivated and more faithful to the spirit of the kind of error-theoretic nominalism that the original proposal was (...) part of. Along the way, I also propose novel treatments of Arthur Pap’s and W.V.O. Quine’s notoriously hard-to-paraphrase sentences about abstracta. (shrink)

In the language of second-order logic, first- and second-order variables are distinguished syntactically and cannot be grammatically substituted. According to a prominent argument for the deployment of these languages, these substitution failures are necessary to block the derivation of paradoxes that result from attempts to generalize over predicate interpretations. I first examine previous approaches which interpret second-order sentences using expressions of natural language and argue that these approaches undermine these syntactic restrictions. I then examine Williamson’s primitivist approach according to which (...) second-order sentences are not offered readings in a previously understood language. I argue that the syntactic restrictions alone do not block the derivation of the paradox, unless they are backed by a principled reason that the language cannot be expanded to allow the grammatical substitution of first- and second- order variables. I argue that there is neither a syntactic nor a semantic principle that prohibits such an expansion. (shrink)

This volume announces a new era in the philosophy of God. Many of its contributions work to create stronger links between the philosophy of God, on the one hand, and mathematics or metamathematics, on the other hand. It is about not only the possibilities of applying mathematics or metamathematics to questions about God, but also the reverse question: Does the philosophy of God have anything to offer mathematics or metamathematics? The remaining contributions tackle stereotypes in the philosophy of religion. The (...) volume includes 35 contributions. It is divided into nine parts: 1. Who Created the Concept of God; 2. Omniscience, Omnipotence, Timelessness and Spacelessness of God; 3. God and Perfect Goodness, Perfect Beauty, Perfect Freedom; 4. God, Fundamentality and Creation of All Else; 5. Simplicity and Ineffability of God; 6. God, Necessity and Abstract Objects; 7. God, Infinity, and Pascal's Wager; 8. God and (Meta-)Mathematics; and 9. God and Mind. (shrink)

Universals are putative objects like wisdom, morality, redness, etc. Although we believe in properties (which, we argue, are not a kind of object), we do not believe in universals. However, a number of ordinary, natural language constructions seem to commit us to their existence. In this paper, we provide a fictionalist theory of universals, which allows us to speak as if universals existed, whilst denying that any really do.

W. V. Quine famously defended two theses that have fallen rather dramatically out of fashion. The first is that intensions are “creatures of darkness” that ultimately have no place in respectable philosophical circles, owing primarily to their lack of rigorous identity conditions. However, although he was thoroughly familiar with Carnap’s foundational studies in what would become known as possible world semantics, it likely wouldn’t yet have been apparent to Quine that he was fighting a losing battle against intensions, due in (...) large measure to developments stemming from Carnap’s studies and culminating in the work of Kripke, Hintikka, and Bayart. These developments undermined Quine’s crusade against intensions on two fronts. First, in the context of possible world semantics, intensions could after all be given rigorous identity conditions by defining them (in the simplest case) as functions from worlds to appropriate extensions, a fact exploited to powerful and influential effect in logic and linguistics by the likes of Kaplan, Montague, Lewis, and Cresswell. Second, the rise of possible world semantics fueled a strong resurgence of metaphysics in contemporary analytic philosophy that saw properties and propositions widely, fruitfully, and unabashedly adopted as ontological primitives in their own right. This resurgence — happily, in my view — continues into the present day. -/- For a time, at any rate, Quine experienced somewhat better success with his second thesis: that higher-order logic is, at worst, confused and, at best, a quirky notational alternative to standard first-order logic. However, Quine notwithstanding, a great deal of recent work in formal metaphysics transpires in a higher-order logical framework in which properties and propositions fall into an infinite hierarchy of types of (at least) every finite order. Initially, the most philosophically compelling reason for embracing such a framework since Russell first proposed his simple theory of types was simply that it provides a relatively natural explanation of the paradoxes. However, since the seminal work of Prior there has been a growing trend to consider higher-order logic to be the most philosophically natural framework for metaphysical inquiry, many of the contributors to this volume being among the most important and influential advocates of this view. Indeed, this is now quite arguably the dominant view among formal metaphysicians. -/- In this paper, and against the current tide, I will argue in §1 that there are still good reasons to think that Quine’s second battle is not yet lost and that the correct framework for logic is first-order and type-free — properties and propositions, logically speaking, are just individuals among others in a single domain of quantification — and that it arises naturally out of our most basic logical and semantical intuitions. The data I will draw upon are not new and are well-known to contemporary higher-order metaphysicians. However, I will try to defend my thesis in what I believe is a novel way by suggesting that these basic intuitions ground a reasonable distinction between “pure” logic and non-logical theory, and that Russell-style semantic paradoxes of truth and exemplification arise only when we move beyond the purely logical and, hence, do not of themselves provide any strong objection to a type-free conception of properties and propositions. -/- Most of my arguments in §1 are largely independent of any specific account of the nature of properties and propositions beyond their type-freedom. However, I will in addition argue that there are good reasons to take propositions, at least, to be very fine-grained. My arguments are thus bolstered significantly if it can be shown that there are in fact well-defined examples of logics that are not only type-free but which comport with such a conception of propositions. It is the purpose of §2 to lay out a logic of this sort in some detail, drawing especially upon work by George Bealer and related work of my own. With the logic in place, it will be possible to generalize the line of argument noted above regarding Russell-style paradoxes and, in §3, apply it to two propositional paradoxes — the Prior-Kaplan paradox and the Russell-Myhill paradox — that are often taken to threaten the sort of account developed here. (shrink)

Although higher-order metaphysics seems prima facie to be a promising new approach to metaphysics, it is nonetheless based on a mistake. This mistake is tied to a misuse of formal languages in metaphysics in general, not just to the use of higher-order rather than lower-order languages. I hope to highlight the mistake by discussing a popular recent example of higher- order metaphysics: the argument that reality is not structured using reasoning inspired by the Russell-Myhill paradox. A key issue will be (...) the relationship between higher-order quantification in formal languages and quantification in English, in particular the questions whether such quantifiers are already to be found in English, and whether they could simply be added to English if not in order to gain expressive strength. I will conclude that little of metaphysical significance follows from this Russell-Myhill inspired argument against structure. The reason why the results of higher-order logic show little of metaphysical significance generalizes to other uses of higher-order logic in metaphysical theorizing, and with it supports that higher-order metaphysics is a flawed research program. I end with some reflections on what positive role formal tools like higher-order logic can have in metaphysics, and what the fundamental mistake is on which higher-order metaphysics is based. (shrink)

This paper argues that we should solve paradoxes for propositions (such as the Russell–Myhill paradox) in essentially the same way that we solve Russellian paradoxes for sets. That is, the standard, iterative approach to sets is extended to include properties, and then the resulting hierarchy of sets and properties is used to construct propositions. Propositions on this account are structured in the sense of mirroring the sentences that express them, and they would seem to serve the needs of philosophers of (...) language and metaphysicians who rely on such propositions. The resulting account has limitations, comparable to those faced by the iterative approach to sets, but it is argued to be in important ways preferable to others that have been proposed. (shrink)

What is absolutely unrestricted quantification? We distinguish two theoretical roles and identify two conceptions of absolute generality: maximally strong generality and maximally inclusive generality. We also distinguish two corresponding kinds of absolute domain. A maximally strong domain contains every potential counterexample to a generalisation. A maximally inclusive domain is such that no domain extends it. We argue that both conceptions of absolute generality are legitimate and investigate the relations between them. Although these conceptions coincide in standard settings, we show how (...) they diverge under more complex assumptions about the structure of meaningful predication, such as cumulative type theory. We conclude by arguing that maximally strong generality is the more theoretically valuable conception. (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)

After a critical presentation of the debate between absolutists and relativists regarding generality where I show that the debate is framed in a way that is bound to be harmful to the relativist’s position, I examine critically one of the customary arguments advanced against the relativist: the expressibility objection (according to which the relativist would be logically unable to express her own position). I then propose a radical way out of this debate-usually centered on semantic paradoxes-by arguing that it rests (...) on an unintelligible notion of “object”. I finally introduce a useful distinction between omnis and totus to elucidate the notions in play. (shrink)

Erik Wielenberg [2014] and Mark Murphy [2017], [2018] have defended a series of arguments for the conclusion that creatures are not good intrinsically. In response, I take two steps. First, I introduce a conception of intrinsic value that makes created intrinsic value unproblematic. Second, I respond to their arguments in turn. The first argument is from the sovereignty-aseity intuition and an analysis of intrinsicality that makes derivative good extrinsic. I challenge the analysis. The second comes from a conception of perfection (...) where God exhausts all possibilities for goodness. I challenge this conception. The third comes from the contingency of creation and the internal logic of intrinsic value. I challenge the assumption that connects the two. I then present an argument from reference magnetism that the only conception of God, God's perfections, and religious language that makes it plausible that the sentence "God is intrinsically good and creatures are not" is true involves a strongly univocalist position. A view that is either unstatable or places strong demands on our conception of God, God's perfections and religious language thereby loses plausibility. (shrink)

This paper develops a novel theory of abstraction—what we call collective abstraction. The theory solves a notorious problem for noneliminative structuralism. The noneliminative structuralist holds that in addition to various isomorphic systems there is a pure structure that can be abstracted from each of these systems; but existing accounts of abstraction fail for nonrigid systems like the complex numbers. The problem with the existing accounts is that they attempt to define a unique abstraction operation. The theory of collective abstraction instead (...) simultaneously defines a collection of distinct abstraction operations, each of which maps a system to its corresponding pure structure. The theory is precisely formulated in an essentialist language. This allows us to throw new light on the question to what extent structuralists are committed to symmetric dependence. Finally, we apply the theory of collective abstraction to solve a problem about converse relations. (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)

Dispositional Essentialism, as commonly conceived, consists in the claims that at least some of the fundamental properties essentially confer certain causal-nomological roles on their bearers, and that these properties give rise to the natural modalities. As such, the view is generally taken to be committed to a realist conception of properties as either universals or tropes, and to be thus incompatible with nominalism as understood in the strict sense. Pace this common assumption of the ontological import of Dispositional Essentialism, the (...) aim of this paper is to explore a nominalist version of the view, Austere Nominalist Dispositional Essentialism. The core features of the proposed account are that it eschews all kinds of properties, takes certain predicative truths as fundamental, and employs the so-called generic notion of essence. As I will argue, the account is significantly closer to the core idea behind Dispositional Essentialism than the only nominalist account in the vicinity of Dispositional Essentialism that has been offered so far—Ann Whittle’s Causal Nominalism—and is immune to crucial problems that affect this view. (shrink)

Is it possible to maintain classical logic, stay close to classical semantics, and yet accept that language might be semantically indeterminate? The article gives an affirmative answer by Ramsifying classical semantics, which yields a new semantic theory that remains much closer to classical semantics than supervaluationism but which at the same time avoids the problematic classical presupposition of semantic determinacy. The resulting Ramsey semantics is developed in detail, it is shown to supply a classical concept of truth and to fully (...) support the rules and metarules of classical logic, and it is applied to vague terms as well as to theoretical or open-ended terms from mathematics and science. The theory also demonstrates how diachronic or synchronic interpretational continuity across languages is compatible with semantic indeterminacy. (shrink)

Some views articulate reality's hierarchical structure using relations from the fundamental to representations of reality. Other views instead use relations from the fundamental to constituents of non-representational reality. This paper argues against the first kind of view.

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)

This paper investigates the connection between two recent trends in philosophy: higher-orderism and conceptual engineering. Higher-orderists use higher-order quantifiers (in particular quantifiers binding variables that occupy the syntactic positions of predicates) to express certain key metaphysical doctrines, such as the claim that there are properties. I argue that, on a natural construal, the higher-orderist approach involves an engineering project concerning, among others, the concept of existence. I distinguish between a modest construal of this project, on which it aims at engineering (...) higher-order analogues of the familiar notion of first-order existence, and an ambitious construal, on which it additionally aims at engineering a broadened notion of existence that subsumes first-order and higher-order existence. After identifying a substantial problem for the ambitious project, I investigate a possible response which is based on adopting a cumulative type theory as the background higher-order logic. While effective against the problem at hand, this strategy turns out to undermine a major reason to embrace higher-orderism in the first place, namely the idea that higher-orderism dissolves a range of otherwise intractable debates in metaphysics. Higher-orderists are therefore best advised to pursue their engineering project on the modest variant and against the background of standard type theory. (shrink)

The Received View of particles in quantum mechanics is that they are indistinguishable entities within their kinds and that, as a consequence, they are not individuals in the metaphysical sense and self-identity does not meaningfully apply to them. Nevertheless cardinality does apply, in that one can have n> 1 such particles. A number of authors have recently argued that this cluster of claims is internally contradictory: roughly, that having more than one such particle requires that the concepts of distinctness and (...) identity must apply after all. A common thread here is that the notion of identity is too fundamental to forego in any metaphysical account. I argue that this argument fails. I then argue that the failure of individuality and identity applies also to macroscopic physical objects, that the problems cannot be constrained to apply only within the microscopic realm. (shrink)

Inspired by Cantor's Theorem (CT), orthodoxy takes infinities to come in different sizes. The orthodox view has had enormous influence in mathematics, philosophy, and science. We will defend the contrary view---Countablism---according to which, necessarily, every infinite collection (set or plurality) is countable. We first argue that the potentialist or modal strategy for treating Russell's Paradox, first proposed by Parsons (2000) and developed by Linnebo (2010, 2013) and Linnebo and Shapiro (2019), should also be applied to CT, in a way that (...) vindicates Countabilism. Our discussion dovetails with recent independently developed treatments of CT in Meadows (2015), Pruss (2020), and Scambler (2021), aimed at establishing the mathematical viability, and therefore epistemic possibility, of Countabilism. Unlike these authors, our goal isn't to vindicate the mathematical underpinnings of Countabilism. Rather, we aim to argue that, given that Countabilism is mathematically viable, Countabilism should moreover be regarded as true. After clarifying the modal content of Countabilism, we canvas some of Countabilism's many positive implications, including that Countabilism provides the best account of the pervasive independence phenomena in set theory, and that Countabilism has the power to defuse several persistent puzzles and paradoxes found in physics and metaphysics. We conclude that in light of its theoretical and explanatory advantages, Countabilism is more likely true than not. (shrink)

In this paper, I propose that 'true’ is polysemous, and thus ambiguous. I suggest that the semantic paradoxes both motivates taking 'true’ to be polysemous and shows that the concept truth is indefinitely extensible. In doing so, I explain that 'true’ is polysemous between the meanings corresponding to the subconcepts of the concept truth generated by such indefinite extensibility. I conclude that the proposal provides satisfying solutions to the semantic paradoxes.

Subverting a once widely held Quinean paradigm, there is a growing consensus among philosophers of logic that higher-order quantifiers (which bind variables in the syntactic position of predicates and sentences) are a perfectly legitimate and useful instrument in the logico-philosophical toolbox, while neither being reducible to nor fully explicable in terms of first-order quantifiers (which bind variables in singular term position). This article discusses the impact of this quantificational paradigm shift on metaphysics, focussing on theories of properties, propositions, and identity, (...) as well as on the metaphysics of modality. (shrink)

Standard Type Theory, STT, tells us that b^n(a^m) is well-formed iff n=m+1. However, Linnebo and Rayo have advocated the use of Cumulative Type Theory, CTT, has more relaxed type-restrictions: according to CTT, b^β(a^α) is well-formed iff β > α. In this paper, we set ourselves against CTT. We begin our case by arguing against Linnebo and Rayo’s claim that CTT sheds new philosophical light on set theory. We then argue that, while CTT ’s type-restrictions are unjustifiable, the type-restrictions imposed by (...) STT are justified by a Fregean semantics. What is more, this Fregean semantics provides us with a principled way to resist Linnebo and Rayo’s Semantic Argument for CTT. We end by examining an alternative approach to cumulative types due to Florio and Jones; we argue that their theory is best seen as a misleadingly formulated version of STT. (shrink)

Necessitism is the controversial thesis that necessarily everything is necessarily something, namely that everything, everywhere, necessarily exists. What is controversial about necessitism is that, at its core, it claims that things could not have failed to exist, while we have a pre-theoretical intuition that not everything necessarily exists. Contingentism, in accordance with common sense, denies necessitism: it claims that some things could have failed to exist. Timothy Williamson is a necessitist and claims that David Lewis is a necessitist too. The (...) paper argues that, granted the assumptions that lead to interpret the Lewisian as a necessitist, she can preserve contingentist intuitions, by genuinely agreeing with the folk that existence is contingent. This is not just the uncontroversial claim that the Lewisian, as a result of the prevalence of restricted quantification in counterpart theoretic regimentations of natural language, can agree with the folk while disagreeing with them in the metaphysical room. Rather, this is the claim that it is in the metaphysical room that the Lewisian can endorse the intuitions lying behind contingentism. (shrink)

Some Presentists—according to whom everything is present—identify instants of time with propositions of a certain kind. However, the view that times are propositions seems to be at odds with Presentism: if there are times then there are past times, and therefore things that are past; but how could there be things that are past if everything is present? In this paper, we describe the Presentist view that times are propositions ; we set out the argument that Presentism is incompatible with (...) the view that times are propositions ; and then we describe three possible responses to that argument on behalf of Presentists who identify times with propositions. We argue that each of these responses comes with significant costs. Finally, we describe a fourth possible response—according to which times are irreducibly higher-order entities—which appears to avoid the costs of the other three. We also describe and respond to two objections to the higher-order strategy. (shrink)

The long-standing dispute between absolutists and relativists traditionally focuses on whether there are absolute truths, absolute epistemic norms, and absolute.

In this paper I take second order-quantification to be a sui generis form of quantification, irreducible to first-order quantification, and I examine the implications of doing so for the debate over the existence of properties. Nicholas K. Jones has argued that adding sui generis second-order quantification to our ideology is enough to establish that properties exist. I argue that Jones does not settle the question of whether there are properties because—like other ontological questions—it is first-order. Then I examine three of (...) the main arguments for the existence of properties. I conclude that sui generis second-order quantification defeats the “one over many” argument and that, coupled with second-order predication, it also defeats the reference and quantification arguments. (shrink)

This paper defends Priorianism, a theory in the philosophy of time which combines three theses: first, that there is a metaphysical distinction between the present time and non-present times; second, that there are temporary propositions, that is, propositions that change in truth-value simpliciter over time; and third, that there is change over time only if there are temporary propositions. Priorianism is accepted by many Presentists, Growing Block Theorists, and Moving Spotlight Theorists. However, it is difficult to defend the view without (...) appealing to premises that those who reject the view find controversial. My aim in this paper is to defend Priorianism in a way that largely avoids appealing to such premises. I do three things: first (Section 1), I describe the component theses of Priorianism and the relations between them. Next (Section 2), I show how Priorians can respond to the argument that the B-theory implies that there are temporary propositions, and therefore satisfies the Priorian condition for there being change over time. Finally (Section 3), I defend the Priorian thesis that there is change over time only if there are temporary propositions against an alternative principle of change defended by Ross Cameron (The Moving Spotlight, 2015). (shrink)

According to the structured theory of propositions, if two sentences express the same proposition, then they have the same syntactic structure, with corresponding syntactic constituents expressing the same entities. A number of philosophers have recently focused attention on a powerful argument against this theory, based on a result by Bertrand Russell, which shows that the theory of structured propositions is inconsistent in higher order-logic. This paper explores a response to this argument, which involves restricting the scope of the claim that (...) propositions are structured, so that it does not hold for all propositions whatsoever, but only for those which are expressible using closed sentences of a given formal language. We call this restricted principle Closed Structure, and show that it is consistent in classical higher-order logic. As a schematic principle, the strength of Closed Structure is dependent on the chosen language. For its consistency to be philosophically significant, it also needs to be consistent in every extension of the language which the theorist of structured propositions is apt to accept. But, we go on to show, Closed Structure is in fact inconsistent in a very natural extension of the standard language of higher-order logic, which adds resources for plural talk of propositions. We conclude that this particular strategy of restricting the scope of the claim that propositions are structured is not a compelling response to the argument based on Russell’s result, though we note that for some applications, for instance to propositional attitudes, a restricted thesis in the vicinity may hold some promise. (shrink)

In many different ontological debates, anti-arbitrariness considerations push one towards two opposing extremes. For example, in debates about mereology, one may be pushed towards a maximal ontology (mereological universalism) or a minimal ontology (mereological nihilism), because any intermediate view seems objectionably arbitrary. However, it is usually thought that anti-arbitrariness considerations on their own cannot decide between these maximal or minimal views. I will argue that this is a mistake. Anti-arbitrariness arguments may be used to motivate a certain popular thesis in (...) the philosophy of mathematics that rules out the maximalist view in many different ontological debates. (shrink)

Prior’s puzzle is standardly taken to be the puzzle of why, given the assumption that that-clauses denote propositions, substitution of “the proposition that P” for “that P” within the complements of many propositional attitude verbs is invalid. I show that Prior’s puzzle is much more general than is ordinarily supposed. There are two variants on the substitutional form of the puzzle—a quantificational variant and a pronominal variant—and all three forms of the puzzle arise in a wide range of grammatical positions, (...) rather than merely in the complements of propositional attitude verbs. The generalized puzzle shows that a range of proposed solutions to the original puzzle fail, or are radically incomplete, and also reveals the connections between Prior’s puzzle and debates over the nature of semantic types and higher-order quantification. I go on to develop a novel, higher-order solution to the generalized form of the puzzle, and I argue that this higher-approach is superior to its first-order alternatives. (shrink)

Higher-order realists about properties express their view that there are properties with the help of higher-order rather than first-order quantifiers. They claim two types of advantages for this way of formulating property realism. First, certain gridlocked debates about the nature of properties, such as the immanentism versus transcendentalism dispute, are taken to be dissolved. Second, a further such debate, the tropes versus universals dispute, is taken to be resolved. In this paper I first argue that higher-order realism does not in (...) fact resolve the tropes versus universals dispute. In a constructive spirit, I then develop higher-order realism in a way that leads to a dissolution, rather than a resolution, of this dispute too. (shrink)

In linguistics, the dominant approach to the semantics of plurals appeals to mereology. However, this approach has received strong criticisms from philosophical logicians who subscribe to an alternative framework based on plural logic. In the first part of the article, we offer a precise characterization of the mereological approach and the semantic background in which the debate can be meaningfully reconstructed. In the second part, we deal with the criticisms and assess their logical, linguistic, and philosophical significance. We identify four (...) main objections and show how each can be addressed. Finally, we compare the strengths and shortcomings of the mereological approach and plural logic. Our conclusion is that the former remains a viable and well-motivated framework for the analysis of plurals. (shrink)

Philosophy and Phenomenological Research, Volume 103, Issue 3, Page 547-566, November 2021.

Some have argued for a division of epistemic labor in which mathematicians supply truths and philosophers supply their necessity. We argue that this is wrong: mathematics is committed to its own necessity. Counterfactuals play a starring role.

Plural expressions found in natural languages allow us to talk about many objects simultaneously. Plural logic — a logical system that takes plurals at face value — has seen a surge of interest in recent years. This book explores its broader significance for philosophy, logic, and linguistics. What can plural logic do for us? Are the bold claims made on its behalf correct? After introducing plural logic and its main applications, the book provides a systematic analysis of the relation between (...) this logic and other theoretical frameworks such as set theory, mereology, higher-order logic, and modal logic. The applications of plural logic rely on two assumptions, namely that this logic is ontologically innocent and has great expressive power. These assumptions are shown to be problematic. The result is a more nuanced picture of plural logic's applications than has been given thus far. Questions about the correct logic of plurals play a central role in the final chapters, where traditional plural logic is rejected in favor of a "critical" alternative. The most striking feature of this alternative is that there is no universal plurality. This leads to a novel approach to the relation between the many and the one. In particular, critical plural logic paves the way for an account of sets capable of solving the set-theoretic paradoxes. (shrink)

The principle of sufficient reason threatens modal collapse. Some have suggested that by appealing to the indefinite extensibility of contingent truth, the threat is neutralized. This paper argues that this is not so. If the indefinite extensibility of contingent truth is developed in an analogous fashion to the most promising models of the indefinite extensibility of the concept set, plausible principles permit the derivation of modal collapse.

Plural Logic is an extension of First-Order Logic which has, as well as singular terms and quantifiers, their plural counterparts. Analogously, Higher-Level Plural Logic is an extension of Plural Logic which has, as well as plural terms and quantifiers, higher-level plural ones. Roughly speaking, higher-level plurals stand to plurals like plurals stand to singulars; they are pluralised plurals. Allegedly, Higher-Level Plural Logic enjoys the expressive power of a simple type theory while committing us to nothing more than the austere ontology (...) of First-Order Logic. Were this true, Higher-Level Plural Logic would be a useful tool, with various applications in philosophy and linguistics. However, while the notions of plural reference and quantification enjoy widespread acceptance today, their higher-level counterparts have been received with a lot of scepticism. In this paper, I argue for the legitimacy of Higher-Level Plural Logic by providing evidence to the effect that natural languages contain higher-level plural expressions and showing that it is likely that they do so in an indispensable manner. Since the arguments I put forward are of the same sort advocates of Plural Logic have employed to defend their position, I conclude that the commonly held view that Plural Logic is legitimate, but not so its higher-level plural extensions is untenable. (shrink)

In explaining the notion of a fundamental property or relation, metaphysicians will often draw an analogy with languages. The fundamental properties and relations stand to reality as the primitive predicates and relations stand to a language: the smallest set of vocabulary God would need in order to write the “book of the world.” This paper attempts to make good on this metaphor. To that end, a modality is introduced that, put informally, stands to propositions as logical truth stands to sentences. (...) The resulting theory, formulated in higher-order logic, also vindicates the Humean idea that fundamental properties and relations are freely recombinable and a variant of the structural idea that propositions can be decomposed into their fundamental constituents via logical operations. Indeed, it is seen that, although these ideas are seemingly distinct, they are not independent, and fall out of a natural and general theory about the granularity of reality. (shrink)

There are two broad approaches to theorizing about ontological categories. Quineans use first-order quantifiers to generalize over entities of each category, whereas type theorists use quantification on variables of different semantic types to generalize over different categories. Does anything of import turn on the difference between these approaches? If so, are there good reasons to go type-theoretic? I argue for positive answers to both questions concerning the category of propositions. I also discuss two prominent arguments for a Quinean conception of (...) propositions, concerning their role in natural language semantics and apparent quantification over propositions within natural language. It will emerge that even if these arguments are sound, there need be no deep question about Quinean propositions’ true nature, contrary to much recent work on the metaphysics of propositions. (shrink)

A number of `no-proposition' approaches to the liar paradox find themselves implicitly committed to a moderate disquotational principle: the principle that if an utterance of the sentence `$P$' says anything at all, it says that $P$ (with suitable restrictions). I show that this principle alone is responsible for the revenge paradoxes that plague this view. I instead propose a view in which there are several closely related language-world relations playing the `semantic expressing' role, none of which is more central to (...) semantic theorizing than any other. I use this thesis about language and the negative result about disquotation to motivate the view that people do say things with utterances of paradoxical sentences, although they do not say the proposition you'd always expect, as articulated with a disquotational principle. (shrink)