Gather -type predicates: massiness over participants Jeremy Kuhn - - PowerPoint PPT Presentation

Overview Monotonicity Stratified Reference Grounded SR Conclusion

Gather-type predicates: massiness over participants

Jeremy Kuhn New York University North East Linguistic Society 45 at MIT November 1, 2014

[slides: https://files.nyu.edu/jdk360/public/papers/Kuhn-gather-slides.pdf]

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Overview Monotonicity Stratified Reference Grounded SR Conclusion

Section 1 Overview

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Two kinds of collective predicates

◮ Predicates like gather and be numerous have both been described as “collective predicates.” (1) a. The students are numerous. b. * Marco is numerous. (2) a. The students gathered. b. * Marco gathered.

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Two kinds of collective predicates

◮ However, the two classes of predicates differ with respect to plural quantifiers (e.g. all, most, several).

(Winter 2001, Champollion 2010)

(3) Gather-type predicates a. The students gathered. b. All the students gathered. c. * Each student gathered. (4) Numerous-type predicates a. The students are numerous. b. * All the students are numerous. c. * Each student is numerous.

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Overview Monotonicity Stratified Reference Grounded SR Conclusion

Two kinds of collective predicates

◮ However, the two classes of predicates differ with respect to plural quantifiers (e.g. all, most, several).

(Winter 2001, Champollion 2010)

(5) Gather-type predicates a. The students gathered. b. All the students gathered. c. * Each student gathered. (6) Numerous-type predicates a. The students are numerous. b. * All the students are numerous. c. * Each student is numerous.

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Two kinds of collective predicates

◮ All the students/puzzle pieces/jurors/axioms... Gather-type predicates Numerous-type predicates gather be numerous be similar be a group of ten meet form a pyramid disperse suffice to defeat the army be consistent (axioms) be inconsistent (axioms) hold hands return a verdict of ‘not guilty’ fit together be a group of less than ten disagree be denser in the middle

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The intuition

◮ Numerous-type predicates generally involve an emergent property of the whole group...

◮ be numerous, be a group of ten — the number of individuals; ◮ return a verdict of ‘not guilty’ — the legality of returning a verdict; ◮ be incompatible (of a set of axioms) — the logical properties of the set.

◮ Gather-type predicates allow ‘distributive sub-entailments’...

(Dowty 1987)

◮ gather — individuals went to the same place as someone else; ◮ fit together — puzzle pieces connect with some other piece.

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The take-home message

◮ The take-home message from this talk: The gather/numerous distinction is analogous to the mass/count distinction in the nominal domain and the atelic/telic distinction in the temporal domain. nouns verbs verbs (w.r.t time) (w.r.t. participants) mass water look at an apple gather count chair eat an apple be numerous

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Solving problems

◮ However, monotonicity (either as divisibility or cumulativity) both overgenerates and undergenerates the class of gather-type predicates, so no account to date has provided a successful entailment-based diagnostic. ◮ Here, I build on these intuitions, repairing the problems with two innovations: I. In the spirit of Champollion 2010, gather-predicates have Stratified Reference, not monotonicity. II. Gather-predicates display a homomorphism between the mereology of individuals and the mereology of situations (as in Kratzer 1989, Fine 2012).

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Section 2 Monotonicity and beyond

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Bounded monotonicity

◮ A good starting point: (7) A predicate P has ε-bounded downward monotonicity iff ∀x[P(x) → ∀y[(µ(y) > ε ∧ y ≤ x) → P(y)]]

“If P holds of x, then P holds of all sufficiently large parts of x.”

◮ Mass nouns: For any entity x, if x is water, then any sufficiently large subpart of x is water. ◮ Gather predicates: For any plurality x, if x gathered, then any sufficiently large subplurality of x gathered.

◮ (Bounded monotonicity avoids the Minimal Parts Problem: a single atom cannot be water; a single person cannot gather.)

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Bounded monotonicity

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Overview Monotonicity Stratified Reference Grounded SR Conclusion

Bounded monotonicity

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Overview Monotonicity Stratified Reference Grounded SR Conclusion

Bounded monotonicity

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Overview Monotonicity Stratified Reference Grounded SR Conclusion

Bounded monotonicity

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Bounded monotonicity

◮ At a first pass, this seems to do quite well! ◮ gather

◮ If the boys gathered (=went to the same place at the same time), then any subset of at least two boys gathered.

◮ be similar

◮ If the boys are similar, then any subset of at least two boys is similar.

◮ In fact, Winter 2001 very briefly considers 2-bounded downward monotonicity as a possible diagnostic.

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◮ But, n-bounded downward monotonicity faces problems. ◮ Undergeneration: incorrectly rejects some predicates from the gather class.

◮ hold hands, fit together, disagree

◮ Overgeneration: incorrectly admits some predicates to the gather class.

◮ be a group of more than three or less than five, be a group of less than ten, be denser in the middle

◮ Today, I will fix these problems while maintaining the congruence between mass nouns and gather-type predicates.

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Section 3 The “Tricky Parts Problem” and Stratified reference

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The “Tricky Parts Problem”

◮ Undergeneration: some gather-type predicates are not 2-bounded monotonic.

◮ hold hands: if all the children held hands, it is not necessarily the case that any two given children held hands with each

• ther.

◮ fit together: if all the puzzle pieces fit together, it is not necessarily the case that any two given puzzle pieces fit together.

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The “Tricky Parts Problem”

◮ Suppose that six students are holding hands as follows:

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The “Tricky Parts Problem”

◮ The following set of three students are not holding hands!

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The “Tricky Parts Problem”

◮ In fact, hold hands is not n-bounded monotonic for any n.

◮ Consider a circle of 2n students holding hands. The set consisting of every other student in the circle is a subset of n students, and yet the predicate hold hands does not hold of it.

◮ Thus, what we have is not a Minimal Parts Problem.

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The “Tricky Parts Problem”

◮ Instead, the “Tricky Parts Problem”:

◮ A universal quantifier in the definition of divisibility is paying attention to irrelevant subparts (the ones a tricky logician would point to as counterexamples).

(8) A predicate P has ε-bounded divisibility iff ∀x[P(x) → ∀y[µ(y) > ε ∧ y ≤ x → P(y)]]

◮ This is essentially the same problem as the “succotash” problem for mass nouns. (Ask me in the question period for details.)

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Stratified Reference

◮ Champollion 2010 (in other domains): Change a universal quantifier to an existential quantifier! ◮ Several equivalent formulations; we will do this using covers. ◮ If x is a plural entity, a cover of x is a set of (possibly

• verlapping) plural entities whose sum is x.

◮ I.e. C is a cover of x if C = x.

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Overview Monotonicity Stratified Reference Grounded SR Conclusion

Stratified Reference

◮ A predicate P has downward monotonicity if, for every event in P, every cover of sufficiently large sub-events has cells that are also in P. ◮ A predicate P has Stratified Reference if, for every event in P, there is some cover of sufficiently small sub-events with cells that are also in P.

(SR: Champollion 2010)

(9)

• a. Div(P)

iff ∀x[P(x) → ∀Ce,t[( C = x ∧∀y ∈ C.|y| ≥ ε) → (∀y ∈ C.P(y))]]

• b. SR(P)

iff ∀x[P(x) → ∃Ce,t[( C = x ∧ ∀y ∈ C.|y| ≤ ε) ∧ (∀y ∈ C.P(y))]]

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Overview Monotonicity Stratified Reference Grounded SR Conclusion

Stratified Reference

◮ Gather-type predicate have SR:

◮ gathered has SR, because any gathering event can be decomposed into (i.e. is the sum of) small gathering events (overlapping groups of two or three). ◮ fit together has SR, because any fitting-together event can be decomposed into small fitting-together events (overlapping pairs of adjacent pieces).

◮ Numerous-type predicate do not have SR:

◮ be numerous does not have SR, because events that have only small subsets of the participants will by definition not be numerous events. ◮ return a verdict of not guilty does not have SR, because any event with only a subset of the participants cannot be (by legal definition) an event of returning a verdict.

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Puzzle pieces

=

w + x x + y x + z w + x + y + z

• +

+

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Examples

(10) a. All the axioms are consistent. b. * All the axioms are inconsistent.

Parallel example: “All the computer programs are (*in)compatible.” (p.c. Benjamin Spector)

consistent inconsistent a > b a > b a > c b > c a > d c > d a > e d > e a > f e > a Consistency holds Inconsistency emerges

• f all subsets.

from the set as a whole!

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Examples

(11)

• a. All the students agreed (about what book to read).
• b. All the students disagreed (about what book to read).

◮ Why is disagree different from inconsistent? agree disagree Aaron thinks “A!” Aaron thinks “A!” Bobby thinks “A!” Bobby thinks “A!” Carol thinks “A!” Carol thinks “B!” David thinks “A!” David thinks “B!” Eve thinks “A!” Eve thinks “B!” ◮ Any disagreeing situation has at least two opinions. There’s a cover that has both opinions in each cell.

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Section 4 Grounded Stratified Reference

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Overgeneration

◮ Overgeneration: Downward monotonicity and SR both incorrectly predict that all will be grammatical with a certain set of tautological predicates (adapted from discussion by Higginbotham 1994 and Lønning 1987). (12) * All the students are a group of more than four or less than five. ◮ Note that the predicate in (10) holds of all groups, so it technically has divisibility and SR. In fact, any diagnostic based purely on possible-world entailment will fail for similar reasons.

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Towards situation-based semantics

(10) * All the students are a group of more than four or less than five. ◮ Intuitively: Divisibility/SR holds, but for the wrong reasons.

◮ A group of six satisfies the predicate for a different reason that its two-person subgroups do.

◮ We want to say: “the predicate holds of the sum by virtue of the fact that it holds for the parts.”

◮ I.e. there is more to an event than simply the individuals who are part of it.

◮ Solution: move to a framework of truth where sentences mean more than just the worlds in which they are true.

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Situation semantics, in a nutshell

◮ Possible world semantics → Situation semantics Kratzer 1989, “Lumps of thought”: ◮ Worlds are complete information states — if you know which w you’re in, you know everything. ◮ Situations are parts of worlds, or incomplete information states. ◮ Because situations have part-relations, they come with a mereology!

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Situation semantics, in a nutshell

◮ E.g., there is a situation which contains the information that it is raining, but no other information about the world.

◮ (We’ll say that this situation “witnesses the proposition” that it is raining.)

◮ This situation is a part of the situation that contains the information that it is raining and that Jeremy is talking. rain jeremy w

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Situation semantics, in a nutshell

◮ Propositional attitudes give insight into situations. (13) John thinks that Mary is talking on the phone and standing next to the window. Valid inference: John thinks that Mary is talking on the phone. (14) John thinks that Mary is talking on the phone. Invalid inference: John thinks that Mary is either talking on the phone or going on a walk. ◮ Why?

◮ The s witnessing ‘p’ is part of the s′ witnessing ‘p and q’. ◮ The s witnessing ‘p or r’ is not part of the s′ witnessing ‘p’.

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Overview Monotonicity Stratified Reference Grounded SR Conclusion

Situation-homomorphic stratified reference

◮ Translating Stratified Reference into state terms: (15) For any plural entity x in P, you can find an ε-cover Cov of x and a set of situations S such that

• a. For all y ∈ Cov, there is an s ∈ S that witnesses P(y),

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Overview Monotonicity Stratified Reference Grounded SR Conclusion

Situation-homomorphic stratified reference

◮ Now, we can add the intuition of ‘by virtue of’ with a single clause... (16) For any plural entity x in P, you can find an ε-cover Cov of x and a set of situations S such that

• a. For all y ∈ Cov, there is an s ∈ S that witnesses P(y),

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Overview Monotonicity Stratified Reference Grounded SR Conclusion

Situation-homomorphic stratified reference

◮ Now, we can add the intuition of ‘by virtue of’ with a single clause... (17) For any plural entity x in P, you can find an ε-cover Cov of x and a set of situations S such that

• a. For all y ∈ Cov, there is an s ∈ S that witnesses P(y),
• b. S witnesses P(x).

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The homomorphism, graphically

y z y + z = x For any plurality x,

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Overview Monotonicity Stratified Reference Grounded SR Conclusion

The homomorphism, graphically

y z y + z = x s3 For any plurality x, if x is in P,

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The homomorphism, graphically

y z y + z = x s1 s2 s3 For any plurality x, if x is in P, then there is an ε-cover of x whose cells are in P,

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Overview Monotonicity Stratified Reference Grounded SR Conclusion

The homomorphism, graphically

y z y + z = x s1 s2 s3 For any plurality x, if x is in P, then there is an ε-cover of x whose cells are in P, and whose witnessing situations have a sum that witnesses P(x).

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Overview Monotonicity Stratified Reference Grounded SR Conclusion

A homomorphism

◮ “gather at 9:00”

◮ s witnesses “John and Mary gathered at 9:00” s′ witnesses “Mary and Sue gathered at 9:00” (s + s′) witnesses “John, Mary and Sue gathered at 9:00”

In contrast... ◮ “be a group of more than four or less than five”

◮ s witnesses “John, Mary, and Sue are a group of less than five” s′ witnesses “Sue, Eve, and Hugh are a group of less than five” s′′ witnesses “J, M, S, E, and H are a group of more than four” ◮ BUT: s + s′ = s′′

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Section 5 Conclusion: Why Stratified Reference?

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The progression of the talk

Divisibility (Grounded SR) Stratified Reference

Progression of this talk Semantic strength

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Quantifying over covers

◮ Why Stratified Reference? ◮ Brisson 2003, Kuhn 2014: plural quantifiers (all, most, etc.) quantify over the cells of a cover of the plural noun.

◮ “All the students disagreed (about what book to read).” = Every pair of students disagreed. ◮ “No students who hung out yesterday hung out today” → Can be true if the total set of students is the same; just with different groupings.

◮ SR ensures the existence of a possible cover to quantify over.

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Insensitivity to accidents

◮ A final theme: natural language is insensitive to accidents. ◮ SR removed a universal quantifier that accidentally considered ‘tricky’ ways of dividing up a group. ◮ The individual-to-state homomorphism ensured that the reason that the predicate holds of a sum is grounded on the reason it holds of the parts.

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The take-home message

◮ The take-home message from this talk: The gather/numerous distinction is analogous to the mass/count distinction in the nominal domain and the atelic/telic distinction in the temporal domain. nouns verbs verbs (w.r.t time) (w.r.t. participants) mass water look at an apple gather count chair eat an apple be numerous

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Thanks!

Special thanks to Lucas Champollion, Carmen Dobrovie-Sorin, Benjamin Spector, and to audiences at New York University and Paris Diderot University. This work was supported in part by the ERC Advanced Grant Project ‘New Frontiers in Formal Semantics.’

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References

Brisson, Christine. 2003. Plurals, all, and the nonuniformity of collective

• predication. Linguistics and Philosophy 26, 129-184.

Champollion, L. (2010). Parts of a whole: distributivity as a bridge between aspect and measurement. PhD Thesis. Dobrovie-Sorin, C. 2013. Most: the View from Mass Quantification. Proceedings of the 19th Amsterdam Colloquium, December 2013. Dowty, D. (1987). Collective predicates, distributive predicates and all. Proc.

• f the 3rd Eastern States Conference on Linguistics. Ohio State.

Fine, K. 2012, “The pure logic of ground”, Review of Symbolic Logic, 25/1, 1-25.

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References

Higginbotham, J., 1994, “Mass and count quantifiers", Linguistics and Philosophy, 17: 447-480. Kuhn, J., 2014, “Massy yet chunky: all quantifies over covers”, Presentation at Paris 7. Kratzer, A. 1989, “Lumps of thought,” Linguistics and Philosophy 12: 607-653. Lønning, J.T., 1987, “Mass terms and quantification", Linguistics and Philosophy, 10: 1-52. Schwarzschild, R. (2006). The role of dimensions in the syntax of noun

• phrases. Syntax.

Winter, Y. (2001). Flexibility Principles in Boolean Semantics. MIT Press. Winter, Y. (2001). Plural Predication and the Strongest Meaning Hypothesis. Journal of Semantics.

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