Higher-Order Metaphysics

Quantificationalism is the view that some true propositions are false at or relative to some domains of quantification, in much the same sense in which propositions can be true or false at times or possible worlds. This paper seeks to give solid formal foundations to quantificationalism, comparable to what modal and tense logics are to the notions of contingent and temporary truth. It has the twofold purpose of understanding what sort of problems arise when we try and make precise sense (...) of Quantificationalism, and of solving these problems by outlining a comprehensive higher-order logical framework in which Quantificationalism can be precisely formulated and shown to be consistent. (shrink)

Eliminativists say that the idea of extending rigidity beyond singular terms must be abandoned, especially because they claim that Kripkean “identifications” involving natural kind terms can be shown to be necessary if true by the standard theory of reference fixing, using at most singular term rigidity. First I note that this claim is false for Cats are animals and analogous identifications involving ordinary natural kind terms of different types. Then I recall that the view of rigidity as necessary application provides (...) a modal-logical derivation of the necessity of those identifications, and is thus supported by this set of facts. I also consider, on behalf of the eliminativist, a rigidity-free derivation of that necessity, using the apparatus of higher-order metaphysics. I note that this derivation does not better motivate eliminativism, in part because its conceptual apparatus is not clearly more fundamental or less controversial than the apparatus of rigidity as necessary application. (shrink)

Special quantifiers are quantifiers like 'something', 'everything', and 'several things'. They are special both semantically and syntactically and play quite an important role in philosophy, in discussions of ontological commitment to abstract objects, of higher-order metaphysics, and of the apparent need for propositions. This paper will review and discuss in detail the syntactic and semantic peculiarities of special quantifiers and show that they are incompatible with substitutional and higher-order analyses that have recently been proposed. It instead defends and develops in (...) formal detail a semantic analysis of special quantifiers as nominalizing quantifiers. On this analysis, special quantifiers involve both singular objectual quantification and implicit on non-singular (higher-order, plural, or mass) quantification. The analysis rests on a range of recent insights and proposals in generative syntactic theory, in particular the recognition of '–thing' as a light noun and a potential classifier as well as recent views of the decomposition of attitudinal and locutionary verbs in syntax. (shrink)

According to intensional approaches to modeling content (such as "possible worlds semantics"), sentences which are logically equivalent express the same proposition. Partisans of hyperintensionality think this is too coarse-grained. Though there has been substantial interest recently in developing hyperintensional approaches to modeling content, we are still in early days: it is not clear how fine-grained propositions are on the various approaches, and we do not have a systematic map of how the various approaches relate to each other. In this paper, (...) I begin to sketch that map: I show that two prominent hyperintensional approaches can be seen not as rivals but, in large part, as two sides of the same coin. One is the truthmaker semantics, and the other, the "two-component" approach, models propositions as ordered pairs of "truth-conditional content'' and "subject matter''. I show, in particular, that certain versions of the two approaches approaches actually converge when it comes to these questions of fineness of grain. (shrink)

The question of Absolute Generality is whether quantifiers are ever as general as can be. Absolutists claim that quantifiers sometimes are absolutely general, while Relativists claim that quantifiers are never absolutely general. Although diverse philosophers have found the Relativist ethos compelling, it has been hard to articulate a consistent thesis which says what the Relativist seems to want to say. In this paper, I offer Relativists a way forward: I argue that what is needed to successfully state Relativism is a (...) way of generalizing that is non-quantificational. After showing how to define such a device of generalization in terms of identity between properties in a higher-order logical language, I use the device to articulate a form of Relativism which I prove to be consistent and which I argue captures the intuitive vision of the Relativist. (shrink)

Could the truths of mathematics have been different than they in fact are? If so, which truths could have been different? Do the contingent mathematical facts supervene on physical facts, or are they free floating? I investigate these questions within a framework of higher-order modal logic, drawing sometimes surprising connections between the necessity of arithmetic and analysis and other theses of modal metaphysics: the thesis that possibility in the broadest sense is governed by a logic of S5, that what is (...) possible holds in some maximally specific possibility, and that every property can be rigidified. The investigation will distinguish sharply between platonic contingency---contingency about whether particular abstract ``platonic'' mathematical objects are arranged in a certain way (e.g. in a natural number or real number structure)---from a deeper variety of structural contingency concerning what holds of objects whenever they *are* arranged in that way. (shrink)

In 'Higher-order quantification and the elimination of abstract objects', Dorr uses sum types to develop a novel higher-orderist account of nominalization in natural languages. Dorr's account is elegant and ingenious, but we argue that it faces some serious problems. Dorr could avoid these particular problems if he replaced sum types with (what we call) union types, but union types would then introduce new problems of their own. Fortunately, our own fictionalist approach to nominalization steers clear of all these difficulties.

There is a longstanding puzzle about empty names. On the one hand, the principles of classical logic seem quite plausible. On the other hand, there would seem to be truths involving empty names---such as: Pegasus doesn't exist---that require rejecting certain classically valid principles. In this paper, I argue that there is a natural and principled way of reconciling the truth of such claims with classical logic, that differs in crucial respects from the standard Russell/Quine reconciliation. While this account is compatible (...) with classical logic, it is not neutral with respect to other principles of a broadly logical character. For example, on this account, the truth of claims like Pegasus doesn't exist are incompatible with the validity of the controversial Barcan Formula. I argue that this fact may be marshalled to provide an argument against this and other related modal principles. In closing, I show how, given this account, we can, like descriptivists, provide a general unified account of both empty and denoting names that, unlike descriptivism, is compatible with the claim that denoting names are rigid designators. (shrink)

The Lewis-Sider argument from vagueness is one of the most powerful objections against restricted composition. Many have resisted the argument by rejecting its key premise, namely that existence is not vague. In this paper, I argue that this strategy is ineffective as a response to vagueness-based objections against restricted composition. To that end, I formulate a new argument against restricted composition: the argument from determinate vagueness. Unlike the Lewis-Sider argument, my argument doesn’t require accepting that existence is not vague, but (...) only that it is not vague in a specific way, which, I argue, is entailed by restricted composition. I show that the rejection of this species of vague existence follows from assumptions even friends of vague existence should be happy to accept. (shrink)

A common way of clarifying the notion of ground is by way of examples from logic: thus a conjunction is grounded in both of its conjuncts; a disjunction in each of its true disjuncts; a double negation in its negatum; and so on. Developing a semantics that accommodates these logical examples in full generality turned out to be a difficult task. In this paper, I develop a novel approach that substitutes fusion for a more discerning relation of combination between states (...) in the framework of truth-maker semantics. (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 concept of essence is a paradigm case of a putatively hyperintensional concept in metaphysics, a concept that can draw distinctions between necessarily equivalent propositions. Yet as several authors have recently emphasized, fine-grained distinctions in metaphysics come with a substantial risk of inconsistency, due to the Russell-Myhill paradox and its variants. This concern is especially pressing for hyperintensionalists about essence because the hyperintensionality of essence has been motivated primarily by examples. This paper presents a novel, more systematic case for the (...) hyperintensionality of essence that shows that the relevant distinctions can be systematically modeled in a natural theoretical framework. I start by offering a general heuristic to motivate a range of examples of fine-grained essentialist distinctions. I then present a higher-order logical framework for regimenting essentialist statements and provide a general argument for hyperintensionality on the basis of minimal assumptions about the logic of essence. Finally, I address the question of consistency by developing a semantic framework that demonstrates that the relevant fine-grained distinctions can indeed be consistently drawn. (shrink)

Purity is the principle that fundamental facts only have fundamental constituents. In recent years, it has played a significant role in metaphysical theorizing—but its logical foundations are underdeveloped. I argue that recent advances in higher-order logic reveal a subtle ambiguity regarding Purity’s interpretation; there are stronger and weaker versions of that principle. The arguments for Purity only support the weaker interpretation, but arguments that employ it only succeed if the stronger interpretation is true. As a result, nearly every metaphysician who (...) has appealed to Purity has made a mistake—in that the inferences that they make are not justified by the arguments that they provide. (shrink)

Monism is the claim that only one object exists. While few contemporary philosophers endorse monism, it has an illustrious history – stretching back to Bradley, Spinoza and Parmenides. In this paper, I show that plausible assumptions about the higher-order logic of property identity entail that monism is true. Given the higher-order framework I operate in, this argument generalizes: it is also possible to establish that there is a single property, proposition, relation, etc. I then show why this form of monism (...) is inconsistent; because all propositions are identical, p is identical to ~p – and so they have the same truth-value. At least one of the assumptions that generate higher-order monism must be rejected. (shrink)

According to traditional universalism, properties are instantiated by objects, where instantiation is a ‘tie’ that binds objects and properties into facts. I offer two arguments against this view. I then develop an alternative higher-order account which holds that properties are primitively predicated of objects yet, unlike traditional nominalism, are nevertheless genuinely real. When it’s a fact that Fo, it’s not because object o instantiates F-ness, but just that Fo – where F still exists. Against orthodox higher-order approaches, however, my arguments (...) against instantiation also serve to support a sparse conception of properties. In developing the view, I introduce an extension of ontological and ideological commitment in the form of syntactic commitment. (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 explores the view that the vocabulary of metaphysical fundamentality is opaque, using Sider’s theory of structure as a motivating case study throughout. Two conceptions of fundamentality are distinguished, only one of which can explain why the vocabulary of fundamentality is opaque.

The purity principle requires that identity truths such as “Hesperus is identical to Phosphorus” are grounded. This argument from purity for the groundedness of identity truths for first-order entities can be naturally generalized to higher-order identities like “to be a vixen is to be a female fox.” In this paper, I will examine various accounts of the grounds of identity truths by taking the cases of higher-order identities into consideration. Drawing on some essentialist insights, I will propose a novel account (...) and argue that it offers an attractive explanation of how identities, including both first-order and higher-order cases, are grounded. (shrink)

David Armstrong claims that a law of nature is both a universal as well as a state of affairs. Unfortunately, this leaves his account vulnerable to a “logician’s protest”, first privately raised by Tichý, that appeals to the type theory of higher-order logic. In this paper, I flesh out the objection and show how Armstrong’s cursory response is inadequate. I argue that the logician’s protest is a very serious problem for Armstrong, as well as any other account of lawhood that (...) posits such a ‘dual nature’ (according to which a law is both a universal and state of affairs). As a result, the logician’s protest serves as an important dilemma for any theory of laws: laws of nature can be identified with universals or with states of affairs, but not both. I conclude that the seriousness of the logician’s protest is an underappreciated constraint on the metaphysics of one's theory. (shrink)

Propositional contingentism is the thesis that there might have been propositions which might have not have been something. Serious actualism is the thesis that it is impossible for a property to be exemplified without there being something which exemplifies it. Both are popular. Likewise, the dominant view in the metaphysics of modality is that metaphysical possibility and necessity can be understood – in some sense – in terms of possible worlds, i.e. total ways the world could have been. Here, I (...) argue that, given some minimal assumptions, the conjunction of propositional contingentism and serious actualism entails that worlds are modally fragile – every world is ontologically dependent on every proposition. I then show that such a consequence is inconsistent with the claim that propositions true at all possible worlds are necessary. (shrink)

This is a comment on Cian Dorr 'Higher-Order Quantification and the Elimination of Abstract Objects'. The aim of this contribution is to clarify and further develop a view (with its empirical generalizations) on which higher-order objects play a highly restricted role in the ontology of natural language. A sharp distinction is drawn between ontologically dependent objects (events, tropes, qualities, attitudinal objects etc.) and higher-order objects (properties, relations, propositions, etc.). Natural language reflects an ontology of the former, rather than of the (...) latter. (shrink)

Prior’s problem consists in the impossibility of replacing clausal complements of most attitude verbs by ‘ordinary’ NPs; only ‘special quantifiers’ that is, quantifiers like 'something' permit a replacement, preserving grammaticality or the same reading of the verb: (1) a. John claims that he won. b. ??? John claims a proposition / some thing. c. John claims something. In my 2013 book Abstract Objects and the Semantics of Natural Language, I have shown how this generalizes to nonreferential complements of various other (...) intensional predicates and argued for a Nominalization Theory of special quantifiers. In this paper, I will review and extend the range of linguistic generalizations that motivate the Nominalization Theory and show that they pose serious problems for higher-order and substitutional analyses of special quantifiers. (shrink)

This paper will propose a novel semantic and syntactic analysis of stative verbs, more specifically abstract state verbs like 'need', 'believe', 'know', 'own', 'owe', and 'lack'. On that analysis, such verbs have an underlying structure on which they are complex predicates consisting of the light verb HAVE and a noun for a trope-like thing (a trope or attitudinal, modal or intensional object), a structure that is also input to semantic interpretation. Thus, 'need a coat' is underlyingly 'have need (of) a (...) coat'. The analysis accounts for restrictions on adverbial modifiers of abstract states verbs and surprising constraints on explicit property-referring terms, NPs of the sort the property of needing a coat. The paper advocates a notion of a property that is fundamentally different from the notions familiar from the philosophical literature (natural, non-natural properties, abundant, sparse properties). It also suggests that that notion of a property is as innate as constraints of universal grammar, given a generative perspective. (shrink)

Facts (and, more generally, states of affairs) are plausibly individuated in an at least moderately coarse-grained way: the fact that this rose is red should not be distinguished from the fact that this rose fails to fail to be red. Something similar can be said for properties and relations. It is relatively easy to formulate principles that entail that the individuation of states of affairs, properties, and relations—in brief, intensional entities—is coarse-grained enough to conform to this idea. It is less (...) easy to devise a complementary principle that imposes an *upper* bound on the coarse-grainedness of intensional entities. A promising approach to this problem is to formulate a principle according to which the *fundamental* entities (including fundamental properties and relations) are freely recombinable or ‘logically independent’; but certain special provisions that have to be made with regard to relations tend to render such a principle unattractively complex. In this paper I put forward a simpler principle, based on a broadly positionalistic conception of relations. (shrink)

In this book I articulate a new view about fundamental reality, an alternative to the familiar opposition between monism and pluralism. On this view, what’s metaphysically fundamental is neither the universal whole nor certain subcosmic entities, but the plurality of all entities, i.e. all entities taken collectively. I call it collective allism. Using higher-order resources, I systematically develop the view and give shape to the robust realism about plurals and the unorthodox ideology of fundamentality upon which it rests. I argue (...) that collective allism is tenable and merits serious consideration in current debates about fundamental reality—it’s a worthy rival to monism and pluralism given considerations about quantum entanglement, mereological infinitism, natural laws, and relations. -/- . (shrink)

This is an introduction to the special issue ‘Higher-order metaphysics of ground’. It provides a general introduction to the topic and summaries of the ten contributed articles.

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 way we answer questions about what there is crucially depends on the language and the logic in which they are framed. This entry introduces the orthodox view on how to carry out such debates, as was formulated by W. V. O. Quine, as well as a number of influential alternatives. A further issue that is explored is whether disagreement about what there is turns on mind-independent features of reality, or it is an artifact of language.

This is a review of Peter Fritz's _Foundations of Modality: From Propositions to Possible Worlds_.

My aim in this paper is to offer a novel justification for β-Equivalence. β-Equivalence is a standard principle of higher-order logic, but it is metaphysically controversial. My argument for β-Equivalence is based on a distinctively Fregean conception of predication. I argue that the Fregean conception motivates a non-standard notation for predicates, which can then be used to show that β-Equivalence is trivial on that conception. I also argue that the Fregean conception motivates a functional individuation of properties, which we can (...) use to extend the Fregean justification for β-Equivalence to cover a more general principle, β-Conversion. (shrink)

According to an influential view in the debate on grounding and essence, there cannot be any cases of ‘upwards essence’, i.e., cases in which a grounding connection flows from the essence of the grounding truth or constituents of it. To use the Finean (2012a) slogan, “it is the fact to be grounded that ‘points’ to its grounds and not the grounds that point to what they may ground”. This paper argues to the contrary. Far from being outright incoherent, potential cases (...) of upwards essence abound. The arguments that have been raised against upwards essence or naturally come to mind turn out to be wanting on closer examination. And we can even derive the existence of certain cases of upwards essence from fairly innocuous general principles. (shrink)

Teitel (Mind 128:39-68, 2019) argues that the following three doctrines are jointly inconsistent: i) the doctrine that metaphysical necessity reduces to essence; ii) the doctrine that possibly something could fail to exist; and iii) the doctrine that metaphysical necessity obeys a modal logic of at least S4. This paper presents a novel solution to Teitel’s puzzle, regimented in a higher-order logical setting, which is crucially based on the idea that the putative reduction of metaphysical necessity to essence should be understood (...) through appealing to some hyperintensional notion—such as grounding or real definition—rather than the notion of identity/identification. Moreover, it will also be shown that the proposed reductive account has a significant advantage over its rival account. (shrink)

What is time? In this paper, I defend the view that time is change. In Section 1, I introduce the view that time is change and contrast it with an alternative view. In Section 2, I draw on recent developments in higher-order metaphysics to defend a particular theory of change, and therefore, of time. Finally, in Section 3, I respond to several objections to the view that time is change.

This paper develops a logic of essence (HLE) in the framework of higher-order logic. The theory aims to provide a general framework for theorizing about the essences of objects, properties, propositions, and logical operations like conjunction, negation, quantification, etc. The first part of the paper presents the formal language and axiom system of HLE. After that, some theorems of the system are proved and it is shown how the logic of metaphysical necessity can be developed within the framework of HLE. (...) The second part of the paper develops a possible worlds semantics for HLE, gives a proof of soundness, and provides examples of models that demonstrate the consistency of some simple essentialist theories. (shrink)

In an anonymous referee report written in 1945, Church suggested a sweeping argument against verificiationism, the thesis that every truth is knowable. The argument, which was published with due acknowledgement by Fitch almost two decades later, has generated significant attention as well as some interesting successor arguments. In this paper, we present the most important episodes in this intellectual history using the logic that Church himself favoured, and we give reasons for thinking that the arguments are less than decisive. However, (...) we present some new arguments against verificationism that we believe to be dialectically effective. (shrink)

Higher‐order metaphysics is in full swing. Its proponents argue that higher‐order logic should replace set theory at the foundations of mathematics and metaphysics. But amid the enthusiasm, surprisingly little attention has been paid to some serious epistemological challenges facing the program—foremost among them a variant of the Benacerraf challenge, developed by Field and Clarke‐Doane. Roughly put, the challenge is to explain the reliability of our higher‐order logical beliefs. A similar problem is familiar from the philosophy of set theory, where it (...) has led to a pluralist reconception of the foundations of mathematics. In this paper, I argue that regardless of whether higher‐order logic is preferable to set theory on abductive grounds, they stand or fall together when faced with this epistemological challenge. They are companions in guilt (or innocence). I conclude that, absent other solutions, a promising path forward is to adopt a pluralist approach to higher‐order logic. The consequences of such a shift are difficult to overstate. (shrink)

I review the volume ``Higher-Order Metaphysics'' edited by Peter Fritz and Nicholas K. Jones.

Sentences that ascribe action are logically related, but it is not always obvious why. According to event semantics, implications and non-implications result from referential relations between unpronounced constituents. Taking as starting point examples including free relative clauses, this paper advances the alternative view that examples as such present logical relations as forms of predicative dependence indicated with pronounced constituents. To this end, I argue that Verbal Phrases and verbal traces follow the pattern of Verbal Phrase Anaphora and, more controversially, that (...) they can be semantically interpreted in terms of higher-order quantification that represent actions as properties, but neither as events nor event-kinds. (shrink)

Peter Fritz and Nicholas K. Jones (eds.), Higher-Order Metaphysics, Oxford University Press, 2024, vii + 547pp., ISBN 978–0–19–289488–5.

Higher‐order metaphysics is in full swing. Its proponents argue that higher‐order logic should replace set theory at the foundations of mathematics and metaphysics. But amid the enthusiasm, surprisingly little attention has been paid to some serious epistemological challenges facing the program—foremost among them a variant of the Benacerraf challenge, developed by Field and Clarke‐Doane. Roughly put, the challenge is to explain the reliability of our higher‐order logical beliefs. A similar problem is familiar from the philosophy of set theory, where it (...) has led to a pluralist reconception of the foundations of mathematics. In this paper, I argue that regardless of whether higher‐order logic is preferable to set theory on abductive grounds, they stand or fall together when faced with this epistemological challenge. They are companions in guilt (or innocence). I conclude that, absent other solutions, a promising path forward is to adopt a pluralist approach to higher‐order logic. The consequences of such a shift are difficult to overstate. (shrink)

Higher-order logic, with its type-theoretic apparatus known as the simple theory of types (STT), has increasingly come to be employed in theorising 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)

How do conjunctions relate to their conjuncts? How, for example, does relate to the propositions and? This paper explores the unorthodox answer that conjunctions are identical to their conjuncts: throwing grammar to the wind, just 'is' the propositions and. I suggest two ways of motivating the view (§1), present my preferred formulation of it (§2), discuss some objections (§3), point out some connections to neighboring issues (§4), and finally, consider how it may be extended to a more comprehensive view of (...) logical complexity (§5). (shrink)

Tracing back to at least Frege, genuine higher-orderism sharply distinguishes between objects and categorically distinct higher-order entities and the expressions that denote them. Against the background of this view, Frege himself felt compelled to regard numbers as objects rather than higher-level entities—concepts, in Frege’s terminology—because number words in arithmetical contexts appear to function as object-denoting expressions. In this paper, I argue that, from the perspective of natural language semantics, a case can be made that number words in such contexts function (...) as concept-denoting rather than object-denoting expressions and that, consequently, numbers should be conceived of as concepts rather than objects. In developing this argument, I will develop a novel semantic analysis of expressions like ‘the number of Mars’ moons’ as higher-order definite descriptions. Moreover, the view developed in this paper will shed new light on so-called easy arguments for the existence of numbers. (shrink)

Higher‐order metaphysicians take facts to be higher‐order beings, i.e., entities in the range of irreducibly higher‐order quantifiers. In this paper, I investigate the impact of this conception of facts on the debate about the reality of tense. I identify two major repercussions. The first concerns the logical space of tense realism: on a higher‐order conception of facts, a prominent version of tense realism, dynamic absolutism, turns out to conflict with the laws of (higher‐order tense) logic. The second concerns our understanding (...) of the positions occupying this logical space: on a higher‐order conception of facts, an attractive interpretation of the central tense realist notion of ‘facts constituting reality’ becomes unavailable. I discuss these results in the context of the more general project of higher‐order metaphysics and the (meta)metaphysics of time, drawing out their implications for the nature of the disputes both between realists and anti‐realists about tense and between different tense realist factions. (shrink)

Higher-order logic augments first-order logic with devices that let us generalize into grammatical positions other than that of a singular term. Some recent metaphysicians have advocated for using these devices to raise and answer questions that bear on many traditional issues in philosophy. In contrast to these 'higher-order metaphysicians', traditional metaphysics has often focused on parallel, but importantly different, questions concerning special sorts of abstract objects: propositions, properties and relations. The answers to the higher-order and the property-theoretic questions may coincide (...) sometimes but will often come apart. I argue that when they do, the higher-order questions are closer to the metaphysical action and so it would be better for these debates to proceed in higher-order terms. (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)

This three-part chapter explores a higher-order logic we call ‘Classicism’, which extends a minimal classical higher-order logic with further axioms which guarantee that provable coextensiveness is sufficient for identity. The first part presents several different ways of axiomatizing this theory and makes the case for its naturalness. The second part discusses two kinds of extensions of Classicism: some which take the view in the direction of coarseness of grain (whose endpoint is the maximally coarse-grained view that coextensiveness is sufficient for (...) identity), and some which take the view in the direction of fineness of grain (whose endpoint is the maximally fine-grained theory containing all distinctness claims compatible with Classicism). The third part introduces some techniques for constructing models of Classicism, and uses them to prove the consistency of many of the extensions of Classicism introduced in the second part. (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.