Generalized Quantifier Theory Research Articles

Convexity and Monotonicity in Language Coordination: Simulating the Emergence of Semantic Universals in Populations of Cognitive Agents

Natural languages vary in their quantity expressions, but the variation seems to be constrained by general properties, so-called universals. Their explanations have been sought among constraints of human cognition, communication, complexity, and pragmatics. In this article, we apply a state-of-the-art language coordination model to the semantic domain of quantities to examine whether two quantity universals—monotonicity and convexity—arise as a result of coordination. Assuming precise number perception by the agents, we evolve communicatively usable quantity terminologies in two separate conditions: a numeric-based condition in which agents communicate about a number of objects and a quotient-based condition in which agents communicate about the proportions. We find out that both universals take off in all conditions but only convexity almost entirely dominates the emergent languages. Additionally, we examine whether the perceptual constraints of the agents can contribute to the further development of universals. We compare the degrees of convexity and monotonicity of languages evolving in populations of agents with precise and approximate number sense. The results suggest that approximate number sense significantly reinforces monotonicity and leads to further enhancement of convexity. Last but not least, we show that the properties of the evolved quantifiers match certain invariance properties from generalized quantifier theory.

Aristotelian Diagrams for the Proportional Quantifier ‘Most’

In this paper, we study the interaction between the square of opposition for the Aristotelian quantifiers (‘all’, ‘some’, ‘no’, and ‘not all’) and the square of opposition generated by the proportional quantifier ‘most’ (in its standard generalized quantifier theory reading of ‘more than half’). In a first step, we provide an analysis in terms of bitstring semantics for the two squares independently. The classical square for ‘most’ involves a tripartition of logical space, whereas the degenerate square for ‘all’ in first-order logic (FOL) involves a quadripartition, due to FOL’s lack of existential import. In a second move, we combine these two squares into an octagon of opposition, which was hitherto unattested in logical geometry, while the meet of the original tri- and quadripartitions yields a hexapartition for this octagon. In a final step, we switch from FOL to a logical system, which does assume existential import. This yields an octagon of the well known Lenzen type, and its bitstring semantics is reduced to a pentapartition.

Formal System of Categorical Syllogistic Logic Based on the Syllogism <i>AEE-4</i>

Adopting a different method from the previous scholars, this article deduces the remaining 23 valid syllogisms just taking the syllogism AEE-4 as the basic axiom. The basic idea of this study is as follows: firstly, make full use of the trichotomy structure of categorical propositions to formalize categorical syllogisms. Then, taking advantage of the deductive rules in classical propositional logic and the basic facts in the generalized quantifier theory, we deduce the remaining 23 valid categorical syllogisms by taking just one syllogism (that is, AEE-4) as the basic axiom. This article not only reveals the reducible relations between the syllogism AEE-4 and the other 23 valid syllogisms, but also establishes a concise formal axiomatic system for categorical syllogistic logic. We hope that the results and methods will provide a good mathematical paradigm for studying other kinds of syllogistic logics, and that the project will appeal to specialists in logic, linguistic semantics, computational semantics, cognitive science and artificial intelligence.

Reduction between Aristotelian Modal Syllogisms Based on the Syllogism ◇I□A◇I-3

In order to provide a consistent explanation for Aristotelian modal syllogistic, this paper reveals the reductions between the Aristotelian modal syllogism ◇I□A◇I-3 and the other valid modal syllogisms. Specifically, on the basis of formalizing Aristotelian modal syllogisms, this paper proves the validity of ◇I□A◇I-3 by means of the truth value definition of (modal) categorical propositions. Then in line with classical propositional logic and modal logic, generalized quantifier theory and set theory, this paper deduces the other 47 valid Aristotelian modal syllogisms from the modal syllogism ◇I□A◇I-3. This study shows that the reasons why these syllogisms are reducible are: 1) any of Aristotelian quantifier can be defined by the other three Aristotelian quantifiers; 2) the Aristotelian quantifiers some and no have symmetry; 3) the possible modality ◇ and necessary modality □ can be mutually defined. This formal study of Aristotelian modal syllogistic not only conforms to the needs of formalization transformation of various information in the era of artificial intelligence, but also provides a unified mathematical research paradigm for other kinds of syllogistic.

Conjoining Meanings: Semantics without Truth Values

<i>Conjoining Meanings: Semantics without Truth Values</i>

Screening out All Valid Aristotelian Modal Syllogisms

It is easy to understand that whether a classical syllogism is valid. That whether a modal syllogism is valid is not so transparent. The prevailing view on Aristotelian modal syllogistic is that the syllogistic is incomprehensible due to its many faults and inconsistencies. Although adequate semantic analysis or reconstruction of the syllogistic have be given by many authors, it is far from obvious how to extend these results so as to consistently cover the whole modal syllogistic developed. The major aim of this paper is to overcome these difficulties, and screen out 384 Aristotelian valid modal syllogisms from 6656 Aristotelian modal syllogisms in natural language. They can be formalized by means of set theory and generalized quantifier theory, and their validity can be proved by possible world semantics and the truth definition of Aristotelian quantifiers defined in generalized quantifier theory. The basic steps of screening out all valid Aristotelian modal syllogisms are as follows: firstly one can get all possible modal syllogisms obtained by adding modal operators to 24 valid classical syllogisms, and secondly eliminate invalid modal syllogisms by characteristic rules of modal syllogisms. It is hoped that these innovative achievements will make contributions to promote the development of Aristotelian and generalized modal syllogistic, natural language information processing, and further research on knowledge representation and knowledge reasoning in computer science.

Axiomatization of Aristotelian Syllogistic Logic Based on Generalized Quantifier Theory

Syllogistic reasoning is important due to the prominence of syllogistic arguments in human reasoning, and also to the role they have played in theory of reasoning from Aristotle onwards. Aristotelian syllogistic logic is a formal study of the meaning of four Aristotelian quantifiers and of their properties. This paper focuses on logical system based on syllogistic reasoning. It firstly formalized the 24 valid Aristotle’s syllogisms, and then has proven that the other 22 valid Aristotle’s syllogisms can be derived from the syllogisms ‘Barbara’ AAA-1 and ‘Celarent’ EAE-1 by means of generalized quantifier theory and set theory, so the paper has completed the axiomatization of Aristotelian syllogistic Logic. This axiomatization needs to make full use of symmetry and transformable relations between/among the monotonicity of the four Aristotelian quantifiers from the perspective of generalized quantifier theory. In fact, these innovative achievements and the method in this paper provide a simple and reasonable mathematical model for studying other generalized syllogisms. It is hoped that the present study will make contributions to the development of generalized quantifier theory, and to bringing about consequences to natural language information processing as well as knowledge representation and reasoning in computer science.

At least not false, at most possible: between truth and assertibility of superlative quantifiers

Generalized Quantifier Theory defines superlative quantifiers at most n and at least n as truth-conditionally equivalent to comparative quantifiers fewer than n+1 and more than n\(-\)1. It has been demonstrated, however, that this standard theory cannot account for various linguistic differences between these two types of quantifiers. In this paper I discuss how the distinction between assertibility and truth-conditions can be applied to explain this phenomenon. I draw a parallel between the assertibility of disjunctions and superlative quantifiers, and argue that those assertibility conditions are essentially modal. I use epistemic logic to formalize the assertibility conditions and revisit some of the linguistic puzzles related to superlative quantification.

The 'Whole' Story of Partitive Quantification

Since early work in Generalized Quantifier Theory (GQT), the traditional approach to DP quantifiers has assumed that their LF mirrors the Q+NP surface structure (cf., among many others, Barwise and Cooper 1981 and Keenan and Stavi 1986). Under this approach, quantifiers behave syntactically as determiners taking NP complements and semantically as relations between sets, as shown in Figure 1 on the following page. Recent literature suggests instead that the LF of DP quantifiers mirrors the Q+PP surface structure such that these expressions c-command an NP embedded in a partitive PP. After briefly reviewing this existing work, the present paper advances two novel arguments in support of this partitive structure. Importantly, where previous proposals limit (to varying degrees) the scope of the partitive structure to only a subset of the expressions in (1), the evidence I contribute suggests that all of these expressions are necessarily partitive at LF. I

Safety and the True–True Problem

AbstractStandard accounts of semantics for counterfactuals confront the true–true problem: when the antecedent and consequent of a counterfactual are both actually true, the counterfactual is automatically true. This problem presents a challenge to safety‐based accounts of knowledge. In this paper, drawing on work by Angelika Kratzer, Alan Penczek, and Duncan Pritchard, we propose a revised understanding of semantics for counterfactuals utilizing machinery from generalized quantifier theory which enables safety theorists to meet the challenge of the true–true problem.

If-Clauses and Probability Operators

Adams’ thesis is generally agreed to be linguistically compelling for simple conditionals with factual antecedent and consequent. We propose a derivation of Adams’ thesis from the Lewis-Kratzer analysis of if-clauses as domain restrictors, applied to probability operators. We argue that Lewis’s triviality result may be seen as a result of inexpressibility of the kind familiar in generalized quantifier theory. Some implications of the Lewis-Kratzer analysis are presented concerning the assignment of probabilities to compounds of conditionals.

On the grammar and processing of proportional quantifiers: most versus more than half

Proportional quantifiers have played a central role in the development of formal semantics because they set a benchmark for the expressive power needed to describe quantification in natural language (Barwise and Cooper Linguist Philos 4:159–219, 1981). The proportional quantifier most, in particular, supplied the initial motivation for adopting Generalized Quantifier Theory (GQT) because its meaning is definable as a relation between sets of individuals, which are taken to be semantic primitives in GQT. This paper proposes an alternative analysis of most that does not treat it as a lexical item whose meaning is accessible without the help of compositional processes. Instead, proportional most is analyzed as the superlative of many (cf. Bresnan Linguist Inq 4(3):274–344, 1973). Two types of empirical evidence are presented in support of this view, both exploiting the fact that only a decompositional analysis of proportional quantifiers provides the means to generate different logical forms for seemingly equivalent statements of the form most A B and more than half of the A B.

Monadic L-fuzzy quantifiers of the type [formula omitted

The aim of this paper is to introduce monadic L -fuzzy quantifiers of the type 〈 1 n , 1 〉 and investigate some of their semantic properties that are well-known in generalized quantifier theory. Relationships between the properties of generalized quantifiers are also studied for monadic L -fuzzy quantifiers. Finally, a property “to be extensional with respect to a given L -permutation equivalence” is proposed and some properties are shown.

On the denotations of anaphors

In the spirit of generalized quantifier theory, we provide direct interpretations for anaphors such as himself, herself, everyone but himself, no student but himself, both himself and the teacher, etc., as they occur in Ss like: Every worker criticized himself, No student criticized every student but himself, Some student criticized both himself and the teacher, etc. Then we provide a syntax independent definition of “anaphor” which enables us to non-circularly identify anaphors in different languages and to test the validity of claims such as “Anaphors are always locally c-commanded by their antecedents.” We also compute the sense in which anaphors increase logical expressive power by computing the size of the denotation set for anaphors as compared with mere non-anaphoric generalized quantifiers.

A characterization of monotonicity with collective quantifiers

This paper studies the monotonicity behavior of plural determiners that quantify over collections. Following previous work, we describe the collective interpretation of determiners such as all, some and most using generalized quantifiers of a higher type that are obtained systematically by applying a type shifting operator to the standard meanings of determiners in Generalized Quantifier Theory. Unlike previous proposals, one unified determiner fitting operator both captures existential quantification with plural determiners and respects their monotonicity properties. However, some previously unnoticed facts indicate that monotonicity of plural determiners is not always preserved when they apply to collective predicates. We show that the proposed operator describes this behavior correctly, and characterize the monotonicity of the collective determiners it derives. It is proved that determiner fitting always preserves monotonicity properties of determiners in their second argument, but monotonicity in the first argument of a determiner is preserved if and only if it is monotonic in the same direction in the second argument.This research was supported by grant no. 1999210 (“Extensions and Implementations of Natural Logic”) from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel. The second author was also partly supported by the fund for the promotion of research at the Technion, research no. 120-042. Part of the research by the second author was curried out during a stay at the Utrecht University, which was supported by an NWO grant no. B30-541. We are indebted to Johan van Benthem for his remarks, which initiated our interest in questions of monotonicity and collectivity. Thanks also to Nissim Francez, Ed Keenan and Shuly Wintner for their comments.

Monotonicity and Collective Quantification

This article studies the monotonicity behavior of plural determiners that quantify over collections. Following previous work, we describe the collective interpretation of determiners such as all, some and most using generalized quantifiers of a higher type that are obtained systematically by applying a type shifting operator to the standard meanings of determiners in Generalized Quantifier Theory. Two processes of counting and existential quantification that appear with plural quantifiers are unified into a single determiner fitting operator, which, unlike previous proposals, both captures existential quantification with plural determiners and respects their monotonicity properties. However, some previously unnoticed facts indicate that monotonicity of plural determiners is not always preserved when they apply to collective predicates. We show that the proposed operator describes this behavior correctly, and characterize the monotonicity of the collective determiners it derives. It is proved that determiner fitting always preserves monotonicity properties of determiners in their second argument, but monotonicity in the first argument of a determiner is preserved if and only if it is monotonic in the same direction in the second argument. We argue that this asymmetry follows from the conservativity of generalized quantifiers in natural language.

Flexibility Principles in Boolean Semantics: The Interpretation of Coordination, Plurality, and Scope in Natural Language

Since the early work of Montague, Boolean semantics and its subfield of generalized quantifier theory have become the model-theoretic foundation for the study of meaning in natural languages. This book uses this framework to develop a new semantic theory of central linguistic phenomena involving coordination, plurality, and scope. The proposed theory makes use of the standard Boolean interpretation of conjunction, a choice-function account of indefinites, and a novel semantics of plurals that is not based on the distributive/collective distinction. The key to unifying these mechanisms is a version of Montagovian semantics that is augmented by flexibility principles: semantic operations that have no counterpart in phonology.This is the first book to cover these areas in a way that is both linguistically comprehensive and formally explicit. On one hand, it addresses questions of primarily linguistic concern: the semantic functions of words like and and or in different languages, the interpretation of indefinites and their scope, and the semantic typology of noun phrases and predicates. On the other hand, it addresses formal questions that are motivated by the treatment of these linguistic problems: the use of Boolean algebras in linguistics, the proper formalization of choice functions within generalized quantifier theory, and the extension of this theory to the domain of plurality. While primarily intended for readers with a background in theoretical linguistics, the book will also be of interest to researchers and advanced students in logic, computational linguistics, philosophy of language, and artificial intelligence.

Reference and Anaphoric Relations (review)

Reviewed by: Reference and anaphoric relations ed. by Klaus von Heusinger, Urs Egli Iván García Reference and anaphoric relations. Ed. by Klaus von Heusinger and Urs Egli. (Studies in linguistics and philosophy 72.) Dordrecht: Kluwer, 2000. Pp. xi, 347. $132.00. This edited collection of fifteen original research articles has its roots in the workshop ‘Reference and Anaphoric Relations’ (University of Konstanz, June 1996). The papers deal with the representation and interpretation of indefinite and definite noun phrases; anaphoric pronouns; and several closely related issues such as reference, scope, quantifier movement, and the relation between context and meaning. The book begins with an overview article by the editors, ‘Introduction: Reference and the semantics of anaphora’, which surveys the motivation for the development of new semantic formalisms in the last decades and gives a general outline of the articles in the volume. The book is divided into five parts. In Part 1, ‘Historical aspects of anaphoric relations’ (15–76), the contributors consider the Stoic and scholastic treatment of reference and anaphora. Egli implements the Stoic view on anaphora and quantified sentences into a dynamic semantics. Reinhard Hülsen shows that medieval logicians advanced a fairly consistent and comprehensive theory of the semantics of ‘relativa grammaticalia’. Jeroen Groenendijk and Martin Stokhof illustrate some basic concepts of dynamic semantics and suggest an analysis of anaphoric (in)definite descriptions as contextually dependent quantificational expressions. The issues of scope and quantifier movement are considered in Part 2, ‘Quantification and scope’ (77–154). Donka Farkas reviews the main tasks that a theory of scope has to solve and develops a nonconfigurational, nonmovement based account of scope. Henriëtte de Swart investigates the relation between scope, lexical composition, and properties of generalized quantifiers. She argues against a lexical decomposition account of negative quantifiers in German and Dutch, suggesting instead a higher order interpretation of the determiners. Elena Paducheva provides an analysis of definiteness effects in English and Russian. Stephen Neale examines an inconsistency which arises from the conjunction of three commonly accepted hypotheses regarding persistence, polarity, and plurality. In Part 3, ‘Anaphoric reference’ (155–90), two papers by Robert van Rooy and Hartley Slater concentrate on the problem of intentional identity and the semantics of Hob-Nob sentences. Part 4, ‘Choice functions and the semantics of indefinites’ (191–266), contains three papers on the representation of indefinites by means of choice functions. Arnim von Stechow compares the in situ approach of choice functions with the classical movement approach of quantifier raising at LF, while Yoad Winter argues convincingly that the semantics of indefinites involves choice functions by appealing to universals of generalized quantifier theory. Klaus von Heusinger contends that the complex referential nature of indefinites must be reflected in their semantic representation. In Part 5, ‘Representation and interpretation’ (267–338), Jaroslav Peregrin investigates the related concepts of reference and inference, arguing that reference is parasitic on inference, contrary to what is commonly assumed. Paul Dekker discusses the relation between interpretation and representation and develops a weakly representational system of dynamic epistemic predicate logic that offers a unified treatment of anaphoric and demonstrative pronouns. Closing the volume, Reinhard Muskens investigates the interaction between representation and interpretation with respect to underspecified structures. This beautifully produced book reflects the increasing interest in the semantics of anaphoric expressions and is highly recommended to all those interested in formal semantics and the syntax-semantics interface. Iván García Stanford University Copyright © 2001 Linguistic Society of America

The Algebraic Semantics of Interrogative NPs

We suggest a new semantic interpretation of interrogative NPs, based on a novel algebraic theory of the semantics of questions. We discuss the role of interrogative NPs in compositionally deriving the semantics of wh-questions, examine the coordination of interrogative NPs, and apply notions from generalized quantifier theory to their interpretation.

Vector Space Semantics: A Model-Theoretic Analysis of Locative Prepositions

This paper introduces a compositional semantics of locative prepositional phrases which is based on a vector space ontology. Model-theoretic properties of prepositions like monotonicity and conservativity are defined in this system in a straightforward way. These notions are shown to describe central inferences with spatial expressions and to account for the grammaticality of preposition modification. Model-theoretic constraints on the set of possible prepositions in natural language are specified, similar to the semantic universals of Generalized Quantifier Theory.