Directions in generalized quantifier theory

The Review of Symbolic Logic, 2008

This note explains the circumstances under which a type 〈1〉 quantifier can be decomposed into a type 〈1, 1〉 quantifier and a set, by fixing the first argument of the former to the latter. The motivation comes from the semantics of Noun Phrases (also called Determiner Phrases) in natural languages, but in this article, I focus on the logical facts. However, my examples are taken among quantifiers appearing in natural languages, and at the end, I sketch two more principled linguistic applications.

1985

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Linguistics and Philosophy, 2010

Quantifiers in Language and Logic (QLL) is a major contribution to natural language semantics, specifically to quantification. It integrates the extensive recent work on quantifiers in logic and linguistics. It also presents new observations and results. QLL should help linguists understand the mathematical generalizations we can make about natural language quantification, and it should interest logicians by presenting an extensive array of quantifiers that lie beyond the pale of classical logic. Here we focus on those aspects of QLL we judge to be of specific interest to linguists, and we contribute a few musings of our own, as one mark of a worthy publication is whether it stimulates the reader to seek out new observations, and QLL does. QLL is long and fairly dense, so we make no attempt to cover all the points it makes. But QLL has a topic index, a special symbols index and two tables of contents, a detailed one and an overview one, all of which help make it user friendly. QLL is presented in four parts: I, ''The Logical Conception of Quantifiers and Quantification'' with an introductory section ''Quantification''. II, ''Quantifiers of Natural Language'', the most extensive section in the book and of the most direct interest to linguists. III, ''Beginnings of a Theory of Expressiveness, Translation, and Formalization'' introduces notions of expressive power and definability, and IV, presents recent work and techniques concerning quantifier definability over finite domains, making accessible to linguists recent work in finite model theory.

The paper provides a logical theory of a specific class of natural language expressions called intermediate quantifiers (most, a lot of, many, a few, a great deal of, a large part of, a small part of), which can be ranked among generalized quantifiers. The formal frame is the fuzzy type theory (FTT). Our main idea lays in the observation that intermediate quantifiers speak about elements taken from a class that is made "smaller" than the original universe in a specific way. Our theory is based on the formal theory of trichotomous evaluative linguistic expressions. Thus, an intermediate quantifier is obtained as a classical quantifier "for all" or "exists" but taken over a class of elements that is determined using an appropriate evaluative expression. In the paper we will characterize the behavior of intermediate quantifiers and prove many valid syllogisms that generalize classical Aristotle's ones. . 1 In , the intermediate quantifiers are called comparative. According to the literature on generalized quantifiers, however, "comparative quantifiers" are quantifiers such as "more than about", "about three times less than", etc. We will, therefore, keep the term intermediate in this paper.

Communications in Computer and Information Science, 2014

In this paper, we will first discuss fuzzy generalized quantifiers and their formalization in the higher-order fuzzy logic (the fuzzy type theory). Then we will briefly introduce a special model of intermediate quantifiers, classify them as generalized fuzzy ones and prove that they have the general properties of isomorphism invariance, extensionality and conservativity. These properties are characteristic for the quantifiers of natural language.

Journal of Logic, Language and Information, 1993

We study several modal languages in which some (sets of) generalized quantifiers can be represented; the main language we consider is suitable for defining any first order definable quantifier, but we also consider a sublanguage thereof, as well as a language for dealing with the modal counterparts of some higher order quantifiers. These languages are studied both from a modal logic perspective and from a quantifier perspective. Thus the issues addressed include normal forms, expressive power, completeness both of modal systems and of systems in the quantifier tradition, complexity as well as syntactic characterizations of special semantic constraints. Throughout the paper several techniques current in the theory of generalized quantifiers are used to obtain results in modal logic, and conversely.

Routledge Companion to Philosophy of Language. Eds. D. Graff Fara & G. Russell. Pp. 579-95, 2012

This chapter offers a logical, linguistic, and philosophical account of modern quantification theory. Contrasting the standard approach to quantifiers (according to which logical quantifiers are defined by enumeration) with the generalized approach (according to which quantifiers are defined systematically), the chapter begins with a brief history of standard quantifier theory and identifies some of its logical, linguistic, and philosophical strengths and weaknesses. It then proceeds to a brief history of generalized quantifier theory and explains how it overcomes the weaknesses of the standard theory. One of the main philosophical advantages of the generalized theory is its philosophically informative criterion of logicality. The chapter describes the work done so far in this theory, highlights some of its central logical results, offers an overview of its main linguistic contributions, and discusses its philosophical significance.

New Frontiers in Artificial Intelligence (JSAI-isAI 2013 Workshops, LENLS, JURISIN, MiMI, AAA, and DDS, Kanagawa, Japan, October 27-28, 2013, Revised Selected Papers), Yukiko Nakano, Ken Satoh, Daisuke Bekki (Eds.), LNAI 8417, pp.115-124, Springer., 2014

This paper proposes a proof-theoretic definition for generalized quantifiers (GQs). Sundholm first proposed a proof-theoretic definition of GQs in the framework of constructive type theory. However, that definition is associated with three problems: the proportion problem, absence of strong interpretation and lack of definitional uniformity. This paper presents an alternative definition for "most" based on polymorphic dependent type theory and shows strong potential to serve as an alternative to the traditional model-theoretic approach.

2008

Hungarian is often claimed to be a remarkably ‘logi cal’ language, perhaps to the point of ‘wearing its LF on its sleeve’ (cf. É. Kiss 1991). Even its topic and focus positions are said to be associated with logi cal rather than discourse functions by É. Kiss (2007), and the quantifier cat egory is certainly one which by its very name seems to render abundant reference to logic inevitable. Once the linguist agrees to work with the notion of quan tifiers, he/she will be hard pressed to refuse their logico-semantic interpretat ion as ‘variable binding operators’, with an inherent bias to syntactic mode ls operating with such processes and objects as overt or covert movements, traces, and so on. In the present paper, I will attempt to show that virtually no formal logic is required for understanding the fundamental syntacti c properties of Hungarian quantifiers. This I will do by exploiting a well-kn own but often neglected parallel in the distribution of quantifiers and ‘qu alifiers’ ...