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# Partial Multiplicity Inferences
In certain quantificational contexts plurals give rise to
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# Multiplicity Inf as Scalar Implicature
The scalar implicature approach to multiplicity inferences
(Spector 2007, Zweig 2009, Ivelieva 2014, Mayr 2015).
- A plural noun is semantically number neutral
- A singular noun is only true of singular entities
- Plural and singular compete and give rise to scalar implicatures = multiplicity inferences
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Issue: Sentences containing singular and plural indefinites are truth-conditionally identical.
(14) Paul is meeting with students
(15) Paul is meeting with a student
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# Finding Semantic Asymmetry
(14) Paul is meeting with students
(15) Paul is meeting with a student
Previous accounts:
- Higher-order SI (Spector 2007):
(14) competes with (15) augmented with its SI
- Embedded SI (Zweig 2009, Ivelieva 2014, Mayr 2015):
(14) and (15) contain subconstituents with different truth-conditional content
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Proposal: Non-propositional asymmetry with discourse referents; no need for additional mechanisms.
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# Example: Simple Positive Sentence
(16) Paul is meeting with studentsx
(17) Paul is meeting with a studentx
(16) and (17) are propositionally equivalent, but not anaphorically.
- Possible values of x in (16) include both singular and plural entities.
- Possible values of x in (17) are all singular entities.
Since (16) is less informative, it generates a scalar implicature that (17) is not what the speaker means. So the possible values of x in (16) must be all plural.
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# Some details
A (secondary) scalar implicature is the negation of a stronger alternative.
Let's assume a particular way of achieving this (an alternative later):
If φ has a dynamically more informative (and relevant) alternative ψ, then an utterance of φ in c amounts to:
c[φ] − c[ψ]
φ is dynamically more informative than ψ if for each c, c[φ]⊆c[ψ]; but for some c', c'[ψ]⊈c'[φ].
(To simplify let's not discuss contextual informativity, primary implicatures, multiple alternatives, etc.)
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# Details (cont.)
(16) Paul is meeting with studentsx
(17) Paul is meeting with a studentx
Dynaic analysis:
c[(16)] = {⟨f[x↦d], w⟩ | ⟨f, w⟩∈c, Paul is meeting with d in w and d is a single student in w or a plurality consisting of students in w}
c[(17)] = {⟨f[x↦d], w⟩ | ⟨f, w⟩∈c, Paul is meeting with d in w and d is a single student in w}
c[(16)] − c[(17)] = {⟨f[x↦d], w⟩ | ⟨f, w⟩∈c, Paul is meeting with d in w and d is a plurality consisting of students in w}
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# Example: Negation
(18) Paul is not meeting with studentsx
(19) Paul is not meeting with a studentx
Negation kills discourse referents, so (18) and (19) are identical both propositionally and anaphorically.
⇒ No scalar implicature
c[¬φ] = {⟨f, w⟩∈c | there is no ⟨f', w⟩∈c[φ] s.t. f≤f'}
Footnote: Double negation is a problem for dynamic theories but there are some solutions; to the extent it introduces discourse referents, we can run the same reasoning as the simple positive case
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# Example: Exactly One
(20) Exactly one of my students have journal papersx
(21) Exactly one of my students have a journal paperx
- (20) introduces a new discourse referent x, whose possible values are singular journal papers or plural journal papers.
- (21) introduces a discourse referent x, whose possible values are singular journal papers only.
As in the simple positive sentence, (20) is less informative than (21), and generates a scalar implicature that the values of x don't include singular journal papers.
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# Interim Summary
Multiplicity inferences of plural indefinites can be derived as scalar implicatures with discourse referents.
- No need for higher order implicatures or embedded implicatures
- If discourse referents are needed and carry information, they should give rise to pragmatic inferences.
Pragmatic reasoning: Exclude the possibilities that the alternative ψ gives rise to, c[ψ], from c[φ]
--
The present system makes slightly different predictions than previous scalar implicature accounts.
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# Quantificational Subordination
Quantificational subordination (Van den Berg 1996, Nouwen 2003, Brasoveanu 2007).
(22) Every student of mine wrote a paperx this term.
a. — *Itx is about Binding Theory.
b. — They all submitted itx to a journal.
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Van den Berg's idea: (22) introduces a structured discourse referent, representing a mapping from my students to their papers. (22b) accesses it.
Student 1 ⟼ Paper A Student 1 ⟼ Paper B
Student 2 ⟼ Paper B Student 2 ⟼ Paper C
Student 3 ⟼ Paper C Student 3 ⟼ Paper A
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# Multiplicity Inference with Quantifiers
(23) Every student of mine wrote papersx this term
(24) Every student of mine wrote a paperx this term
(23) introduces a structured discourse referent where a value of the paper discourse referent for each student is either singular or plural.
Student 1 ⟼ Paper A Student 1 ⟼ Papers A+D
Student 2 ⟼ Paper B Student 2 ⟼ Paper B
Student 3 ⟼ Paper C Student 3 ⟼ Papers C+E
The scalar implicature is that whatever (23) can mean is not what the speaker means, i.e. at least one student of mine is mapped to multiple papers.
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# ∨ under ∀: Problem
(25) Everyone speaks German or French
Crnič, Chemla & Fox (2015) ponit out that the prediction of the standard account for (25) is too strong.
(26) Everyone speaks German
(27) Everyone speaks French
(28) Everyone speaks German and French
(25) is compatible with (26), e.g., as long as some speak French and some don't speak both.
☞ The SI of (25) is that someone speaks German; someone speaks French; and not everyone speaks both.
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# ∨ under ∀: Analysis
(29) Everyone speaks German or Frenchx
(30) Everyone speaks Germanx
(29) introduces a structured discourse referent mapping people to one of the two languages.
(30) represents a mapping that maps everyone to German, and the SI is that that's not what the speaker means. (Similarly for French)
But what's negated is (30) with a particular mapping. It doesn't entail that not everyone speaks German.
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# ∨ under ∀: Analysis (cont.)
(29) Everyone speaks German or Frenchx
(30) Everyone speaks Germanx
Suppose that everyone speaks German in w and some speak French in w.
Person 1 ⟼ German Person 1 ⟼ German
Person 2 ⟼ German Person 2 ⟼ French
Person 3 ⟼ German Person 3 ⟼ German
SI will eiminate w paired with the mapping on the left
But not w paired with the mapping on the right
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# ∨ under ∀: prediction
(31) Everyone speaks German or Frenchx
They all learned itx at school
Context 1
- Klaus speaks Gr natively, learned Fr at school
- Wataru learned Gr at school
- Paul speaks Fr natively, learned Gr at school
--
Context 2
- Sophie speaks Fr natively, learned Gr at school
- Wataru learned Gr at school
- Paul speaks Fr natively, learned Gr at school
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# Summary
Multiplicity inferences can be accounted for straightforwardly as quantity implicatures about discourse referents without additional mechanisms.
- Nice predictions about quantificational contexts (cf. quantificational subordination).
- It derives standard scalar implicatures (e.g. most ⤳ not all) (not shown today).
Further extensions and remaining issues:
- Modal subordination
- Primary implicatures (cf. Dieuleveut, Chemla, Spector 2019)
- Plural definites (Mayr 2015)
- Notion of entailment
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class: center, middle
PART 2
EMBEDDED
IMPLICATURES
UNDER INDEFINITES
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# A Consequence on SI's under Indefinites
If φ has a dynamically more informative (and relevant) alternative ψ, then an utterance of φ in c amounts to:
c[φ] − c[ψ]
φ is dynamically more informative than ψ if for each c, c[φ]⊆c[ψ]; but for some c', c'[ψ]⊈c'[φ].
☞ This predicts that when a SI is triggered under an indefinite, it takes scope under the indefinite.
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# Example
(32) Paul solved most of the problems
⤳ ¬(Paul solved all the problems)
--
(31) Someonex solved most of the problems
c[(31)] = {⟨f[x↦d], w⟩ | ⟨f,w⟩∈c, d solved most or all of the problems in w}
--
(32) Someonex solved all of the problems
c[(32)] = {⟨f[x↦d], w⟩ | ⟨f,w⟩∈c, d solved all of the problems in w}
c[(32)]−c[(31)] = {⟨f[x↦d], w⟩ | ⟨f,w⟩∈c, d solved most but not all of the problems in w}
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# SI under Indefinites
The 'embedded SI' is possible with these examples (Geurts 2009, Sudo 2016).
(33) There's a student that solved most of the problems
G) No student solved all the problems
L) That student didn't solve all the problems
(34) At least one student solved most of the problems
G) No student solved all
L) The relevant student(s) didn't solve all
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It's less clear if the 'global SI' is unavaialble. But cf:
(35) Paul can read most of these papers by tomorrow.
(36) You are allowed to eat most of the cake.
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# Sequential Update with SI
There is an alternative way of getting the same results (Geurts 2006, Sudo 2016).
Compute the SI sequentially after the assertion, c[A][SI].
--
(37) Paul solved most of the problems.
(38) Paul solved all of the problems.
Utterance of (37) in context c ➠ c[(37)][¬(38)]
To obtain 'embedded SIs' under indefinites with sequential updates, we have to assume a certain analysis of indefinites.
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# Indefinites and Novelty Condition
Heim's (1982) idea: Indefinites are variables (no ∃!), and they must be new variables (Novelty Condition).
- For any context c the assignemnts {f| ⟨f, w⟩ ∈c} have the same domain, dom(c)
- x is a new variable in c if x ∉ dom(c); and it's old otherwise.
c[P(x)]
= {⟨f[x↦d], w⟩ | ⟨f, w⟩∈c, d∈Iw(P)}
if x is a new variable
= {⟨f, w⟩ | ⟨f, w⟩∈c, f(x)∈Iw(P)}
if x is an old variable
(39) A studentx left. (40) The studentx left.
student(x) ∧ left(x)
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# Novelty Condition
Suppose that the Novelty Condition is satisfied as soon as the first occurrence of the variable satisfies it, and it doesn't apply to those occurrences in scalar alternatives.
--
(45) There's a studentx who solved most of the problems
student(x) ∧ solvedMost(x)
(46) There's a studentx who solved all of the problems
student(x) ∧ solvedAll(x)
In c[(45)][¬(46)], only the first occurrence of x is new, but that satisfies the Novelty Condition.
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# Two Ways to Achieve the Same Thing
1. Reasoning about hypothetical utterance: The effects that ψ could have brought about are excluded.
c[φ] − c[ψ]
2. Sequential update Process the negation of ψ after φ; Variables in ψ can have antecedents in φ.
c[φ][¬ψ]
No difference for multiplicity inference and SIs under indefinites.
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class: center, middle
PART 3
IGNORANCE
IMPLICATURE
WITH INDEFINITES
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Peirce's Puzzle
The Contest
A promotional contest is being held. Contestants are phoned in random order and asked a simple trivia question. The contest stops if someone gets it wrong. If every contestant gets it right, a winner is chosen randomly and awarded $1,000. Hearing the contest is over, John says:
(43) Either someone got the question wrong, or someone is $1,000 richer.
(44) #Someone either got the question wrong, or is $1,000 richer.
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# Why This is a Puzzle
(45) Someone married John or someone married Sue.
(46) Someone married John or Sue.
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In Classical Logic:
(∃xJx ∨ ∃xSx) ⟺ ∃x(Jx ∨ Sx)
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Classical Quantity Implicature:
- Speaker didn't say 'Someone married John'
- Speaker lacks enough evidence for its truth
- Similarly for 'Someone married Sue'
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# Why This is a Puzzle (cont.)
It's not about the use of indefinites (contrary to what Dekker 2001 suggests).
(47) Someone married John or Sue.
(48) Someone became my in-law.
The literal meanings of these two sentences can be contextually equivalent.
☞ The ignorance implicature comes from disjunction.
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# Standard Reasoning
The standard pragmatic reasoning doesn't yield the results we want.
(47) Someone married John or Sue.
- Speaker said '∃x(Jx ∨ Sx)'
- By Maxim of Quality, ◻∃x(Jx ∨ Sx)︎
- By Maxim of Quantity, ¬◻∃x(Jx), ¬◻∃x(Sx), ¬◻∃x(Jx ∧ Sx)
- Epistemic step: ◻¬∃x(Jx ∧ Sx)
These inferences are too weak.
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In particular, ◻∃x(Jx ∨ Sx) doesn't require John and Sue's potential partners to be the same person.
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# Sequential Reasoning
The only way out we can think of is sequential reasoning about the same value of x.
(48) Someone married John or Sue.
- Speaker said '(Jx ∨ Sx)' for a new variable x
- By Maxim of Quality, ◻(Jx ∨ Sx)︎
- By Maxim of Quantity, ¬◻(Jx), ¬◻(Sx), ¬◻(Jx∧Sx)
- Epistemic step: ◻¬(Jx∧Sx)
To obtain the results, it's crucial that there's no rebinding in ◻(Jx ∨ Sx).
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# Two Existential Quantifiers
When there are two existential quantifiers:
(49) Someone married John or someone married Sue.
- Speaker said '(Fx ∨ Gy)' for new variables x, y
- By Maxim of Quality, ◻(Fx ∨ Gy)︎
- By Maxim of Quantity, ¬◻(Fx), ¬◻(Gy), ¬◻(Fx∧Gy)
- Epistemic step: ◻¬(Fx∧Gy)
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We must assume that (50) has an LF similar to (49).
(50) John or Sue married someone.
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class: center, middle
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Conclusions
We need discourse referents for anaphora.
They bear information, so should participate in pragmatics (scalar implicatures, ignorance implicatures).
This hasn't been discussed much, although both Gricean Pragmatics and discourse referents have been with us since the 1960s.
Multiplicity inferences are given a straightforward account under this view.
Ignorance inferences under indefinites seem to require sequential reasoning for the same value of the variable.
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ADVERT
Sinn und Bedeutung 25 at
Queen Mary University of London and
University College London
Special themed sessions: Wed 2 Sept 2020
❖ Semantics and Typology
❖ Anaphora in Super Linguistics
Main session: Thu 3–Sat 5 Sept 2020
Invited speakers
Ana Arregui (Umass)
Lisa Bylinina (Leiden)
Clemens Mayr (Göttingen)
Abstract submission deadline: Sun 1 March 2020
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count: false
# E-type Pronouns
Situation semantics with E-type pronouns (Heim 1990, Elbourne 2005).
- Pronouns are disguised definite descriptions
- Definite descriptions have uniqueness presuppositions
- Uniqueness with respect to situations
If there's a mango, Jacopo will eat it.
Rothschild & Mandelkern (2019): situations need to be passed on in a way similar to dynamic binding.
Jacopo has a mango. He will eat it.
Every man had a mango and ate it.