Algebraic Semantics in Language and Philosophy

CSLI Lecture Notes No. 74

ALGEBRAIC SEMANTICS IN LANGUAGE AND PHILOSOPHY

GODEHARD LINK

PUBLICATIONS] CENTER FOR THE STUDY OF LANGUAGE AND INFORMATION STANFORD. CALIFORNIA

Copyright © 1998 CSLI Publications Center for the Study of Language and Information Leland Stanford Junior University Printed in the United States 02 01 00 99 98 54321 Library of Congress Cataloging-in-Publication Data Link, Godehard. Algebraic semantics in language and philosophy / Godehard Link. p. cm. — (CSLI lecture notes ; no. 74) Collection of 14 essays, seven of which have been previously published. Includes bibliographical references (p. ) and index. Contents: The logical analysis of plurals and mass terms — Plural — Hydras : on the logic of relative clause constructions with multiple heads — Generalized quantifiers and plurals — Je drei Apfel = Three apples each : quantification and the German je — First-order axioms for the logic of plurality — Ten years of research on plurals : where do we stand? — Algebraic semantics for natural language — Quantity and number — The French Revolution—a philosophical event? — Algebraic semantics of event structures — The ontology of individuals and events — The philosophy of plurality — Mereology, second-order logic, and set theory — Appendix : a chapter in lattice theory. ISBN 1-57586-091-0 (alk. paper). — ISBN 1-57586-090-2 (pbk. : alk. paper) 1. Grammar, Comparative and general—Number. 2. Grammar, Comparative and general—Quantifiers. 3. Semantics (Philosophy) 4. Language and logic. 5. Language and languages—Philosophy. I. Title. II. Series. P240.8.L56 1997 415—dc21 97-46523 CIP oo The acid-free paper used in this book meets the minimum requirements of the American National Standard for Information Sciences—Permanence of Paper for Printed Library Materials, ANSI Z39.48-1984. CSLI was founded early in 1983 by researchers from Stanford University, SRI International, and Xerox PARC to further research and development of integrated theories of language, information, and computation. CSLI headquarters and CSLI Publications are located on the campus of Stanford University. CSLI Publications reports new developments in the study of language, information, and computation. In addition to lecture notes, our publications include monographs, working papers, revised dissertations, and conference proceedings. Our aim is to make new results, ideas, and approaches available as quickly as possible. Please visit our web site at http://csli-www.stanford.edu/publications/

for comments on this and other titles, as well as for changes and corrections by the author and publisher.

Contents Preface

ix

Introduction

1

1 The Logical Analysis of Plurals and Mass Terms: A Latticetheoretical Approach 1.1 Introduction 1.2 The Logic of Plurals and Mass Terms (LPM) 1.3 Applications to Montague Grammar 2 Plural 2.1 Introduction 2.2 Types of Plural Constructions 2.2.1 Indefinite Plural NPs; Bare Plurals (BP) 2.2.2 Definite Plural NPs 2.2.3 Universally Quantified PNPs; Quantifier Floating . . 2.2.4 Numerals and Other Plural Quantifiers 2.2.5 Partitive Constructions 2.2.6 Coordinate Conjoined NP Structures 2.2.7 Collective Nouns, Predicates, and Adverbs 2.2.8 Distributive vs Collective Predication 2.2.9 Relational Plural Sentences 2.2.10 Reciprocal Constructions and respectively 2.3 Ontology 2.3.1 The Power Set Model; Collections; Mereology . . . . 2.3.2 The Algebraically Structured Universe of Individual Sums and Groups 2.4 Data Explained. Translation into Logical Form 2.4.1 Indefinite PNPs -. 2.4.2 Bare Plurals

11 11 22 29 35 35 36 37 41 44 46 48 49 49 50 54 59 61 61 66 70 70 71

VI

2.5

2.4.3 Definite PNPs 2.4.4 all 2.4.5 Partitive 2.4.6 Conjoined NPs 2.4.7 RP Sentences 2.4.8 Reciprocals 2.4.9 respectively Outlook and Special Problems

71 72 72 72 73 74 74 75

3 Hydras. On the Logic of Relative Clause Constructions with Multiple Heads 77 3.1 Introduction 77 3.2 The Frame of Analysis 79 3.3 Analysis of the Hydras (4) - (10) 83 3.4 Conclusion 88 4 Generalized Quantifiers and Plurals 89 4.1 The Logic of Plurals, LP: Review of the Basic Ideas . . . . 89 4.2 Lifting LP into the Generalized Quantifier Framework . . . 94 4.3 Plural Quantification 97 4.4 The Treatment of Numerals 101 4.5 Floated Quantifiers 109 4.6 The Case of the German je 113 5 Je drei Apfel—three apples each: Quantification and the German je 117 5.1 Different Uses of je 118 5.2 The Distributional Domain 119 5.3 The Distributive Share 122 5.4 Accessability 123 5.5 je vs jeweils: Events as Distributional Domain 127 5.6 The Semantics of je Constructions 129 6 First-Order Axioms for the Logic of Plurality 6.1 The original LP arsenal and its use 6.2 LP axioms 6.2.1 The logical basis of free logic 6.2.2 Proper axioms for LP 6.2.3 Structuring the plural semilattice 6.2.4 L0nning's system rephrased in LP 6.3 Metatheory

133 135 138 138 141 143 153 156

Vll

7 Ten Years of Research on Plurals - Where Do We Stand? 163 7.1 Introduction 163 7.2 Current Areas of Research 167 7.3 Groups and the Problem of Over-Representation 172 7.4 Distributivity 176 8 Algebraic Semantics for Natural Language: Some Philosophy, some Applications 189 8.1 Introduction 189 8.2 Algebraic Semantics in Logic 191 8.3 Algebraic Semantics for Natural Language 195 8.3.1 Plural lattices 197 8.3.2 Mass Terms 199 8.3.3 Events 200 8.4 Some Applications to Plural Theory 204 8.5 Ontology of Plurals 208 8.6 Conclusion 211 9 Quantity and Number 9.1 Ontological Relativity 9.2 Structural Relativity in the Notion of Number

213 213 221

10 The French Revolution — a Philosophical Event? 10.1 Introduction 10.2 The Structure of Events 10.2.1 Events in Time and Space 10.2.2 Roles 10.2.3 The Structure of Complex Events 10.3 Uniformities: Types of Events 10.4 Algebraic Semantics for Events 10.4.1 Modeling 10.4.2 The Aether Model

231 231 232 232 237 240 241 244 244 245

11 Algebraic Semantics of Event Structures 11.1 Introduction: The Project of Algebraic Semantics 11.2 The Model Structure 11.3 Translation and Truth 11.4 Examples

251 251 257 261 264

Vlll

12 The 12.1 12.2 12.3

Ontology of Individuals and Events General Remarks Metaphysical Methodology Individuals 12.3.1 Constitution 12.3.2 Temporal Parts 12.3.3 Processes 12.3.4 The Notion of Individual 12.4 Events 12.4.1 The Classified Process View 12.4.2 Formal Properties of Events 12.4.3 Questions about Events

269 269 273 277 279 280 284 287 294 297 301 305

13 The 13.1 13.2 13.3 13.4 13.5

Philosophy of Plurality 311 Introduction 311 Ontological Commitment 312 The Counting Fallacy 318 Is There a Problem with Denotational Plural Semantics? . . 321 Cross-linguistic Evidence 326

14 Mereology, Second-Order Logic, and Set Theory 331 14.1 Introduction 331 14.2 A mereological interpretation of monadic second-order logic 332 14.3 LP as a framework for mereological set theory 344 Appendix: A Chapter in Lattice Theory 15.1 Ordering Relations; Preordered Sets; Posets 15.2 Semilattices; Lattices 15.3 Boolean Lattices 15.4 Plural Lattices

353 353 359 370 375

Bibliography

383

Subject Index

417

Name Index

429

Preface This volume contains a collection of the author's papers on the algebraic approach to the characterisation of objects in language and philosophy. The common philosophical perspective of the essays can be described as follows. Language is able to refer to a wide array of objects of different sorts, which differ from each other in their characteristic structural properties. The universe of linguistic or philosophical discourse is thus most naturally taken as a multi-sorted domain containing all those objects. However, the traditional set-up of Tarskian semantics, where the domain of individuals is just a non-empty set, does not per se represent the various relations that connect these kinds of object both internally and across different kinds. Algebraic semantics, then, deals with structured domains instead of flat ones: various algebraic relations model the structural properties of the entities in the domain. The kinds of object that are investigated systematically in the essays are concrete particulars such as plural entities, mass objects, and events. However, the algebraic approach naturally applies also to other philosophical entities, like propositions, properties, relations, or situations. Thus, the volume complements algebraic work that has been done on the latter kinds of object. The essays give a detailed motivation, both linguistic and philosophical, of the need to introduce concrete particulars of the above kind into the domain, and describe the structural properties that govern their behavior. A first-order logical framework is developed in which it is possible to quantify over these objects. Its usefulness in providing a compositional semantics for them is demonstrated. The framework exhibits most clearly the structural similarity between plurals and mass terms. Also, the interplay between plural objects and (plural) events is discussed. The part of the theory which is developed most in the volume is the Logic of Plurality, LP. LP is a first-order mereology which can be-viewed as a nominalistic version of monadic second-order logic. Its philosophical significance is explored in the last two essays. IX

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PREFACE

Many chapters deal with applications in linguistic semantics. The main topics, which are treated extensively, are (i) plural determiners and plural quantification, and (ii) collective, distributive, and intermediate predication. Three essays are concerned with the notion of event, both from a linguistic and a philosophical point of view. In particular, a separate chapter is devoted to the outline of a unified ontology of individuals and events. The 14 essays have been written over a period of 15 years. Seven of them have already been published, but at widely different places with varying accessibility, and one of them in German. Inclusion in this volume should render them readily accessible now. Another chapter appeared before as preprint in a conference proceedings. The remaining chapters contain both unpublished material and recent new work that has been written for this book. Finally, a technical Appendix provides the necessary concepts from Lattice Theory that are used throughout the book. The published essays has been left unchanged, except for minor corrections. Here is a list of the original sources. Chapter 1: 'The logical analysis of plurals and mass terms: A lattice-theoretical approach', in: Meaning, Use, and Interpretation of Language, R. Bauerle et al. (eds.), de Gruyter, Berlin 1983, 302-323, reprinted with kind permission from Walter de Gruyter & Co.; Chapter 2: 'Plural', in: Semantik. Em Internationales Handbuch der zeitgenossischen Forschung, A. von Stechow and D. Wunderlich (eds.), de Gruyter, Berlin 1991, 418-440, English translation by the author of the German original, reprinted with kind permission from Walter de Gruyter & Co.; Chapter 3: 'Hydras: On the logic of relative constructions with multiple heads', in: Varieties of Formal Semantics. Proceedings of the Fourth Amsterdam Colloquium, F Landman and F. Veltman (eds.), Foris, Dordrecht 1984, 245-257, reprinted with kind permission from Walter de Gruyter & Co.; Chapter 4: 'Generalized quantifiers and plurals', with 1 figure, in: Generalized Quantifiers. Linguistic and Logical Approaches, P. Gardenfors (ed.), Reidel, Dordrecht 1987, 151-180, reprinted with kind permission from Kluwer Academic Publishers; Chapter 7: 'Ten years of research on plurals — Where do we stand?', in: Plurality and Quantification, F. Hamm and E. Hinrichs (eds.), Kluwer, Dordrecht 1997, 19-54, reprinted with kind permission from Kluwer Academic Publishers; Chapter 8: 'Algebraic semantics for natural language: some philosophy, some applications', in: International Journal of Human-Computer Studies, 43, 1995, 765-784, reprinted with kind permission from Academic Press, Ltd., London; Chapter 9: 'Quantity and number', in: Semantic Umversals and Universal Semantics, D. Zaefferer (ed.), Foris, Dordrecht 1991, 133-149, reprinted with kind permission from Walter de Gruyter & Co.

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xi

Chapter 11, 'Algebraic semantics of event structures' appeared in the preprint volume of the Proceedings of the Sixth Amsterdam Colloquium, J. Groenendijk et al. (eds.), ITLI, Amsterdam 1987, 243-262. For compatibility of reference, it has also been left unchanged. Chapter 2 was actually written in 1984, and had been around as an underground for some time. Chapter 5 dates back to 1986, and Chapter 10 to 1988. A first version of Chapter 6, together with the Appendix, was used as material for a joint course (with Jan Tore L0nning) on Algebraic Semantics at the Third European Summer School in Language, Logic and Information, Saarbriicken, Germany, in August 1991. The chapter has been considerably reworked since then. Chapter 12 is the most recent one and was written a few weeks ago. The first part of Chapter 13 is new, too, while the second part goes back to 1994. Finally, the content of Chapter 14 was presented at a symposium on second-order logic and plural quantification at the Annual Meeting of the American Philosophical Association (APA), Pacific Division, in April 1996. Since the single chapters originated as independent papers there is a certain amount of inevitable overlap, which is true in particular of the exposition of the system LP of plural logic. I apologize for that; the reader is asked to skip repetitions as soon as he or she has picked up the sometimes varying logical notation that is used in a given chapter. However, the advantage is that the chapters can to a large degree be read separately. The Bibliography has become rather long. There is a general and a specific reason for this. The general reason is that I had to include references from three fields: linguistics, logic, and philosophy; the specific reason is that the Bibliography contains all titles both from the survey article in the Handbook of Semantics (Chapter 2) and from a list of titles that was compiled for the European Summer School mentioned before. Also, I have at occasion included titles in the Bibliography that are actually not cited in the text; so the reader might want to look through it for additional relevant literature. Since it is my conviction that research is basically a collective enterprise it is only natural for me to acknowledge my intellectual debt to the many colleagues, students, and friends who have helped me over the years to become clearer about the ideas developed in this book. Many of their names are listed in the acknowledgements to some of the individual chapters (Chapters 1, 4, 9, 11, 14 and the Appendix). In addition to and partial repetition of those individuals let me mention here Emmon Bach, John Barwise, Johan van Benthem, Ulrich Blau, George Boolos (f), John Burgess, Jaap van der Does, Marcel Erne, Claudia Gerstner-Link, Volker Halbach, Fritz Hamm, Stefan Iwan, Hans Kamp, Ed Keenan, Manfred Krifka, Fred

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Landman, Hans Leifi, Jan Tore L0nning, Alice ter Meulen, Karl-Georg Niebergall, Barbara Partee, Michael Resnik, Mats Rooth, Remko Scha, Roger Schwarzschild, Hinrich Schiitze, Peter Sells, Peter Simons, A. von Stechow, Wolfgang Stegmiiller (f), Holger Sturm, Pat Suppes, Matthias Varga von Kibed, Dietmar Zaefferer, and Ed Zalta. I wish to thank the Deutsche Forschungsgemeinschaft, Bonn, Germany, for repeatedly sponsoring my research, most recently with a three months' grant for my stay at the Center for the Study of Language and Information (CSLI), Stanford University, from August through October, 1996. I had intended to finish the book by then, but it took me another year to be finally done with it. Due to the generous hospitality of the CSLI I was able to return this year, again for late summer and early fall, and to finish my work. I wish to express my sincere thanks to the staff of the CSLI, in particular, Michele King, and its director, John Perry, for granting me the privilege to be a frequent visitor there. During my last two visits at the CSLI I had the pleasure to be a member of the Metaphysics Research Lab run by my friend Ed Zalta. I wish to thank him for being my continued discussion partner in philosophical and logical matters, for freely giving his time to help me with the intricacies of the NeXT workstation, and for having been such an enjoyable host. Writing this book was an adventure for me in a particular respect: this is my first book that I wrote in English. At occasion I felt the pain of not being able to express myself in my mother tongue; but on the whole I hope that I managed to produce a text that doesn't sound outlandish at every turn. There are two chapters (10 and 12) which I felt I wanted to have checked by a native speaker. I wish to thank Peter Sells and Ed Zalta for carefully going through the material and suggesting stylistic improvements. Many people have also provided technical advice and assistance in the course of the production of the book. I'd like to thank them all, in particular: Cleo Condoravdi for compiling a large part of the bibliography, CSLI's systems administrator Emma Pease for general computational support, and Tony Gee of CSLI Publications for the production of the figures in Chapter 2 as well as the splendid realization of the cover idea. Finally I wish to thank the editor of CSLI Publications, Dikran Karagueuzian, for his enthusiastic support and continued encouragement. Last but not least, my special thanks go to my wife Claudia for all her love and support. Stanford, California October, 1997

G.L.

ALGEBRAIC IN PHILOSOPHY

Introduction If you walk up the steep path to the Acropolis in Athens and finally pass through the Propylaea you are not only struck by the majestic view of the Parthenon temple but also by the unrivaled beauty of the famous Caryatid porch of the Erechtheum. Those six Caryatids, also called korai, are draped female figures which are used as columns to support the roof of the porch. They seem to carry their load through the ages, in classical contrapposto and complete serenity. The caryatids exemplify a collective property. A single one of them wouldn't be able to uphold the structure. There are two theoretical questions connected with this, one linguistic and one philosophical: First, how is collective predication to be treated in the semantic analysis of natural language, and second, is collective exemplification of properties a phenomenon that survives the logical-philosophical treatment of regimentation which natural language locutions are commonly submitted to? Are there "plural objects"? To begin with the latter, imagine that, contrary to fact, the roof of the porch of the Erechtheum temple had originally been supported by a regular brick wall. Then the property of supporting the roof, call it P, would have been had by a single entity, the wall, which is as concrete an object as anything could be. Now subsequent architects found the structure rather unappealing and clumsy, so they took out more and more parts of the wall, first putting in windows, then creating more open space, and eventually ending up with replacing the remaining bricks with beautiful Ionian columns and finally with the caryatid figures. What exemplifies the property P now? According to some philosophers and linguists, it is now the set of those figures, which is an abstract entity. This is the set account of pluralities. The question is at which point in the history of the building the concrete thing turned into an abstract object. It has been clear to me for a long time that the set account just couldn't be right. It is true that for a few decades now philosophers have used set theory as a modeling tool in their theorizing, but as a conceptual instrument

2

INTRODUCTION

in the history of ideas set theory is a real late-comer. Philosophers had typically been concerned with concepts rather than sets. While the notion of an object falling under a concept or property has always been in use, the proper abstractions necessary to arrive at a viable notion of extension (let alone one that could be used as a point of departure for the modern iterative notion of set) was hardly arrived at before Frege. It seems natural then to ask how pluralities had been conceptualized in the past, and how they should be treated today in a way that regains the pre-set-theoretic innocence. When we look at the rules of classical syllogistics we see that they constituted some kind of relational calculus. Much later it was Leibniz who formalized the relation of subsumption in such a way as to result in a regular algebraic framework.1 Common to all those approaches is the notion of part. It is also basic in dealing with concrete particulars of our everyday world, an obvious fact that is reflected in the all-pervasive part locution in language speaking about things and their location in space and time. One of my claims in this book is that the part relation is also the proper tool for the analysis of pluralities. That involves the decision to treat pluralities together with particulars rather than with universals. The latter option would entail the assimilation of pluralities to properties and hence, in the language of set theory, to sets. By contrast, allowing "singular" objects being parts rather than elements or members of pluralities makes it possible to avoid the mysterious transition from the concrete to the abstract which was highlighted by our example above. Thus pluralities are viewed as objects of the same kind as individual things. The ontological claim here is what could be called "relative nominalism:" No matter what kind of entity you take to constitute your basic individuals (concrete things, properties, sets, events, qualia, tropes, or what have you) no extra commitment will be produced when admitting pluralities of those individuals in your ontology. Also, under such a view, an important parallelism between plural objects and mass-like objects can be perceived that will otherwise go unnoticed. A typical principle exhibiting this parallelism is the cumulative reference property that will be dealt with in the beginning chapters of the book. The notion of part is of course highly ambiguous. I would like to distinguish between a rich and an austere notion. The rich notion is typically paraphrased by the term "constituent" hinting at the fact that parts in this sense are elements in a complex whole which is organized by a number of extra-logical relations. The austere version is the logical notion of part; its meaning doesn't go beyond what can be expressed in purely logical terms, so it doesn't involve extra-logical relations. It is this latter notion of part that I will be mostly concerned with in the J

On Leibniz see, e.g., Lenzen (1990); Swoyer (1994); Zalta (1997a).

INTRODUCTION

3

book. It lends itself to a mathematical treatment as an algebraic relation which obeys certain axioms. This is the source of the algebraic outlook underlying the present work. Since algebra is the general theory of structures, the algebraic perspective also entails a kind of structuralist attitude towards reality as far as metaphysics is concerned. It doesn't deny that rich and highly sophisticated empirical knowledge is available about our external world. But that is just not part of metaphysics or ontology. Philosophers should confine themselves to a formal view of metaphysics. This theme is taken up in Chapter 12. There are several places in the book where I try to explain the algebraic approach; see in particular Sections 1.1, 2.3, 11.1, and Chapter 8. This book doesn't consist, however, of a number of purely philosophical reflections about topics of the kind just alluded to; quite the opposite is true. In fact, having arrived at a certain philosophical view concerning them I found it necessary to foster it by some serious research into the linguistic mechanisms of reference to individuals of various sorts, in particular, the mechanisms of plural reference. I also collected some cross-linguistic evidence that I thought were supportive of my view. To bear out the structural perspective I also made extensive use of all the formal tools from logic and mathematics that seemed to me relevant to my task. In the final chapters of the book I return to the philosophical issues involving problems of metaphysics, ontology, and the philosophy of mathematics. I will now describe the content of the individual chapters, with some comments on their interrelations and, regarding the published essays, their significance in the light of current research. Chapter 1 contains the my first account of the algebraic research program in linguistic semantics and philosophy. It develops a unified logic of plural and mass expressions, LPM, stressing the structural similarity between plural denotation and homogeneous mass denotation in terms of concepts from lattice theory. In its philosophical part it argues for what I here called relative nominalism with respect to pluralities; it also contains a contribution to the ontology of concrete objects, by specifying a relation of constitution that admits of distinguishing a physical object from the portion of matter making it up. That involves a rejection of the ontological principle of No Coincidence, viz. that no two objects can be at the same place at the samatime. The intuition that the stuff and the object it composes are really one thing is accounted for by introducing a homomorphic relationship between the

4

INTRODUCTION

algebraic structures of plural individuals and portions of matter which allows for "ignoring" information in so-called invariant contexts. There were some problems with this account that are discussed in Chapter 12, where the issue of constitution is taken up again in a purely ontological setting. On the linguistic side, the chapter is written in the spirit of compositional semantics which was inherited from the Montagovian paradigm. In fact, it constitutes the attempt to save the principle of compositionality in the light of recalcitrant data involving plural constructions. Thus from a position of "pluralist minimalism" one could say that the theory is just a technical tool for securing compositionality. A central idea in this context was that of a compositional plural operator which later led to the introduction of other operators on verb phrases, in particular, the distributivity operator (see below). The semantics that was given for the logic LPM contains a complete atomic Boolean algebra (with its zero element removed) as modeling the structure of pluralities. CAB algebras are known to be the models of Classical Mereology, so there is an obvious connection to the mereological theories in philosophy that were used in connection with the modern nominalist program, the outlook of which was quite different, though. The formation of pluralities involves taking joins of individuals only, so I found it more accessible to isolate this feature in later work in terms of the notion of semilattice. Such a semilattice, however, has to obey certain additional principles in order to conform with our intuitions about pluralities. It turns out that when these principles are specified we essentially come back to the CAB algebras. The logical issues involved here are described at length in Chapter 6. Chapter 2 is a handbook article which tries to systematize what had been known about plural semantics at the time the article was written. Naturally, it heavily draws on the material presented in Chapter 1, leaving out, however, the mass term part of the system LPM; the remaining logic was henceforth called LP. A wide array of plural constructions is described in the chapter, which are eventually given explicit logical forms in the first-order framework of LP. The focus is on the various readings a plural sentence can get. In particular, there is the example of a construction involving two plural noun phrases, for which I gave seven different readings generated from scope ambiguity and the collective/distributive distinction. Distributive readings are marked in their logical form with the distributivity operator, D, which appeared here for the first time. The analysis of these examples led to a number of interesting developments later on, in particular, to the discovery of so-called partitional readings and their interaction with the monotonicity properties of various determiners; a discussion of these issues can be found in Chapter 7.

INTRODUCTION

5

In Chapter 3 a special problem for the compositional analysis is addressed which concerns certain relative clause constructions with more than one head noun and with a collective predicate in the relative clause. I call those constructions "hydras." It is shown in the chapter how these cases can be given a satisfactory treatment in the framework of plural logic. Chapter 4 is one of the first papers in which plurality is treated in the framework of generalized quantifier theory. It is noticed there that in order to incorporate collective predication a new kind of filter construction is needed for the denotation of conjoined noun phrases like John and Mary. Instead of taking the usual intersection filter which actually represents the properties that are common to John and Mary individually the new filter is the (principal) filter containing all collective properties of the plurality John ® Mary. Regarding the main theme of quantification, a distinction is made between spurious and genuine plural quantification. A construction is genuinely plural only when the quantifier ranges over pluralities and not just singular objects. An asymmetry is noticed between existential and universal quantification: while the former is quite common, the latter doesn't seem to be easily expressible in language. The chapter also contains an adjectival analysis of numerals which is different in conception from the generalized quantifier view where numerals are treated as determiners. Finally, the phenomenon of "floated quantifiers" is addressed and analyzed in terms of the distributivity operator. Chapter 5 is a case study of the German je-locution which acts like a distributivity operator. What is interesting here is that the quantificational element is not the determiner of the noun phrase specifying the quantifier scope but rather a particle attached to the "dependent" noun phrase. This device is called "anti-quantifier" by Choe (1987a) who discovered a similar phenomenon in Korean. The German data exhibit rather involved quantificational structures which can be expressed quite effortlessly in the language by using the je particle. Their analysis is given in terms of the D-operator. Chapter 6 focuses on the purely logical properties of the theory of plurals, LP. A "genetic" approach is chosen in which a number of plausible axioms for pluralities are introduced and motivated in turn. First, the logical basis of LP is presented explicitly; it is a first-order free logic with abstraction and description terms. Then the axioms for characterizing pluralities are given; the resulting structures are called plural lattices. Apart from the axioms for the general notion of semilattice, there are four_more axioms that seem to be specific to our conception of pluralities: (i) an axiom which excludes bottom elements (there is no "null entity"); (ii) an axiom called atomic separation which says that any two incomparable elements

6

INTRODUCTION

are "separated" by an atom, thereby making sure that every element has an atom below it; (iii) an axiom of definable completeness which guarantees a supremum for every non-empty set of individuals which can be defined in the theory; and finally (iv) an axiom which expresses the property that the semilattice is "maximally unfolded." A number of properties of the resulting system is discussed, and a comparison is made with a similar system of J.T. L0nning's. Furthermore, a connection is established with the logical and philosophical literature on mereology by discussing the relationship between the confusingly large number of axioms that can be found in various accounts. Finally, some important metalogical issues of the system are discussed. It is shown in particular that LP doesn't admit of a complete axiomatization with respect to full standard models. That means that LP, albeit formulated in a first-order language, is really second-order in nature. This will become important in the last two chapters of the book. The proofs to a number of theorems are deferred to the Appendix. Chapter 7 contains a survey of the semantic literature on plurals in the 1980s and early 1090s. From early on there was one problem with the adequacy of the plural structures of LP concerning the so-called intermediate level readings. A sentence like The Leitches and the Latches met in the park is ambiguous because it could mean either that the two families came together for a common activity or else that each family met separately in a place that happened to be the same park. Now the first, collective, reading can be expressed in LP whereas for the second there doesn't seem to be a device available because in a plurality there are no discernible subcollections "half-way down" towards the atoms. In order to cope with this problem Landman (1989) proposed a certain conception of "group formation" which in effect amounted to a usual set-theoretic solution. In this chapter I give reasons against adopting such a treatment. The other main issue that is dealt with in the chapter is a discussion of readings for plural sentences, in particular the so-called partitional readings. The discussion focuses on interesting work by J. van der Does, in particular his (1992), where a number of new verb phrase operators are introduced. Since the analysis is carried out in the theory of generalized quantifiers, the interaction of plurality with determiners of varying monotonicity types is discussed. I give a representation of generalized quantifiers in the algebraic framework and defend an earlier analysis of mine that had been contested by van der Does. Chapter 8 was written for a conference on the role of formal ontology in conceptual analysis and knowledge representation, the audience being mainly researchers working in artificial intelligence and knowledge representation. I took the opportunity to present the algebraic approach to se-

INTRODUCTION

7

mantles and ontology and to give a number of applications that have been dealt with elsewhere in the book in one form or another. Thus the chapter can serve as a kind of introduction to the ideas underlying the approach. There is also a short section on the ontology of plurals that was prompted by the publication of David Lewis's book Parts of Classes, which proved highly relevant to my work. Lewis's theory and the nominalistic interpretation of second-order logic underlying it, which is due to Boolos, are discussed at greater length in Chapter 14. Chapter 9 originated in a paper prepared for quite a different conference in 1988 which brought together logicians, semanticists and universal linguists to discuss issues of the universality of semantic concepts across languages. The present text focuses on ways in which the concept of quantity is realized in various languages, and on different modes of linguistic reference to concrete particulars. The following three chapters deal with the notion of event and its characterization in the algebraic framework. Chapter 10 gives a gentle introduction to the event conception whose technical semantic development is contained in Chapter 11. Its individual components are motivated in terms of examples drawn from a single and highly complex historical event, the French Revolution. The central feature is an algebraically structured domain of events in which individuals can play various roles, both severally and collectively; in this way the collective/distributive distinction can be reproduced in event theory by the contrast between a single event involving a collective agent and a sum of events involving atomic agents only. Apart from particular events, which can occur only once, also repeatable events are considered under the name event type. Event types are conceived here not as extensional classes of particular events, but as fine-grained intensional entities. The various components constitute what I call, for rather contingent reasons, the Aether model. The technical details of this event conception are provided in Chapter 11. This chapter constitutes a variation on two themes that were prominent at the time the text was written, Situation Theory and Discourse Representation Theory (DRT). A notion of truth is defined for an event type in a given concrete but complex event which is called a chunk of the world. Then a translation procedure is sketched leading from natural language sentences to event types; this is done in a variable-free way by using "dependence levels" rather than DRT-style accessibility information. Instead of specifying a set of rules I illustrate the procedure with a number of examples where translations are given and evaluated semantically." I realize that the paper reproduced in this chapter is unsatisfactory in various ways. I have included the text unaltered here since it keeps being

8

INTRODUCTION

referred to on occasion. I would like to point out, however, that I no longer defend the details of the interplay between events and their types as they were developed there although I still believe that the overall picture has some merit. The amendments and additions that I made are laid out, among other things, in Chapter 12. Chapter 12 offers a unified philosophical account of the notion of concrete particular which draws on several ideas from previous chapters of the book. These entities come in two major varieties, as individuals and as events. Although in general not especially fond of reductionism I found it useful to view these two kinds of object as being special manifestations of just one underlying kind of entity, processes. Individuals are what I call stationary processes, whereas events are classified processes. Regarding the former, this view entails that individuals have temporal parts, and indeed I give reasons for embracing this much debated consequence. Also, I give a modified account of the relation of constitution that was first introduced in Chapter 1, in a way that criticisms of the earlier version are met. Concerning events, I try to argue that events are both coarse-grained and fine-grained depending on the context in which reference is made to them. Under the coarse-grained view, two events can be identified as having the same underlying process. The fine-grained view is similar to Kim's notion of event in that it involves an event type which corresponds to a property in Kim's theory. The transition from the fine-grained to the coarse-grained view is effected by "leaving out" information about the event which is irrelevant in the given context. The algebraic method is again applied throughout the chapter. A list with a number of commonly asked questions and puzzles about events, complete with detailed answers from the present perspective, concludes the chapter. The chapter begins, however, with a reflection on philosophical methodology. In line with the general outlook of the book I argue for a completely formal conception of metaphysics which is concerned only with our most general conceptual tools that underlie the very structure of our thought. In the opening section I name a few of those tools and shortly comment on them in turn. Among them I count mereology which is the central theme in this book. While the previous chapter was concerned with basic particulars and neglected the superstructure of pluralities, Chapter 13 deals with some important philosophical problems of the concept of plural object. First the problem of ontological commitment is addressed. Penelope Maddy's view of concrete sets is discussed, and also the question of ontological innocence of forming mereological fusions. Regarding the free-lattice fusions that constitute pluralities I agree with David Armstrong that they are an on-

INTRODUCTION

9

tological "free lunch," but doubts are raised against putting into the same category what I call substantive fusions (e.g., fusions of regions of space that describe our notion of physical space). The next section addresses the counting fallacy according to which what are two cannot be one. The counting fallacy has been a major obstacle for gaining a clear conception about the nature of pluralities. In another section the claim of an alleged inconsistency of the denotational view underlying plural theory, put forth by Barry Schein in a recent book, is refuted. The chapter concludes with a short discussion of cross-linguistic data from Chinese giving evidence for the fundamental nature of the part locution in language and hence, I submit, human thought. The final Chapter 14 deals with a topic in the philosophy of mathematics that has recently drawn wider attention, viz. the nominalistic interpretation of monadic second-order logic in terms of plural quantification. This interpretation was first advanced by George Boolos but was put to use by David Lewis (1991) in his reconstruction of classical set theory from purely mereological principles, which he calls megethology. The chapter first discusses Boolos's views, together with criticisms that have been advanced against them by Michael Resnik. It is shown that those criticisms can be answered by offering the concept of plural quantification available in plural logic. Drawing on the results from the discussion of the counting fallacy, and clearing up a point in Lewis about the supposed untenability of "singularism," an argument is given to the effect that the denotational view of pluralities is compatible with Boolos's position, and that it is the logic of plurality that is missing as a formal tool from Lewis's account. In support of this claim Lewis's concepts and principles of megethology are formalized in LP, showing that the usual standards of formal rigor can also be applied to the phenomenon of plural quantification, contrary to what seems to be suggested in Lewis's book. The Appendix contains in a dense form elementary notions from lattice theory, such as relations, partially ordered sets, suprema and infima, semilattices and lattices, homomorphisms, and Boolean algebras. In a second part the structure of the plural lattice as it was defined in Chapter 6 is further explored. A proof is provided for the basic representation theorem of plural lattices according to which a plural lattice has as standard model the powerset algebra over the set of individual atoms minus the empty set (the result is well-known). Also, the property of being the most general lattice that can be spanned over a given set of atoms is characterized by a certain universal property in the sense of category theory, the extension property. Its equivalence with the axiom of "sup-prime atomicity," which is one of the plural lattice axioms, is proved in detail.

Chapter 1

The Logical Analysis of Plurals and Mass Terms: A Lattice-theoretical Approach 1.1

Introduction

The weekly Magazine of the German newspaper Frankfurter Allgemeine Zeitung regularly issues Marcel Proust's famous questionnaire which is answered each time by a different personality of West German public life. One of those recently questioned was Rudolf Augstein, editor of Der Spiegel; his reply to the question: "Which property do you appreciate most with your friends?" was (1)

"that they are few."

Clearly, this is not a property of any one of Augstein's friends; yet, even apart from the esprit it was designed to display the answer has a straightforward interpretation. The phrase (1) predicates something collectively of a group of objects, here: Augstein's friends. As it is well known, collective predication is a rather pervasive phenomenon in natural language, as the following sample of sentences shows: (2)

The children built the raft.

(3)

The Romans built the bridge. 11

12

PLURALS AND MASS TERMS

(4)

Tom, Dick, and Harry carried the piano upstairs.1

(5)

The playing cards are scattered all over the floor.

(6)

The members of the committee will come together today.

(7)

Mary and Sue are room-mates.

(8)

The girls hated each other.

There is a striking similarity between collective predication and predication involving mass nouns. (9)

a.

The children gather around their teacher,

b.

The water gathers in big pools.

Moreover, a characteristic feature of mass terms, their cumulative reference property2 can be imitated by plurals. (10)

a.

If a is water and b is water then the sum of a and b is water.

b.

If the animals in this camp are horses, and the animals in that camp are horses, then the animals in both camps are horses.

All this has been observed and discussed in the literature although the noted parallelism has perhaps not been stressed too much.3 As it can be seen from Pelletier's 1979 volume, however, there is much disagreement about the proper way of attacking the logical problems posed by plurals and mass terms. Prom a semantic point of view the basic question is: what do mass terms and plural expressions denote? Some have thought that in order to be able to give a satisfactory answer to this question it is necessary to give up or at last extend the underlying set theory and to define new kinds of objects, for instance ensembles (Bunt 1979) or collections (Blau (1981b)). I think, however, that we can retain the usual set-theoretic metalanguage and simply enrich the structure of our models as to account for properties like cumulative reference. On my view, such properties are also not secured by defining some plural or mass term denotations out of others through set-theoretic manipulations; they all should be recognized as 1

Massey's example; see Massey 1976. See Quine 1960, p. 91; Bunt 1979. The main source for mass terms is Pelletier 1979; furthermore see Bennett 1979, Bunt 1979, Ter Meulen 1980, 1981. For the treatment of plurals and collective terms, see Massey 1976, Burge 1977, Blau 1981b, Hoepelman/Rohrer 1980, Scha 1981. The parallelism referred to is explicitly expressed in Bunt 1979. 2

3

INTRODUCTION

13

simply being there. What we rather should try to discover, then, is the network of the various relations which they enter and through which they are tied together. In the case of group and mass objects this picture naturally leads to the notion of lattice structure,4 an idea which is, again, not new: it is inherent in mereological predicate logic and the Calculus of Individuals as developed by Leonard/Goodman 1940 and Goodman/Quine 1947. However, its possible use in the present context has perhaps been obscured by reductionist ontological considerations which are, in my opinion, quite alien to the purpose of logically analyzing the inference structures of natural language.5 Our guide in ontological matters has to be language itself, it seems to me. So if we have, for instance, two expressions a and b that refer to entities occupying the same place at the same time but have different sets of predicates applying to them, then the entities referred to are simply not the same. From this it follows that my ring and the gold making up my ring are different entities; they are, however, connected by what I shall call the constitution relation: There is exactly one portion of matter making up my ring at a time. A constitution relation C has been explicitly introduced into the discussion by Parsons 1979. Sharing his intuitions on this point I shall provide a similar 2-place relation, '>', for constitutes or makes up. Its semantic counterpart in the theory to be presented below, the "materialization" function h, lies at the heart of my reconstruction of the ontology of plurals and mass terms: individuals are created by linguistic expressions involving different structures even if the portion of matter making them up is the same. Consider the example from Blau (1981b) (imagine that there is a deck of playing cards on the table): (11)

a. b.

the cards the deck of cards

While the portions of matter denoted by (lla) and (lib) are the same, I consider the individuals as being distinct.6 (lla) refers to the pure collection of objects, and in many contexts (lib), too, refers just to this collection. In general, however, the introduction of a collective term like (lib) is 4 Recently, Keenan and Faltz 1985 and Keenan 1981 advanced a "Boolean approach" to the semantics of natural language. I feel very sympathetic with this enterprise which, unfortunately, I became aware of only a year ago (January 1981). It is reassuring to see similar techniques be successfully applied in other areas of semantics, too. I have to refer a concrete evaluation of these ideas to another occasion. 5 On this point I agree with Ter Meulen 1981, I think. But I do not follow her in the conclusions she draws from this observation. The inherent lattice structure is independent of the philosophical motives that gave rise to the construction of mereological systems. For the role of nominal mass noun denotations see the remarks below, also footnote 11. 6 I guess that in German, with die Karten vs. das Kartenspiel, the point might come out more clearly.

14

PLURALS AND MASS TERMS

indicative of connotations being added enough for it to refer to a different individual; for instance, a committee is not just the collection of its members, etc. Note, by the way, that the transition to an intension function would be of no help here. There might be two different committees which necessarily consist of exactly the same members.7 It might be thought, then, that collective predication is just the context in which pairs of expressions like (lla,b) refer to collections and thus are coreferential. This is not so, however, as can be seen from the following example. Imagine that there are several decks of cards, a blue one, a green one, etc., lying on a table.8 Then the two following sentences do not mean the same although number consecutively is a collective predicate (the German word is durchnumerieren). (12)

a.

The cards on the table are numbered consecutively.

b.

The decks of cards on the table are numbered consecutively.

By contrast, spatio-temporal collective predicates do refer to the pure collection, or, as I conceive it, the portion of matter making up the individual in question. Examples are be-on-the-table, occupy, etc. I call such predicates invariant. So the following a) sentences are indeed equivalent to their corresponding b) sentences. (13)

(14)

a.

The cards are on the table.

b.

The decks of cards are on the table.

a.

The stars that presently make up the Pleiades galactic cluster occupy an area that measures 700 cubic light years.9

b.

The Pleiades galactic cluster occupies an area etc.

In the following I shall distinguish between pure plural individuals involved in (12) and collections in the portions-of-matter sense referred to in (13) and (14). The former I call (individual) sums or plural objects; they respect levels of "linguistic comprehension" as shown by (12). By contrast, collections do not, they typically merge those levels. Sums and collections are similar, however, in that they both are just individuals, as concrete as the individuals which serve to define them, and of the same logical type as these. The latter feature is important because there is no systematic type ambiguity inherent in predicates like carry, build, demolish, defend, etc. As to the question of concreteness of sums of concrete objects, I agree with 7

The point was apparently made first by David Kaplan as Bennett 1979 reports. This is the situation originally analyzed by Blau in his 1981b paper. 9 This is Surge's example, see Burge 1977. 8

I

INTRODUCTION

15

the intuitions of those who say that an aggregate of objects like a heap of playing cards which can be shuffled, burned, etc , is simply not an abstract entity like a set (see Burge 1977, Blau 1981b) What is more important, however, is the fact that the set approach to plural objects10 does not carry over to the case of mass terms, thus missing the structural analogy between the two cases Inherent in the notions of a set is atomicity which is not present in the linguistic behavior of mass terms Before I go on to present my own approach I want to say something about what Ter Meulen calls nominal mass terms Typical examples are stuff names like gold m sentences like gold has the atomic number 79 But there is also the time-honored sentence water is widespread in which the term water has apparently a somewhat different status It seems to refer to the concrete "scattered individual" that you just find everywhere, hence Quine's analysis in terms of mereology In this sense the sentence should be synonymous to the water (on earth) is widespread Of the same type is the use of gold in America's gold is stored in Fort Knox Here, again, a concrete object is referred to by America's gold, namely the material fusion of all quantities of the US gold reserve So there can be no doubt that some notion of fusion is needed to account for definite descriptions involving predicative mass terms (the /i-operator defined below does just this) Genuine stuff names, however, are something else Substances are abstract entities and cannot be defined in terms of their concrete manifestations The question, then, is of what kind the connections are that are intuitively felt between substances and their quantities Take water, for instance A quantity is water if it displays the internal structure of water, that is t^O But this relation is not a logical one Or else we might look for substance properties which carry over to the quantities of the substance in question Water is a liquid and yet, all concrete water might be frozen So we have to go over to dispositional properties, getting more and more involved into our knowledge of the physical world What I am getting at is that nominal mass terms do not seem to have a proper logic Be this as it may, this issue is completely independent of the lattice structure that governs the behavior of predicative mass terms and plural expressions, it is only this structure that I want address myself to in the present chapter u 10

This approach is the traditional one In one form or another it can be found, for instance, in Bennett 1975 Hausser 1974 v Stechow 1980 Hoepelman/Rohrer 1980 Scha 1981 11 Contributions to the problem of substance names can be found in Pelletier 1979 (in particular, Parsons 1979), Bunt 1979 and Ter Meulen 1980 1981 Let me comment on the latter work, which is formulated in a Montague framework The few remarks I made here will make it evident that I fully agree with ter Meulen in that nominal mass nouns cannot be reduced to predicatne mass nouns But for this very reason I fail to see any cogent argument for the kind of denotation ter Meulen wants to assign to these

16

PLURALS AND MASS TERMS

Now what I am going to propose, then, is basically the following. First of all, let us take seriously the morphological change in pluralization, which is present in many natural languages, and introduce an operator, '*,' working on 1-place predicates P, which generates all the individual sums of members of the extensions of P. Such a starred predicate now has the same cumulative reference property as a mass predicate, it is closed under sum formation: any sum of parts which are *P is again *P. This property gives rise to the introduction of a Boolean structure on the domain of discourse, E: technically, E becomes an atomic Boolean algebra which is taken to be complete so that every subset of E possesses a sum. Now let || • || be the denotation function in a model, and ||P|| the extension of P. Then ||*P||, the extension of *P, can be defined in terms of ||P|| as the complete join-subsemilattice12 in E generated by ||P||. This construction is the mathematical expression of the closure property referred to above. The set A of atoms of E consists of the "singular objects," like this card, that deck of cards, etc. Among them are all the different portions of matter, like the water making up this ice cube. Now, let a and 6 denote two atoms in A. Then there are two more individuals to be called below a + b and a ® b. a + b is still a singular object in A, the material fusion of a and b; a © b is the individual sum or plural object of a and b. The theory is such that a + b constitutes, but is not identical with, a ® 6. This looks like a wild platonistic caprice strongly calling for Occam's Razor. Language, however, seems to function that way. Take for a, b two rings recently made out of some old Egyptian gold. Then the rings, a($b, are new, the stuff, a + b, is old. terms at a given reference point (i.e., intension functions denoting in each world the set of concrete quantities of the substance in question). As it turns out, the arguments she puts forth in Ter Meulen 1981 really lend support only to the first, the critical, point (viz. that reduction is impossible). But what she then goes on to call a nominal mass noun's "extensional reference to an intensional object" (viz. the intension function referred to above) seems to me both syntactically and semantically misguided. For the inevitable doubling of syntactic rules is certainly unwelcome, to begin with. But what is more, those intension functions, even when lifted to still another intensional level as ter Meulen wants to have it, are simply not well motivated as substance name denotations. The statement, for instance, that two fictional substances can be differentiated (op. cit., p. 438) is not compatible with the principle of rigid designation introduced earlier (op. cit., p. 424). More generally, there are no rules that could justify intuitively valid inferences from contexts involving nominal mass nouns to contexts with their corresponding predicative terms — it is my view, anyway, that such inferences are not based on pure logic alone. I conclude from this that the problem of nominal mass nouns is best approached in a spirit of logical abstinence. Nominal mass nouns denote abstract entities, to be sure, and as such they are names of individuals just like John, Munich, and the rest. Beyond this minimal account things become notoriously vague. 12 For this concept see, for instance, Gratzer 1978.

INTRODUCTION

17

Sums are partially ordered through the intrinsic ordering relation '< s ' on E which is expressed in the object language by the 2-place predicate 'II'. It is called the individual part relation (i-part relation, for short) and satisfies the biconditional (15)

allfc axPx]

The /^-operator has the effect of merging levels of comprehension, as I called it above. Thus we will have fixPx = /j,xQx, if P stands for is a card of one of the card decks, and Q for is a card deck. p,xPx and fixQx are not the same, however. If we have two decks of Bridge cards, for instance, axPx contains 104 atoms whereas axQx contains only 2 atoms. There is a second ordering relation, called the material part (m-part) relation and denoted by 'T'. It establishes a partial order on portions of matter, but only a preorder, called < m , on the whole domain of individuals. Objects which are m-parts of one another are materially equivalent in that they have the same portion of matter constituting them. If a is an i-part of 6 then a is an m-part of b; symbolically: (19)

allfc -» aJb

18

PLURALS AND MASS TERMS

In order to explain the meaning of' T' more precisely let me supply the remaining concepts of the model structure to be denned below. In addition to the domain of individuals, E, there is a set D which is endowed with a join operation 'LJ' making D into a complete, but not necessarily atomic, join-semilattice. D is partially ordered by its intrinsic ordering relation, Qx)

For non-distributive predicates we have, of course, no such result, witness carry: a® b might be in the extension of carry while a or b alone is not. Notice that these predicates enter formalizations unstarred. To illustrate, (31)

a. b.

The children gather around their teacher. Q (a*xPx) (with Q for gather etc.)

I think there is no harm in accepting this systematic difference: i.e., distributive predicates working on plural terms have to be starred, all the other predicates must not be. For distributivity seems to be a lexical feature. If we have a formal translation procedure the predicates have to be subcategorized accordingly. There is one more operator, , in the logic LPM to be presented below. T P, for a predicate P, is to be read as partakes in P. 1 introduce this operator in order to be able to distinguish between the plural terms the children and all the children. It seems to me that in all the children built the raft it is claimed that every child took part in the action whereas in the children built the raft it is only said that the children somehow managed to build the raft collectively without presupposing an active role in the action for every single child. This intuition enters the formalization of the two phrases given below. I want to stress, however, that the operator can only be partially characterized in view of the essentially pragmatic nature of its intended interpretation. So let me formulate just the following meaning postulates for which seem to be plausible: (32)

a. b.

V x ( J P x -^By(xUyAPy)) Distr (P) -> MX (JPx o Px)

I conclude the informal part of the chapter with some more formalizations of natural language sentences involving plurals and mass terms. Most of the principles governing their logic that have been mentioned in

INTRODUCTION

21

the literature come out valid in the system LPM below. Some of them are instantiated here, like Massey's plurality principles of symmetry, expansion, and contraction (see Massey 1976). What I did not treat in LPM, however, are any "downward" closure properties that are somehow felt to be present in the behavior of mass terms ("a part of water is still water"). Such principles can be added when a careful linguistic analysis has succeeded in giving them a form that takes care, in particular, of the problem of minimal parts (is every m-part of lemonade really lemonade again?). For some discussion of downward closure properties see, e. g., Bunt 1979 and Hoepelman/Rohrer 1980. (33)

(34)

(35)

(36)

(37)

(38)

(39)

(40)

(41)

a.

A child built the raft.

b.

3x (Px A Qx)

a.

Children built the raft.

b.

3x (*Px A Qx)

a.

The child built the raft.

b.

3y (y = ixPx A Qy)

a.

The children built the raft.

b.

3y (y = cr*xPx A Qy)

a.

Every child saw the raft.

b.

Vx ( P x —» Qx)

a.

All the children built the raft.

b.

Vy ( y = a*xPx -» Qx)

a.

Tom and Dick carried the piano upstairs.

b.

P (a ® b)

a.

Tom and Dick carried the piano upstairs, so Dick and Tom carried it upstairs.

b.

P (a ® b) =>• P (b ® a)

a.

John and Paul are pop stars and George is a pop star, so John, Paul, and George are pop stars.

b.

*P (a 9 b) A PC =>• *P (a ® b ® c)

Px: x is a child; Qx: x built the raft

Px, Qx: dto.

Px, Qx: dto.

Px, Qx: dto.

Px: dto., Qx: x saw the raft

Px, Qx as in (33); Q := Xx(Qx/\\/z(zUx -^ Qz))

a: Tom, 6: Dick; Px: x carried the piano upstairs

(symmetry)

(expansion)

22 (42)

PLURALS AND MASS TERMS a.

John, Paul, George, and Ringo are pop stars, so Paul and Ringo are pop stars. (see (28))

b.

*P(a®b®c®d) ^ *P(b®d)

a. b.

(All) water is wet. \/x (Px —> Qx)

(44)

a. b.

The water of the Rhine is dirty. Q (p,xPx) Px: x is (a quantity of) Rhine water

(45)

a. b.

This ice cube is water. P LX(X > a)

(46)

a. b.

The gold in Smith's ring is old, but Smith's ring is not old. Q LX (Px A x t> a) A -> Qa Px: x is gold; Qx: x is old; a: Smith's ring

1.2

The Logic of Plurals and Mass Terms (LPM)

(43)

(contraction)

Px: x is (a quantity of) water

a: this ice cube; Px: as in (43)

LPM is a first order predicate calculus with the usual logical constants '-i', 'V, 'A', '->•', 'o', 'V, '3', the description operator denoted by V, and the abstraction operator 'A'15 The syntactic variables are, for formulas of LPM, '$'> '^'5 'x'; for individual terms, 'a', '&', 'c'; for variables, 'a;', 'j/', V; for (1-place) predicates, 1P\ 1Q\ These symbols can also appear with primes and indices. For metalinguistic (definitional) identity the symbol '=' (':=') will be used. The set of 1-place predicate constants contains two specified subclasses: the set MT of predicative mass terms and the set DP of distributive predicates. MT and DP are taken to be disjoint sets. As special primitive symbols of LPM we have a 1-place predicate symbol 'E', three 2-place predicate constants, 'II', 'T', '>', and two operators on 1-place predicates, '*' and 'T'. The intended interpretations are the following. E a stands for "a exists"; a II b for "a is an individual part (i-part) of b;" aT6 for "a is a material part (m-part) of b," a\>b for "a constitutes or makes up b-" *P for "the plural predicate of P;" TP for "partakes in P." Now I introduce a number of defined expressions. 15 For an outline of such systems ("PLIIKA") see Link 1979; notice that, in LPM, identity of individual terms need not be taken as primitive, but can be defined in terms of'IT

THE LOGIC LPM

23

(D.I)

PcQ •

(D.2)

P = Q ^ PcQ/\QcP

(D.3)

P®Q :=

(DA)

Ia

Xx(PxVQx)

:= \x(x = a)

Furthermore, the usual definitions for the following formulas involving R are assumed: Refl(R) ("R is reflexive"), Trans (R) ("R is transitive"), Sym (R) ("R is symmetric"), Antisym (R) ("R is antisymmetric"). We then have, with PrO (R) for "R is a preordering relation," PO (R) for "R is a partial ordering relation," and Equ (R) for "R is an equivalence relation:" (D.5)

PrO(R)

o Refl(R) A Trans (R)

(D.6)

PO(R) o Refl(R) A Trans (R) A Antisym (R)

(D.7)

E