Definability of Polyadic Lifts of Generalized Quantifiers

2001

We study definability in terms of monotone generalized quantifiers satisfying Isomorphism closure, Conservativity and Extension. Among the quantifiers with the latter three properties -here called CE quantifiers -one finds the interpretations of determiner phrases in natural languages. The property of monotonicity is also linguistically ubiquitous, though some determiners like an even number of are highly non-monotone. They are nevertheless definable in terms of monotone CE quantifiers: we give a necessary and sufficient condition for such definability. We further identify a stronger form of monotonicity, called smoothness, which also has linguistic relevance, and we extend our considerations to smooth quantifiers. The results lead us to propose two tentative universals concerning monotonicity and natural language quantification. The notions involved as well as our proofs are presented using a graphical representation of quantifiers in the so-called number tree. 3 See, for example, Barwise and Cooper [1], Keenan and Stavi [6], van Benthem [2], Westerståhl [11], Keenan and Westerståhl [7]. 4 We follow the convenient practice of using the corresponding English determiners as names of various NL quantifiers. So, for example, all denotes the quantifier defined by allM (A, B) ⇐⇒ A ⊆ B, for all M and all A, B ⊆ M .

Journal of Computer and System Sciences, 2014

We study definability of second-order generalized quantifiers. We show that the question whether a second-order generalized quantifier Q 1 is definable in terms of another quantifier Q 2 , the base logic being monadic second-order logic, reduces to the question if a quantifier Q 1 is definable in FO(Q 2 , <, +, ×) for certain first-order quantifiers Q 1 and Q 2 . We use our characterization to show new definability and non-definability results for second-order generalized quantifiers. We also show that the monadic second-order majority quantifier Most 1 is not definable in second-order logic.

Journal of Symbolic Logic, 2006

We study definability of second-order generalized quantifiers. We show that the question whether a second-order generalized quantifier Q 1 is definable in terms of another quantifier Q 2 , the base logic being monadic second-order logic, reduces to the question if a quantifier Q 1 is definable in FO(Q 2 , <, +, ×) for certain first-order quantifiers Q 1 and Q 2 . We use our characterization to show new definability and nondefinability results for second-order generalized quantifiers. In particular, we show that the monadic second-order majority quantifier Most 1 is not definable in second-order logic.

Studia Logica, 1995

Johan van Benthem (Amsterdam) Dag Westerståhl (Stockholm) * This paper was inspired by the symposium on Generalized Quantifiers held at the 5th European Summer School in Logic, Language and Information in Lisbon, August 1993. We feel that the work presented there motivates a survey of recent research areas and research problems in the field of generalized quantifiers. The speakers at the symposium, Natasha Alechina, Jaap van der Does,

The Journal of Symbolic Logic, 1996

The concept of a generalized quanti er of a given similarity type was de ned in Lin66]. Our main result says that on nite structures di erent similarity types give rise to di erent classes of generalized quanti ers. More exactly, for every similarity type t there is a generalized quanti er of type t which is not de nable in the extension of rst order logic by all generalized quanti ers of type smaller than t. This was proved for unary similarity types by Per Lindstr om Wes] with a counting argument. We extend his method to arbitrary similarity types.

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.

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.

1997

In this paper all quanti ers are assumed to be so called simple unary quanti ers, and all models are assumed to be nite. We give a necessary and su cient condition for a quanti er to be de nable in terms of monotone quanti ers. For a monotone quanti er we give a necessary and su cient condition for being de nable in terms of a given set of bounded monotone quanti ers. Finally, we give a necessary and su cient condition for a monotone quanti er to be de nable in terms of a given monotone quanti er. Our analysis shows that the quanti er \at least one half" and its relatives behave di erently than other monotone quanti ers.

2011

This paper provides an overview of recent work by the authors and others on two topics in the model theory of finite structures. The point of view here differs from that usually associated with the term ‘finite model theory’, as presented for example in [21] or [46], in which the emphasis and motivation come primarily from computer science. Instead, the inspiration for this work has its origins in contemporary (infinite) model theoretic themes such as dimension, independence, and various measures of the complexity of definable sets. Each of the topics deals with classes of finite structures for first-order logic that are isolated by conditions that are drawn from these model-theoretic considerations. Moreover, in both cases, connections exist to areas in infinite model theory such as stability and simplicity theory, and o-minimality. This survey is intended for both mathematical logicians and computer scientists whose work focuses on logical aspects of the subject. The first theme c...