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).

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)

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.

Initially, the semantics of quantifiers seems to suggest a context-insensitive account of their interpretation unlike, say, indexical or demonstrative noun phrases. However, most philosophers think that incomplete quantifiers and the need for domain restriction shows that all quantifiers are, in fact, context sensitive. This chapter carefully establishes a set of intuitions prevalent in the literature in favour of a context-sensitive analysis of the semantics of quantifiers. These intuitions are shown to be misguided through a Kripke test, (...) a technique which establishes the consistency of a set of assumed counter-examples and a given phenomenon by showing that in a language in which this phenomenon is stipulated, these counter-examples persist. This task is achieved by using the connection (or rather lack thereof) between reports of what is said/asserted/claimed and semantic content to show that the context-sensitive intuitions are the result of the _non-interpretive_, _non-semantic_ effects of true indirect reports. (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)

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)

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)

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)

There is no unanimity in the literature about what logicality is, but there is almost universal agreement, it seems, that topic neutrality or subject independence, that is, isomorphism closure (ISOM), is at least a necessary condition. This chapter presents the general version of ISOM for arbitrary operations and considers some of its variations. It expresses the significance of ISOM for natural language quantification in two claims: first, that simple quantirelations — including determiners and quantificational adverbs — denote ISOM quantifiers, (...) and second, that those denoting non-ISOM quantifiers all arise from more general ISOM operations by freezing some arguments. Constancy, a property that differs from logicality, is also discussed, focusing on what appears to be the empirical manifestation of that property. This property concerns the behavior of certain expressions in intuitively valid arguments, and which are called inference constancy. (shrink)

Some philosophers, notably Professors Quine and Geach, have stressed the analogies they see between pronouns of the vernacular and the bound variables of quantification theory. Geach, indeed, once maintained that ‘for a philosophical theory of reference, then, it is all one whether we consider bound variables or pronouns of the vernacular'. This slightly overstates Geach's positition since he recognizes that some pronouns of ordinary language do function differently from bound variables; he calls such pronouns ‘pronouns of laziness'. Geach's characterisation of (...) pronouns of laziness has varied from time to time, but the general idea should be clear from a paradigm example: A man who sometimes beats his wife has more sense than one who always gives in to her.The pronouns ‘one’ and ‘her’ go proxy for a noun or a noun phrase in the sense that the pronoun is replaceable in paraphrase by simple repetition of its antecedent. (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)

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)

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, it 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)

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)

Many of the motivations in favor of contextualism about knowledge apply also to a contextualist approach to counterfactuals. I motivate and articulate such an approach, in terms of the context-sensitive 'all cases', in the spirit of David Lewis's contextualist view about knowledge. The resulting view explains intuitive data, resolves a puzzle parallel to the skeptical paradox, and renders safety and sensitivity, construed as counterfactuals, necessary conditions on knowledge.

Here, I combine the semantics of Mares and Goldblatt [20] and Seki [29, 30] to develop a semantics for quantified modal relevant logics extending ${\bf B}$. The combination requires demonstrating that the Mares–Goldblatt approach is apt for quantified extensions of ${\bf B}$ and other relevant logics, but no significant bridging principles are needed. The result is a single semantic approach for quantified modal relevant logics. Within this framework, I discuss the requirements a quantified modal relevant logic must satisfy to be (...) “sufficiently classical” in its modal fragment, where frame conditions are given that work for positive fragments of logics. The roles of the Barcan formula and its converse are also investigated. (shrink)

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)

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.

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.

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)

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)

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)

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.

This paper presents analyses of natural language quantifiers and determiners. In doing so, the paper pays special attention to _plural determiners_, determiners that can combine with plural noun phrases (e.g., _all, some, any, the, most_), and argues that Generalized Quantifier Theory gives clearly incorrect accounts of those determiners by assuming the traditional bias against plural constructions. The paper gives a sketch of a recent approach to plural constructions, the _pluralist approach_, that regards them as peers of their singular cousins (...) that have autonomous semantic functions, and presents analyses of plural determiners that result from taking the approach. The paper also presents analyses of Bach-Peters sentences that use polyadic quantifiers or determiners, and gives a sketch of a treatment of donkey anaphora based on the view that indefinite noun phrases by themselves have no quantificational component. (shrink)

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)

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)

It is occasionally tempting, after climbing a mountain, to use the elevation one has gained to dash up to the top of a connected peak which does not have sufficient interest to induce one to climb so high for its sake alone. It is in this spirit that I turn to Geach's Latin Prose theory of relative clauses. The matter itself is of no very great moment, and some new ground will have to be covered in dealing with Geach's arguments. (...) Nevertheless we shall primarily be applying the theory constructed in the body of the paper, and when one is in a position to expose bad arguments relatively rapidly, it is perhaps a good idea not to leave them unchallenged, especially when they appear to be gaining currency. (shrink)

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

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)

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.

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.

Quine introduced a famous distinction between the ‘notional’ sense and the ‘relational’ sense of certain attitude verbs. The distinction is both intuitive and sound but is often conflated with another distinction Quine draws between ‘dyadic’ and ‘triadic’ (or higher degree) attitudes. I argue that this conflation is largely responsible for the mistaken view that Quine’s account of attitudes is undermined by the problem of the ‘exportation’ of singular terms within attitude contexts. Quine’s system is also supposed to suffer from the (...) problem of ‘suspended judgement with continued belief’. I argue that this criticism fails to take account of a crucial presupposition of Quine’s about the connection between thought and language. The aim of the paper is to defend the spirit of Quine’s account of attitudes by offering solutions to these two problems. (shrink)

✤ It is the characterization of those forms of reasoning that lead invariably from true sentences to true sentences, independently of the subject matter.

I examine three ‘anti-object’ metaphysical views: nihilism, generalism, and anti-quantificationalism. After setting aside nihilism, I argue that generalists should be anti-quantificationalists. Along the way, I attempt to articulate what a ‘metaphysically perspicuous’ language might even be.

We characterize the generalized quantifiers Q which satisfy the scheme $QxQy\phi \leftrightarrow QyQx\phi$ , the so-called self-commuting quantifiers, or quantifiers with the Fubini property.

Quantified NPs in questions may lead to an interpretation in which the NP quantifies into the question. Which dish did every guest bring? can be understood as: 'For every guest x: which dish did x bring?'. After a review of previous approaches that tried to capture this quantification formally or to explain it away, it is argued that such readings involve quantification into speech acts. As the algebra of speech acts is more limited than a Boolean algebra – it only (...) contains conjunction, not disjunction or negation – it is predicted that only universal quantifiers can scope out of questions or other speech acts. The approach is extended to indirect questions, which either are embedded speech acts or coerced to denote the true answers, depending on the embedding verb; in the latter case a Boolean structure results, and we find wide-scope readings of non-universal quantifiers. (shrink)