# Linguistic applications of mereology

### Lucas Champollion

### Language and Logic (Introductory)

### First week, from 9:00 to 10:30, room G

## Abstract

## Description

This course roughly corresponds to Chapter 2 of my book, "Parts of a Whole - Distributivity as a Bridge between Aspect and Measurement", which has appeared recently at Oxford University Press. For more information on the book and excerpts, see http://nyu.edu/projects/champollion/book/. The lecture notes are available at http://www.nyu.edu/projects/champollion/esslli-2017-course.pdf.

**Day 1: Mereology and Set Theory.** After an overview of the content to be covered, the course starts by presenting the conceptual notions of parthood and sum that underlie mereology, followed by a gentle introduction to the axioms of classical extensional mereology. The models for this system are isomorphic to complete boolean algebras with the bottom elements removed, such as the powerset of a given set with the empty set removed. This means that there is a close correspondence between mereology and set theory. This lecture exploits this connection to draw on students’ knowledge concerning the familiar concepts of subsethood and union in order to strengthen their intuitions concerning the new relations of parthood and sum.

**Day 2: Nouns and Measurement.** The new concepts from the previous lecture are applied to the singular-plural distinction. Following Link (1983), we define the algebraic closure (star operator) in terms of mereological sum. Different theories of the meaning of the plural are empirically motivated, and formalized using algebraic closure. Zweig (2009)’s theory of dependent plurals is briefly introduced as an example of a recent linguistic application. The notions cumulative, divisive, quantized reference (e.g. Krifka, 1998) are motivated. This leads to a discussion of the count- mass distinction. Finally, measure nouns are discussed and the associated concepts degree and measure function are motivated.

**Day 3: Measurement and Verbs.** The lecture starts by considering constructions in which measure nouns are used, with a focus on pseudopartitives. The constraints on these constructions lead to the relevant notion monotonic/extensive measure function (Krifka, 1998; Schwarzschild, 2006). We then zoom in on the two measure functions *temporal trace* and *spatial trace* and on their uses in the semantics of *for*-adverbials and *in*-adverbials. This leads to a discussion of lexical aspect, with special attention to the parallel between the telic/atelic opposition and the count/mass opposition. The parallel between the two oppositions is formalized via the concepts of cumulative, divisive, and quantized reference discussed in the previous lecture. To carry out this formalization, Neo-Davidsonian event semantics is briefly introduced. This naturally leads into the topic of thematic roles, which are related to measure functions and trace functions. The lecture ends with a discussion of the notions homomorphism and incremental theme and their relevance to lexical aspect.

**Day 4: Verbs and Distributivity.** The phemonenon of aspectual composition (e.g. Krifka, 1998) is used to motivate a compositional approach to lexical aspect. This is compared with other approaches. The collective-distributive opposition is introduced and linked with the related telic-atelic, singular-plural, and count-mass oppositions via cumulative and divisive reference. Two approaches to the modeling of distributive predicates are contrasted with each other: the meaning postulate approach, and the compositional (D operator) approach (e.g. Link, 1998). The lecture ends by drawing out the parallel between the compositional approach to aspect and the compositional approach to distributivity.

**Day 5: Putting it all together.** This lecture (from the slides at http://tinyurl.com/unifieddist) summarizes my work and recent book, focusing on distributivity. My work brings together several strands of research on semantic and pragmatic phenomena in order to propose a comprehensive theory of distributivity, atelicity, plurality, and the count/mass distinction. I start by answering the question *What is distributivity? *in several related ways: distributivity is a property of predicates, of quantifiers, of constructions, or a relation between constituents, and of constructions. I build on these notions to argue that the term *distributivity *applies to a variety of constructions, such as adverbial *each, **for*-adverbials, and pseudopartitives. Building on a common thread that runs through these constructions, I propose a constraint that explains why they reject collective and telic predicates, singular count nouns, and nonmonotonic measure functions. To capture this constraint, I develop *stratified reference, *a parametrized notion whose core idea is that if a predicate applies to something, it must also apply to its parts along some dimension and down to a certain level of granularity. Stratified reference generalizes and relates two core concepts in algebraic semantics: higher-order properties such as Krifka's divisive reference and Dowty's subinterval property, and distributive operators such as Link's D and Schwarzschild's Part. On this basis, I develop a formal account of distance-distributive constructions across languages, such as English adnominal *each, *German *jeweils*,* *or distributive numerals. The common core of these constructions as well as their crosslinguistic variation can be captured by varying the dimension and granularity parameters of stratified reference.