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Parts of a Whole: Distributivity As a Bridge Between Aspect and Measurement”

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Parts of a Whole: Distributivity As a Bridge Between Aspect and Measurement” Algebraic Semantics - Linguistic Applications of Mereology Script to the lecture series Lucas Champollion University of Tübingen [email protected] Tübingen, Germany Apr >E-?=, ?=>? Preface Expressions like ’John and Mary’ or ’the water in my cup’ intuitively involve reference to col- lections of individuals or substances. The parthood relation between these collections and their components is not modeled in standard formal semantics of natural language (Montague >FDA; Heim and Kratzer >FFE), but it plays central stage in what is known as mereological or algebraic semantics (Link >FFE; Krifka >FFE; Landman ?===). In this mini-lecture series I will present in- troductions into algebraic semantics and selected applications involving plural, mass reference, measurement, aspect, and distributivity. I will discuss issues involving ontology and philosophy of language, and how these issues interact with semantic theory depending on how they are resolved. This script is subject to change. Comments welcome. Lucas Champollion, Tübingen, April >Cth, ?=>? ? Contents > Mereology: Concepts and axioms B >.> Introduction . B >.? Mereology . C >.?.> Parthood . C >.?.? Sums....................................... E >.@ Mereology and set theory . >= >.A Selected literature . >= ? Nouns: count, plural, mass >? ?.> Algebraic closure and the plural . >? ?.? Singular count nouns . >B ?.@ Mass nouns and atomicity . >C @ Homomorphisms and measurement >E @.> Introduction . >E @.? Trace functions and intervals . >F @.@ Measure functions and degrees . ?= @.A Unit functions . ?> @.B The measurement puzzle . ?? A Verbs and events ?B A.> Introduction . ?B A.? Thematic roles . ?C A.@ Lexical cumulativity . ?E A.A Aspectual composition . ?F @ B Distributivity and scope @? B.> Introduction . @? B.? Lexical and phrasal distributivity . @? B.?.> Reformulating the D operator . @A B.?.? The leakage problem . @B B.@ Atomic and nonatomic distributivity . @C A Lecture > Mereology: Concepts and axioms >.> Introduction • Mereology: the study of parthood in philosophy and mathematical logic • Mereology can be axiomatized in a way that gives rise to algebraic structures (sets with binary operations deVned on them) Figure >.>: An algebraic structure a ⊕ b ⊕ c a ⊕ b a ⊕ c b ⊕ c a b c • Algebraic semantics: the branch of formal semantics that uses algebraic structures and parthood relations to model various phenomena B >.? Mereology >.?.> Parthood • Basic motivation (Link >FFE): entailment relation between collections and their members (>) a. John and Mary sleep. ) John sleeps and Mary sleeps. b. The water in my cup evaporated. ) The water at the bottom of my cup evaporated. • Basic relation ≤ (parthood) – no consensus on what exactly it expresses • Table >.> gives a few interpretations of the relation ≤ in algebraic semanitcs Table >.>: Examples of unstructured parthood Whole Part some horses a subset of them a quantity of water a portion of it John, Mary and Bill John some jumping events a subset of them a running event from A to B its part from A halfway towards B a temporal interval its initial half a spatial interval its northern half • All these are instances of unstructured parthood (arbitrary slices). • Individuals, spatial and temporal intervals, events, and situations are each generally taken to be mereologies (Link >FFE; Krifka >FFE). • Various types of individuals: – Ordinary individuals: John, my cup, this page – Mass individuals: the water in my cup – Plural individuals: John and Mary, the boys, the letters on this page – Group individuals: the Senate, the Beatles, the boys as a group C Table >.?: Examples of structured parthood from Simons (>FED) Whole Part a (certain) man his head a (certain) tree its trunk a house its roof a mountain its summit a battle its opening shot an insect’s life its larval stage a novel its Vrst chapter • Link (>FE@) on reductionist considerations: they are “quite alien to the purpose of logically analyzing the inference structure of natural language . [o]ur guide in ontological matters has to be language itself” • But reductionism is still possible, just at a later stage. • Compare this with structured parthood (Simons >FED; Fine >FFF; Varzi ?=>=) in Table >.? (cognitively salient parts) • In algebraic semantics one usually models only unstructured parthood. • This contrasts with lexical semantics, which concerns itself with structured parthood (e.g. Cruse >FEC). • Mereology started as an alternative to set theory; instead of 2 and ⊆ there is only ≤. • In algebraic semantics, mereology and set theory coexist. • The most common axiom system is classical extensional mereology (CEM). • The order-theoretic axiomatization of CEM starts with ≤ as a partial order: (?) Axiom of reWexivity 8x[x ≤ x] (Everything is part of itself.) D (@) Axiom of transitivity 8x8y8z[x ≤ y ^ y ≤ z ! x ≤ z] (Any part of any part of a thing is itself part of that thing.) (A) Axiom of antisymmetry 8x8y[x ≤ y ^ y ≤ x ! x = y] (Two distinct things cannot both be part of each other.) • The proper-part relation restricts parthood to nonequal pairs: (B) DeVnition: Proper part x < y =def x ≤ y ^ x 6= y (A proper part of a thing is a part of it that is distinct from it.) • To talk about objects which share parts, we deVne overlap: (C) DeVnition: Overlap x ◦ y =def 9z[z ≤ x ^ z ≤ y] (Two things overlap if and only if they have a part in common.) >.?.? Sums • Pretheoretically, sums are that which you get when you put several parts together. • The classical deVnition of sum in (D) is due to Tarski (>F?F). There are others. (D) DeVnition: Sum sum(x; P ) =def 8y[P (y) ! y ≤ x] ^ 8z[z ≤ x ! 9z0[P (z0) ^ z ◦ z0]] (A sum of a set P is a thing that consists of everything in P and whose parts each overlap with something in P . “sum(x; P )” means “x is a sum of (the things in) P ”.) Exercise >.> Prove the following facts! (E) Fact 8x8y[x ≤ y ! x ◦ y] (Parthood is a special case of overlap.) E (F) Fact 8x[sum(x; fxg)] (A singleton set sums up to its only member.) The answers to this and all following exercises are in the Appendix. • In CEM, two things composed of the same parts are identical: (>=) Axiom of uniqueness of sums 8P [P 6= ? ! 9!z sum(z; P )] (Every nonempty set has a unique sum.) • Based on the sum relation, we can deVne some useful operations: (>>) DeVnition: Generalized sum For any nonempty set P , its sum L P is deVned as ιz sum(z; P ): (The sum of a set P is the thing which contains every element of P and whose parts each overlap with an element of P .) (>?) DeVnition: Binary sum x ⊕ y is deVned as Lfx; yg. (>@) DeVnition: Generalized pointwise sum L For any nonempty n-place relation Rn, its sum Rn is deVned as the tuple hz1; : : : ; zni such that each zi is equal to L fxij9x1; : : : ; xi−1; xi+1; : : : ; xn[R(x1; : : : ; xn)]g: (The sum of a relation R is the pointwise sum of its positions.) • Two applications of sum in linguistics are conjoined terms and deVnite descriptions. – For Sharvy (>FE=), the water = L water J K – For Link (>FE@), John and Mary = j ⊕ m J K • Another application: natural kinds as sums; e.g. the kind potato is L potato. • But this needs to be reVned for uninstantiated kinds such as dodo and phlogiston. One answer: kinds are individual concepts of sums (Chierchia >FFEb). See Carlson (>FDD) and Pearson (?==F) on kinds more generally. F >.@ Mereology and set theory • Models of CEM (or “mereologies”) are essentially isomorphic to complete Boolean algebras with the bottom element removed, or equivalently complete semilattices with their bottom element removed (Tarski >F@B; Pontow and Schubert ?==C). • CEM parthood is very similar to the subset relation (Table >.@). • Example: the powerset of a given set, with the empty set removed, and with the partial order given by the subset relation. Exercise >.? If the empty set was not removed, would we still have a mereology? Why (not)? Table >.@: Correspondences between CEM and set theory Property CEM Set theory > ReWexivity x ≤ x x ⊆ x ? Transitivity x ≤ y ^ y ≤ z ! x ≤ z x ⊆ y ^ y ⊆ z ! x ⊆ z @ Antisymmetry x ≤ y ^ y ≤ x ! x = y x ⊆ y ^ y ⊆ x ! x = y A InterdeVnability x ≤ y $ x ⊕ y = y x ⊆ y $ x [ y = y S B Unique sum/union P 6= ? ! 9!z sum(z; P ) 9!z z = P C Associativity x ⊕ (y ⊕ z) = (x ⊕ y) ⊕ z x [ (y [ z) = (x [ y) [ z D Commutativity x ⊕ y = y ⊕ x x [ y = y [ x E Idempotence x ⊕ x = x x [ x = x x < y ! F Unique separation x ⊆ y ! 9!z[z = y − x] 9!z[x ⊕ z = y ^ :x ◦ z] >.A Selected literature – Textbooks: * Mathematical foundations: Partee, ter Meulen, and Wall (>FF=) * Montague-style formal semantics: Heim and Kratzer (>FFE) * Algebraic semantics: Landman (>FF>) – Books on algebraic semantics: Link (>FFE); Landman (?===) >= – Seminal articles on algebraic semantics: Landman (>FFC); Krifka (>FFE) – Mereology surveys: Simons (>FED); Casati and Varzi (>FFF); Varzi (?=>=) – My dissertation: Champollion (?=>=) >> Lecture ? Nouns: count, plural, mass ?.> Algebraic closure and the plural • Link (>FE@) has proposed algebraic closure as underlying the meaning of the plural. (>) a. John is a boy. b. Bill is a boy. c. ) John and Bill are boys. • Algebraic closure closes any predicate (or set) P under sum formation: (?) DeVnition: Algebraic closure (Link >FE@) The algebraic closure ∗P of a set P is deVned as fx j 9P 0 ⊆ P [x = L P 0]g. (This is the set that contains any sum of things taken from P .) • Link translates the argument in (>) as follows: (@) boy(j) ^ boy(b) ) ∗boy(j ⊕ b) • This argument is valid. Proof: From boy(j) ^ boy(b) it follows that fj; bg ⊆ boy. Hence 9P 0 ⊆ boy[j ⊕ b = L P 0],
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