Quantifier variance is a well-known view in contemporary metaontology, but it remains very widely misunderstood by critics. Here we briefly and clearly explain the metasemantics of quantifier variance and distinguish between modest and strong forms of variance (Section I), explain some key applications (Section II), clear up some misunderstandings and address objections (Section III), and point the way toward future directions of quantifier-variance-related research (Section IV).

People accept conclusions of valid conditional inferences (e.g., if p then q, p therefore q) less, the more disablers (circumstances that prevent q to happen although p is true) exist. We investigated whether rules that through their phrasing exclude disablers evoke higher acceptance ratings than rules that do not exclude disablers. In three experiments we re-phrased content-rich conditionals from the literature as either universal or existential rules and embedded these rules in Modus Ponens and Modus Tollens inferences. In Experiments 2 (...) and 3, we also used abstract rules. The acceptance of conclusions increased when the rule was phrased with “all” instead of “some” and the number of disablers had a higher impact on existential rules than on universal rules. Further, the effect of quantifier was more pronounced for abstract rules and when tested within subjects. We discuss the relevance of phrasing, quantifiers and knowledge on reasoning. (shrink)

The growth of self-tracking and personal surveillance has given rise to the Quantified Self movement. Members of this movement seek to enhance their personal well-being, productivity, and self-actualization through the tracking and gamification of personal data. The technologies that make this possible can also track and gamify aspects of our interpersonal, romantic relationships. Several authors have begun to challenge the ethical and normative implications of this development. In this article, we build upon this work to provide a detailed ethical analysis (...) of the Quantified Relationship. We identify eight core objections to the QR and subject them to critical scrutiny. We argue that although critics raise legitimate concerns, there are ways in which tracking technologies can be used to support and facilitate good relationships. We thus adopt a stance of cautious openness toward this technology and advocate the development of a research agenda for the positive use of QR technologies. (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)

Attempting to deflate ontological debates, the proponent of Quantifier Variance (QV) claims that there are multiple quantifier meanings of equal metaphysical merit. According to Hirsch—the main proponent of QV—metaphysical merit should be understood intensionally: two languages have equal merit if they allow us to express the same possibilities. I examine the notion of metaphysical merit and its purported link to intensionality. That link, I argue, should not be supported by adopting an intensional theory of semantic content. Rather, I give a (...) general strategy for supporting claims about metaphysical merit and examine whether that strategy can be used to link merit and intensionality. Though I don’t deliver a definitive verdict, the discussion provides a clearer framework for articulating and evaluating claims about metaphysical merit. (shrink)

Quantification is a topic which brings together linguistics, logic, and philosophy. Quantifiers are the essential tools with which, in language or logic, we refer to quantity of things or amount of stuff. In English they include such expressions as no, some, all, both, many. Peters and Westerstahl present the definitive interdisciplinary exploration of how they work - their syntax, semantics, and inferential role.

Propositional quantifiers are quantifiers binding proposition letters, understood as variables. This Element introduces propositional quantifiers and explains why they are especially interesting in the context of propositional modal logics. It surveys the main results on propositionally quantified modal logics which have been obtained in the literature, presents a number of open questions, and provides examples of applications of such logics to philosophical problems.

We provide an overview of the meta-ontological position known as "Quantifier Variance".

In this paper, we provide a comprehensive survey of the space of possible analyses of the phenomenon of quantifier domain restriction, together with a set of considerations which militate against all but our own proposal. Among the many accounts we consider and reject are the ‘explicit’ approach to quantifier domain restric‐tion discussed, for example, by Stephen Neale, and the pragmatic approach to quantifier domain restriction proposed by Kent Bach. Our hope is that the exhaustive discussion of this special case of (...) context‐dependence will provide guidelines for how to decide, for an arbitrary case of context‐dependent discourse, whether it should be treated syntactically, semantically, or pragmatically. (shrink)

The formalisation of Natural Language arguments in a formal language close to it in syntax has been a central aim of Moss’s Natural Logic. I examine how the Quantified Argument Calculus (Quarc) can handle the inferences Moss has considered. I show that they can be incorporated in existing versions of Quarc or in straightforward extensions of it, all within sound and complete systems. Moreover, Quarc is closer in some respects to Natural Language than are Moss’s systems – for instance, is (...) does not use negative nouns. The process also sheds light on formal properties and presuppositions of some inferences it formalises. Directions for future work are outlined. (shrink)

Timothy Williamson has argued that in the debate on modal ontology, the familiar distinction between actualism and possibilism should be replaced by a distinction between positions he calls contingentism and necessitism. He has also argued in favor of necessitism, using results on quantified modal logic with plurally interpreted second-order quantifiers showing that necessitists can draw distinctions contingentists cannot draw. Some of these results are similar to well-known results on the relative expressivity of quantified modal logics with so-called inner and (...) outer quantifiers. The present paper deals with these issues in the context of quantified modal logics with generalized quantifiers. Its main aim is to establish two results for such a logic: Firstly, contingentists can draw the distinctions necessitists can draw if and only if the logic with inner quantifiers is at least as expressive as the logic with outer quantifiers, and necessitists can draw the distinctions contingentists can draw if and only if the logic with outer quantifiers is at least as expressive as the logic with inner quantifiers. Secondly, the former two items are the case if and only if all of the generalized quantifiers are first-order definable, and the latter two items are the case if and only if first-order logic with these generalized quantifiers relativizes. (shrink)

I show that the contemporary dominant analysis of natural language quantifiers that are one-place determiners by means of binary generalized quantifiers has failed to explain why they are, according to it, conservative. I then present an alternative, Geachean analysis, according to which common nouns in the grammatical subject position are plural logical subject-terms, and show how it does explain that fact and other features of natural language quantification.

Animals, like humans, experience different levels of well-being depending on decisions made by others. As a result, the well-being of animals must be included in any full accounting of the well-being consequences of decisions. However, this is almost never done in large-scale policy and investment analyses, even though it is common to quantify the consequences for human welfare in these decision analyses. This is partly due to prejudice, but increasingly also because we do not currently have good methods for quantifying (...) animal well-being consequences and putting them on the same scale as quantified human well-being consequences. We might call this ‘the problem of interspecies comparisons.’ This important barrier to including animal well-being in decision-making is the result of an insufficiently developed theory and practice of animal well-being and its relation to human well-being. This chapter explains the problem of interspecies comparisons, explains recent research that develops methods to overcome this problem. (shrink)

Quantified expressions in natural language generally are taken to act like quantifiers in logic, which either range over entities that need to satisfy or not satisfy the predicate in order for the sentence to be true or otherwise are substitutional quantifiers. I will argue that there is a philosophically rather important class of quantified expressions in English that act quite differently, a class that includes something, nothing, and several things. In addition to expressing quantification, such expressions act like (...) nominalizations, introducing a new domain of objects that would not have been present in the semantic structure of the sentence otherwise. The entities those expressions introduce are of just the same sort as those that certain ordinary nominalizations refer to (such as John's wisdom or John's belief that S), namely they are tropes or entities related to tropes. Analysing certain quantifiers as nominalizing quantifiers will shed a new light on philosophical issues such as the status of properties and the nature of propositional attitudes. (shrink)

This essay clarifies quantifier variance and uses it to provide a theory of indefinite extensibility that I call the variance theory of indefinite extensibility. The indefinite extensibility response to the set-theoretic paradoxes sees each argument for paradox as a demonstration that we have come to a different and more expansive understanding of ‘all sets’. But indefinite extensibility is philosophically puzzling: extant accounts are either metasemantically suspect in requiring mysterious mechanisms of domain expansion, or metaphysically suspect in requiring nonstandard assumptions about (...) mathematical objects. Happily, the view of quantifier meanings that underwrites quantifier variance can be used to provide an account of indefinite extensibility that is both metasemantically and metaphysically satisfying. Section 1 introduces the puzzle of indefinite extensibility; section 2 develops and clarifies the metasemantics of quantifier variance; section 3 solves section 1's puzzle of indefinite extensibility by applying section 2's account of quantifier meanings; and section 4 compares the theory developed in section 3 to several other theories in the literature. (shrink)

This volume on the semantic complexity of natural language explores the question why some sentences are more difficult than others. While doing so, it lays the groundwork for extending semantic theory with computational and cognitive aspects by combining linguistics and logic with computations and cognition. -/- Quantifier expressions occur whenever we describe the world and communicate about it. Generalized quantifier theory is therefore one of the basic tools of linguistics today, studying the possible meanings and the inferential power of quantifier (...) expressions by logical means. The classic version was developed in the 1980s, at the interface of linguistics, mathematics and philosophy. Before this volume, advances in "classic" generalized quantifier theory mainly focused on logical questions and their applications to linguistics, this volume adds a computational component, the third pillar of language use and logical activity. This book is essential reading for researchers in linguistics, philosophy, cognitive science, logic, AI, and computer science. (shrink)

This paper asks: Is the quantifier variantist committed to metaphysical vagueness? My investigation of this question goes via a study of vague existence. I’ll argue that the quantifier variantist is committed to vague existence and that the vague existence posited by the variantist requires a puzzling sort of metaphysical vagueness. Specifically, I distinguish between (what I call) positive and negative metaphysical vagueness. Positive metaphysical vagueness is (roughly) the claim that there is vagueness in the world; negative metaphysical vagueness is (roughly) (...) the claim that there is vagueness that is not in our language or thought. I’ll argue that the quantifier variantist’s commitment to vague existence comes with positive metaphysical vagueness -- even if it doesn't require negative metaphysical vagueness. (shrink)

Quantifier variance faces a number of difficulties. In this paper we first formulate the view as holding that the meanings of the quantifiers may vary, and that languages using different quantifiers may be charitably translated into each other. We then object to the view on the basis of four claims: (i) quantifiers cannot vary their meaning extensionally by changing the domain of quantification; (ii) quantifiers cannot vary their meaning intensionally without collapsing into logical pluralism; (iii) quantifier (...) variance is not an ontological doctrine; (iv) quantifier variance is not compatible with charitable translation and as such is internally inconsistent. In light of these troubles, we recommend the dissolution of quantifier variance and suggest that the view be laid to rest. (shrink)

We present an embedding of quantified multimodal logics into simple type theory and prove its soundness and completeness. A correspondence between QKπ models for quantified multimodal logics and Henkin models is established and exploited. Our embedding supports the application of off-the-shelf higher-order theorem provers for reasoning within and about quantified multimodal logics. Moreover, it provides a starting point for further logic embeddings and their combinations in simple type theory.

Recently a number of works in meta-ontology have used a variant of J.H. Harris's collapse argument in the philosophy of logic as an argument against Eli Hirsch's quantifier variance. There have been several responses to the argument in the literature, but none of them have identified the central failing of the argument, viz., the argument has two readings: one on which it is sound but doesn't refute quantifier variance and another on which it is unsound. The central lesson I draw (...) is that arguments against quantifier variance must pay strict attention to issues of translation and interpretation. The paper also has a substantial appendix in which I prove the equivalence of plural mereological nihilism and standard first-order atomistic mereology; results of this kind are often appealed to in the literature on quantifier variance but without many details on the nature or proof of the result. (shrink)

I develop a formal logic in which quantified arguments occur in argument positions of predicates. This logic also incorporates negative predication, anaphora and converse relation terms, namely, additional syntactic features of natural language. In these and additional respects, it represents the logic of natural language more adequately than does any version of Frege’s Predicate Calculus. I first introduce the system’s main ideas and familiarize it by means of translations of natural language sentences. I then develop a formal system built on (...) these principles, the Quantified Argument Calculus or Quarc. I provide a truth-value assignment semantics and a proof system for the Quarc. I next demonstrate the system’s power by a variety of proofs; I prove its soundness; and I comment on its completeness. I then extend the system to modal logic, again providing a proof system and a truth-value assignment semantics. I proceed to show how the Quarc versions of the Barcan formulas, of their converses and of necessary existence come out straightforwardly invalid, which I argue is an advantage of the modal Quarc over modal Predicate Logic as a system intended to capture the logic of natural language. (shrink)

I defend a neo-Lewisean form of contextualism about knowledge attributions. Understanding the context-sensitivity of knowledge attributions in terms of the context-sensitivity of universal quantifiers provides an appealing approach to knowledge. Among the virtues of this approach are solutions to the skeptical paradox and the Gettier problem. I respond to influential objections to Lewis’s account.

We present a general approach to quantified modal logics that can simulate most other approaches. The language is based on operators indexed by terms which allow to express de re modalities and to control the interaction of modalities with the first-order machinery and with non-rigid designators. The semantics is based on a primitive counterpart relation holding between n-tuples of objects inhabiting possible worlds. This allows an object to be represented by one, many, or no object in an accessible world. Moreover (...) by taking as primitive a relation between n-tuples we avoid some shortcoming of standard individual counterparts. Finally, we use cut-free labelled sequent calculi to give a proof-theoretic characterisation of the quantified extensions of each first-order definable propositional modal logic. In this way we show how to complete many axiomatically incomplete quantified modal logics. (shrink)

In this paper, I introduce a quantifier-pluralist theory of time, temporal quantifier relativism. Temporal quantifier relativism includes a restricted quantifier for every instantaneous moment of time. Though it flies in the face of orthodoxy, it compares favorably to rival theories of time. To demonstrate this, I first develop the basic syntax and semantics of temporal quantifier relativism. I then compare the theory to its rivals on three issues: the passage of time, the analysis of change, and temporal ontology.

Quantification is a topic which brings together linguistics, logic, and philosophy. Quantifiers are the essential tools with which, in language or logic, we refer to quantity of things or amount of stuff. In English they include such expressions as no, some, all, both, many. Peters and Westerstahl present the definitive interdisciplinary exploration of how they work - their syntax, semantics, and inferential role.

The purpose of this paper is to look at some existing methods of semantic information quantification and suggest some alternatives. It begins with an outline of Bar-Hillel and Carnap’s theory of semantic information before going on to look at Floridi’s theory of strongly semantic information. The latter then serves to initiate an in-depth investigation into the idea of utilising the notion of truthlikeness to quantify semantic information. Firstly, a couple of approaches to measure truthlikeness are drawn from the literature and (...) explored, with a focus on their applicability to semantic information quantification. Secondly, a similar but new approach to measure truthlikeness/information is presented and some supplementary points are made. (shrink)

In the work of both Matti Eklund and John Hawthorne there is an influential semantic argument for a maximally expansive ontology that is thought to undermine even modest forms of quantifier variance. The crucial premise of the argument holds that it is impossible for an ontologically "smaller" language to give a Tarskian semantics for an ontologically "bigger" language. After explaining the Eklund-Hawthorne argument (in section I), we show this crucial premise to be mistaken (in section II) by developing a Tarskian (...) semantics for a mereological universalist language within a mereological nihilist language (a case which we, and Eklund and Hawthorne, take as representative). After developing this semantics we step back (in section III) to discuss the philosophical motivations behind the Eklund- Hawthorne argument’s demand for a semantics. We ultimately conclude that quantifier variantists can meet any demand for a semantics that might reasonably be imposed upon them. (shrink)

In the dissertation we study the complexity of generalized quantifiers in natural language. Our perspective is interdisciplinary: we combine philosophical insights with theoretical computer science, experimental cognitive science and linguistic theories. -/- In Chapter 1 we argue for identifying a part of meaning, the so-called referential meaning (model-checking), with algorithms. Moreover, we discuss the influence of computational complexity theory on cognitive tasks. We give some arguments to treat as cognitively tractable only those problems which can be computed in polynomial (...) time. Additionally, we suggest that plausible semantic theories of the everyday fragment of natural language can be formulated in the existential fragment of second-order logic. -/- In Chapter 2 we give an overview of the basic notions of generalized quantifier theory, computability theory, and descriptive complexity theory. -/- In Chapter 3 we prove that PTIME quantifiers are closed under iteration, cumulation and resumption. Next, we discuss the NP-completeness of branching quantifiers. Finally, we show that some Ramsey quantifiers define NP-complete classes of finite models while others stay in PTIME. We also give a sufficient condition for a Ramsey quantifier to be computable in polynomial time. -/- In Chapter 4 we investigate the computational complexity of polyadic lifts expressing various readings of reciprocal sentences with quantified antecedents. We show a dichotomy between these readings: the strong reciprocal reading can create NP-complete constructions, while the weak and the intermediate reciprocal readings do not. Additionally, we argue that this difference should be acknowledged in the Strong Meaning hypothesis. -/- In Chapter 5 we study the definability and complexity of the type-shifting approach to collective quantification in natural language. We show that under reasonable complexity assumptions it is not general enough to cover the semantics of all collective quantifiers in natural language. The type-shifting approach cannot lead outside second-order logic and arguably some collective quantifiers are not expressible in second-order logic. As a result, we argue that algebraic (many-sorted) formalisms dealing with collectivity are more plausible than the type-shifting approach. Moreover, we suggest that some collective quantifiers might not be realized in everyday language due to their high computational complexity. Additionally, we introduce the so-called second-order generalized quantifiers to the study of collective semantics. -/- In Chapter 6 we study the statement known as Hintikka's thesis: that the semantics of sentences like ``Most boys and most girls hate each other'' is not expressible by linear formulae and one needs to use branching quantification. We discuss possible readings of such sentences and come to the conclusion that they are expressible by linear formulae, as opposed to what Hintikka states. Next, we propose empirical evidence confirming our theoretical predictions that these sentences are sometimes interpreted by people as having the conjunctional reading. -/- In Chapter 7 we discuss a computational semantics for monadic quantifiers in natural language. We recall that it can be expressed in terms of finite-state and push-down automata. Then we present and criticize the neurological research building on this model. The discussion leads to a new experimental set-up which provides empirical evidence confirming the complexity predictions of the computational model. We show that the differences in reaction time needed for comprehension of sentences with monadic quantifiers are consistent with the complexity differences predicted by the model. -/- In Chapter 8 we discuss some general open questions and possible directions for future research, e.g., using different measures of complexity, involving game-theory and so on. -/- In general, our research explores, from different perspectives, the advantages of identifying meaning with algorithms and applying computational complexity analysis to semantic issues. It shows the fruitfulness of such an abstract computational approach for linguistics and cognitive science. (shrink)

The purpose of this paper is to describe a set of quantified temporal alethic-deontic systems, i.e., systems that combine temporal alethicdeontic logic with predicate logic. We consider three basic kinds of systems: constant, variable and constant and variable domain systems. These systems can be augmented by either necessary or contingent identity, and every system that includes identity can be combined with descriptors. All logics are described both semantically and proof theoretically. We use a kind of possible world semantics, inspired by (...) the so-called T × W semantics, to characterize them semantically and semantic tableaux to characterize them proof theoretically. We also show that all systems are sound and complete with respect to their semantics. (shrink)

We introduce generalized quantifiers, as defined in Tarskian semantics by Mostowski and Lindström, in logics whose semantics is based on teams instead of assignments, e.g., IF-logic and Dependence logic. Both the monotone and the non-monotone case is considered. It is argued that to handle quantifier scope dependencies of generalized quantifiers in a satisfying way the dependence atom in Dependence logic is not well suited and that the multivalued dependence atom is a better choice. This atom is in fact (...) definably equivalent to the independence atom recently introduced by Väänänen and Grädel. (shrink)

There are two main routes to a concept of (generalized) quantifier. The first starts from first‐order logic, FO, and generalizes from the familiar ∀ and ∃ occurring there. The second route begins with real languages, and notes that many so‐called noun phrases, a kind of phrase which occurs abundantly in most languages, can be interpreted in a natural and uniform way using quantifiers.

Henkin quantifiers have been introduced in Henkin (1961). Walkoe (1970) studied basic model-theoretical properties of an extension $L_{*}^{1}$ (H) of ordinary first-order languages in which every sentence is a first-order sentence prefixed with a Henkin quantifier. In this paper we consider a generalization of Walkoe's languages: we close $L_{*}^{1}$ (H) with respect to Boolean operations, and obtain the language L¹(H). At the next level, we consider an extension $L_{*}^{2}$ (H) of L¹(H) in which every sentence is an L¹(H)-sentence prefixed (...) with a Henkin quantifier. We repeat this construction to infinity. Using the (un)definability of truth - in - N for these languages, we show that this hierarchy does not collapse. In addition, we compare some of the present results to the ones obtained by Kripke (1975), McGee (1991), and Hintikka (1996). (shrink)

Quantifier variance holds that different languages can have unrestricted quantifier expressions that differ in meaning, where an expression is a “quantifier” just in case it plays the right inferential role. Several critics argued that J.H. Harris’s “collapse” argument refutes variance by showing that identity of inferential role is incompatible with meaning variance. This standard, syntactic collapse argument has generated several responses. More recently, Cian Dorr proved semantic collapse theorems to generate a semantic collapse argument against variance. The argument is significantly (...) different from standard collapse, so it requires a new response. Here I clarify and analyze the semantic collapse argument, and explain how variantists can and should respond to it. The paper also includes an appendix showing the difficulties of positing identity variance without quantifier variance. The argument in the appendix has yet to appear in print, but is familiar to specialists. (shrink)

Section 1 provides a brief summary of the pair-list literature singling out some points that are particularly relevant for the coming discussion. -/- Section 2 shows that the dilemma of quantifi cation versus domain restriction arises only in extensional complement interrogatives. In matrix questions and in intensional complements only universals support pairlist readings, whence the simplest domain restriction treatment suffices. Related data including conjunction, disjunction, and cumulative readings are discussed -/- Section 3 argues that in the case of extensional complements (...) the domain restriction treatment is inadequate for at least two independent reasons. One has to do with the fact that not only upward monotonic quantifi ers support pairlist readings, and the other with the derivation of apparent scope out readings. The reasoning is supplemented with some discussion of the semantic properties of layered quantifi ers.  The above will establish the need for quantifi cation, so the question arises how the objections explicitly enlisted in the literature against quantifi cation can be answered. Section 4 considers the de dicto reading of the quantifi er s restriction, quanti cational variability, and the absence of pairlist readings with whether questions, and argues that they need not militate against the quanti ficational analysis. -/- Section 5 summarizes the emergent proposal -/- Finally, section 6 discusses the signifi cance of the above findings for the behavior of weak islands. (shrink)

The thesis of quantifier variance is consistent and cannot be refuted via a collapse argument.

Recent research has shown that the superlative quantifiers at least and at most do not have the same type of truth conditions as the comparative quantifiers more than and fewer than. We propose that superlative quantifiers are interpreted at the level of speech acts. We relate them to denegations of speech acts, as in I don’t promise to come, which we analyze as excluding the speech act of a promise to come. Calling such conversational acts that affect (...) future permissible speech acts “meta-speech acts,” we introduce the meta-speech act of a GRANT of a proposition as a denial to assert the negation of that proposition. Superlative quantifiers are analyzed as quantifiers over GRANTS. Thus, John petted at least three rabbits means that the minimal number n such that the speaker GRANTs the proposition that John petted n rabbits is n = 3. We formalize this interpretation in terms of commitment states and commitment spaces, and show how the truth conditions that are derived from it are partly entailed and partly conversationally implicated. We demonstrate how the theory accounts for a wide variety of distributional phenomena of superlative quantifiers, including the contexts in which they can be embedded. (shrink)

Eternalists say that non-present entities (for instance dinosaurs) exist; presentists say that they do not. But some sceptics deny that this debate is genuine, claiming that presentists simply represent eternalists' quantifiers over non-present entities in different notation. This scepticism may be refuted on purely logical grounds: one of the leading candidate ‘presentist quantifiers’ over non-present things has the inferential role of a quantifier. The dispute over whether non-present objects exist is as genuine and non-verbal as the dispute over (...) whether there is life on other planets. (shrink)

A sense of unity -- Basic objects : a reply to Xu -- Objectivity without objects -- The vagueness of identity -- Quantifier variance and realism -- Against revisionary ontology -- Comments on Theodore Sider's four dimensionalism -- Sosa's existential relativism -- Physical-object ontology, verbal disputes, and common sense -- Ontological arguments : interpretive charity and quantifier variance -- Language, ontology, and structure -- Ontology and alternative languages.

This paper argues that English quantifier phrases of the form ‘every F’ admit of a literal referential interpretation, contrary to the standard semantic account of this expression, according to which it denotes a set and a second-order relation. Various arguments are offered in favor of the referential interpretation, and two likely objections to it are forestalled.