Generalized Quantifiers

In this article, logical concepts are defined using the internal syntactic and semantic structure of language. For a first-order language, it has been shown that its logical constants are connectives and a certain type of quantifiers for which the universal and existential quantifiers form a functionally complete set of quantifiers. Neither equality nor cardinal quantifiers belong to the logical constants of a first-order language.

We discuss a well-known puzzle about the lexicalization of logical operators in natural language, in particular connectives and quantifiers. Of the many logically possible operators, only few appear in the lexicon of natural languages: the connectives in English, for example, are conjunction _and_, disjunction _or_, and negated disjunction _nor_; the lexical quantifiers are _all, some_ and _no_. The logically possible nand (negated conjunction) and Nall (negated universal) are not expressed by lexical entries in English, nor in any natural language. Moreover, (...) the lexicalized operators are all upward or downward monotone, an observation known as the Monotonicity Universal. We propose a logical explanation of lexical gaps and of the Monotonicity Universal, based on the dynamic behaviour of connectives and quantifiers. We define update potentials for logical operators as procedures to modify the context, under the assumption that an update by \( \phi \) depends on the logical form of \( \phi \) and on the speech act performed: assertion or rejection. We conjecture that the adequacy of update potentials determines the limits of lexicalizability for logical operators in natural language. Finally, we show that on this framework the Monotonicity Universal follows from the logical properties of the updates that correspond to each operator. (shrink)

The need for the definite article to express a singular kind ("the cat") in the Germanic languages is predicted by Borer's (2005) structural approach to the mass-count distinction. Chierchia's (1998) "down" operator can apply to nPs to derive mass kinds ("rice") and to DivPs to derive plural kinds ("cats"), but there is no determinerless structure that exclusively denotes properties of atomic individuals to which this same operator can apply to derive singular kinds. The only alternative is the process that Chierchia (...) proposes for plural kinds in Romance, where the definite article returns a maximal individual that can be intensionalized into a kind. In articleless languages like Mandarin, this account allows for a universal property-denoting denotation of nP and simultaneously captures the fact that singular kinds have the same distribution as mass kinds. (shrink)

Generic statements (‘Tigers have stripes’) are pervasive and early-emerging modes of generalization with a distinctive linguistic profile. Previous experimental work found that generics display a unique asymmetry between their acceptance conditions and the implications that are typically drawn from them. This paper presents evidence against the hypothesis that only generics display an asymmetry. Correcting for limitations of previous designs, we found a generalized asymmetry effect across generics, various kinds of explicitly quantified statements (‘most’, ‘some’, ‘typically’, ‘usually’), and variations in types (...) of predicated properties (striking vs. neutral). We discuss implications of these results for our understanding of the source of asymmetry effects and whether and in which ways these effects might introduce biased beliefs into social networks. (shrink)

Universalist and nihilist answers to philosophical questions may be extreme, but they are clear enough. Aliquidist answers, by contrast, are typically caught between the Scylla of vagueness and indeterminacy and the Charybdis of ungroundedness and arbitrariness, and steering a proper middle course—saying exactly where in the middle one is going to settle—demands exceptional navigating powers. I myself tend to favor extreme answers precisely for this reason. Here, however, I consider one sense in which Something may claim superiority over its polar (...) competitors, Everything and Nothing, when it comes to answerting the ontological question, “What is there?”. (shrink)

Kripke [1975] gives a formal theory of truth based on Kleene's strong evaluation scheme. It is probably the most important and influential that has yet been given—at least since Tarski. However, it has been argued that this theory has a problem with generalized quantifiers such as All—that is, All ϕs are ψ—or Most. Specifically, it has been argued that such quantifiers preclude the existence of just the sort of language that Kripke aims to deliver—one that contains its own truth predicate. (...) In this paper I solve the problem by showing how Kleene's strong scheme, and Kripke's theory based on it, can in a natural way be extended to accommodate the full range of generalized quantifiers. (shrink)

This paper deals with the question of what it is for a quantifier expression to be vague. First it draws a distinction between two senses in which quantifier expressions may be said to be vague, and provides an account of the distinction which rests on independently grounded assumptions. Then it suggests that, if some further assumptions are granted, the difference between the two senses considered can be represented at the formal level. Finally, it outlines some implications of the account provided (...) which bear on three debated issues concerning quantification. (shrink)

This paper deals with the logical form of quantified sentences. Its purpose is to elucidate one plausible sense in which quantified sentences can adequately be represented in the language of first-order logic. Section 1 introduces some basic notions drawn from general quantification theory. Section 2 outlines a crucial assumption, namely, that logical form is a matter of truth-conditions. Section 3 shows how the truth-conditions of quantified sentences can be represented in the language of first-order logic consistently with some established undefinability (...) results. Section 4 sketches an account of vague quantifier expressions along the lines suggested. Finally, section 5 addresses the vexed issue of logicality. (shrink)

The problem concerns quantifiers that seem to hover between universal and existential readings. I argue that they are neither, but a different quantifier that has features of each. NOTE the published paper has a mistake. I have corrected this in the version on this site. A correction note will appear in Analysis.

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)

In this paper I propose a set-theoretical interpretation of the logical square of opposition, in the perspective opened by generalized quantifier theory. Generalized quantifiers allow us to account for the semantics of quantificational Noun Phrases, and of other natural language expressions, in a coherent and uniform way. I suggest that in the analysis of the meaning of Noun Phrases and Determiners the square of opposition may help representing some semantic features responsible to different logical properties of these expressions. I will (...) conclude with some consideration on scope interactions between quantifiers. (shrink)

This book surveys research in quantification starting with the foundational work in the 1970s. It paints a vivid picture of generalized quantifiers and Boolean semantics. It explains how the discovery of diverse scope behavior in the 1990s transformed the view of quantification, and how the study of the internal composition of quantifiers has become central in recent years. It presents different approaches to the same problems, and links modern logic and formal semantics to advances in generative syntax. A unique feature (...) of the book is that it systematically brings cross-linguistic data to bear on the theoretical issues, discussing French, German, Dutch, Hungarian, Russian, Japanese, Telugu (Dravidian), and Shupamem (Grassfield Bantu), and pointing to formal semantic literature involving quantification in around thirty languages. -- -/- 1. What this book is about and how to use it; 2. Generalized quantifiers and their elements: operators and their scopes; 3. Generalized quantifiers in non-nominal domains; 4. Some empirically significant properties of quantifiers and determiners; 5. Potential challenges for generalized quantifiers; 6. Scope is not uniform and not a primitive; 7. Existential scope versus distributive scope; 8. Distributivity and scope; 9. Bare numeral indefinites; 10. Modified numerals; 11. Clause-internal scopal diversity; 12. Towards a compositional semantics of quantifier words. (shrink)

A number of authors in favor of a unitary account of singular descriptions have alleged that the unitary account can be extrapolated to account for plural definite descriptions. In this paper I take a closer look at this suggestion. I argue that while the unitary account is clearly onto something right, it is in the end empirically inadequate. At the end of the paper I offer a new partitive account of plural definite descriptions that avoids the problems with both the (...) unitary account and standard Russellian analyses. (shrink)

In this paper I revisit the main arguments for a predicate analysis of descriptions in order to determine whether they do in fact undermine Russell's theory. I argue that while the arguments without doubt provide powerful evidence against Russell's original theory, it is far from clear that they tell against a quantificational account of descriptions.

The paper revisits Sharvy's theory of plural definite descriptions. An alternative account of plural definite descriptions building on the ideas of plural quantification and non-distributive plural predication is developed. Finally, the alternative is extrapolated to account for generic uses of definite descriptions.

This paper offers a unified, quantificational treatment of incomplete descriptions like ‘the table’. An incomplete quantified expression like ‘every bottle’ (as in “Every bottle is empty”) can feature in true utterances despite the fact that the world contains nonempty bottles. Positing a contextual restriction on the bottles being talked about is a straightforward solution. It is argued that the same strategy can be extended to incomplete definite descriptions across the board. ncorporating the contextual restrictions into semantics involves meeting a complex (...) array of desiderata, yet the apparently simpler pragmatic alternative faces severe problems and is therefore a nonstarter. (shrink)

The semantic function of the modifier 'together' in adnominal position has generally been considered to be that of preventing a distributive reading of the predicate. This paper will argue that this view is mistaken. The semantic function of adnominal 'together' rather is that of inducing a cumulative measurement of the group that together is associated with. The measurement-based analysis of adnominal together that I propose can also, with some modifications, be extended to adverbial occurrences of together.

This is a review of Quantifiers, Logic, and Language, edited by Jaap van der Does and Jan van Eijk, published by CSLI (Center for the Study of Language and Information) Publications, Stanford, CA, in 1996.

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)

This continues from Part 1. It is shown how an intensional interpretation of physics object languages can be formalised, and how a syntactic compositional time reversal operator can subsequently be defined. This is applied to solve the problems used as examples in Part 1. A proof of a general theorem that such an operator must be defineable is sketched. A number of related issues about the interpretation of theories of physics, including classical and quantum mechanics and classical EM theory are (...) discussed. (shrink)