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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
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Issue: Sentences containing singular and plural indefinites are truth-conditionally identical.
(11) Paul is meeting with students
(12) Paul is meeting with a student
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# Finding Semantic Asymmetry
Issue: Sentences containing singular and plural indefinites are truth-conditionally identical.
(11) Paul is meeting with students
(12) Paul is meeting with a student
Previous accounts:
- Higher-order SI (Spector 2007)
- Embedded SI (Zweig 2009, Ivelieva 2014, Mayr 2015)
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Proposal: Non-propositional asymmetry with discourse referents; no need to postulate these additional mechanisms.
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# Example: Simple Positive Sentence
(11) Paul is meeting with studentsx
(12) Paul is meeting with a studentx
(11) and (12) are propositionally equivalent, but not anaphorically.
- Possible values of x in (11) include both singular and plural entities.
- Possible values of x in (12) are all singular entities.
Since (11) is less informative, it generates a (secondary) scalar implicature that (12) is not what the speaker means. So the possible values of the discourse reference in (11) are all plural.
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# Details
Scalar implicature amounts to the negation of the stronger alternative.
Here let's assume a particular way of achieving this (an alternative way 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]
(We'll ignore primary implicatures, mutually incompatible SIs, etc. for now)
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# Details (cont.)
(11) Paul is meeting with studentsx
(12) Paul is meeting with a studentx
Dynaic analysis:
c[(11)] = {⟨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 multiple students in w}
c[(12)] = {⟨f[x↦d], w⟩ | ⟨f, w⟩∈c, Paul is meeting with d in w and d is a single student in w}
The multiplicity inference is accounted for by subtracting c[(12)] from c[(11)].
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# Example: Negation
(13) Paul is not meeting with studentsx
(14) Paul is not meeting with a studentx
Negation kills discourse referents, so (13) and (14) are semantically completely identical.
⇒ No scalar implicature
c[¬φ] = {⟨f, w⟩∈c | there is no ⟨f', w⟩∈c[φ] s.t. f≤f'}
Fn: Double negation is a problem for dynamic theories but 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
(15) Exactly one of my students have journal papersx
(16) Exactly one of my students have a journal paperx
- (15) introduces a new discourse referent x, whose possible values are singular journal papers or plural journal papers.
- (16) introduces a discourse referent x, whose possible values are singular journal papers only.
As in the simple positive sentence, (15) is less informative than (16), 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]
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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).
(17) 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: (17) introduces a structured discourse referent, representing a mapping from my students to papers. Sentences like (17b) can access 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
(18) Every student of mine wrote papersx this term
(19) Every student of mine wrote a paperx this term
(18) 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 (19) 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
(20) Everyone speaks German or French
Crnic, Chemla & Fox (2015) ponit out that the prediction of the standard account for (20) is too strong.
(21) Everyone speaks German
(22) Everyone speaks French
(23) Everyone speaks German and French
(20) is compatible with (21), for example, as long as some speak French and some don't speak both.
The SI of (20) is taht someone speaks German; someone speaks French; and not everyone speaks both.
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# ∨ under ∀: Analysis
Crnic et al. uses embedded double exhaustification. Bar-Lev (2018) uses Innocent Inclusion.
(20) Everyone speaks German or Frenchx
(21) Everyone speaks Germanx
My account: (20) introduces a structured discourse referent mapping people to one of the two languages.
Alternative (21) 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 (21) with particular mappings. It doesn't entail that not everyone speaks German.
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# ∨ under ∀: prediction
Everyone speaks German or Frenchx
They all learned itx at school
Context 1
- Klaus speaks Gr natively, learned Fr at school
- Wataru speaks Gr, learned Gr school
- Paul speaks Fr natively, learned Gr at school
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Context 2
- Sophie speaks Fr natively, learned Gr at school
- Wataru speaks Gr, learned Gr 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. No need for additional mechanisms.
The account makes nice predictions about quantificational contexts (cf. quantificational subordination).
The present account derives the usual scalar implicatures (e.g. most ⤳ not all) (not shown today).
Further extensions:
- Modal subordination
- Primary implicatures (cf. Dieuleveu, Chemla, Spector 2019)
- Plural definites (Mayr 2015)
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class: center, middle
2
EMBEDDED
IMPLICATURES
UNDER INDEFINITES
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# Scalar Implicatures under Indefinites
When SI is triggered under an indefinite the global SI seems to be unavailable (Geurts 2009, Sudo 2016).
(24) There's a student that solved most of the problems
❌ No student solved all of the problems
✅ That student solved not all of the problems
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(25) At least one student solved most of the problems
❌ No student solved all
✅ The relevant student(s) solved not all
(26) They have furniture that most of us like
❌ They don't have furniture that all of us like
✅ That furniture is not liked by all of us
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# New and Old Variables
One way to obtain these results (Sudo 2016; Geurts 2006):
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
Novelty Condition: Indefinites must introduce a new variable.
- For any context c the assignemnts {f| ⟨f, w⟩ ∈c} have the same domain, dom(c)
- x is a new variable in c iff x ∈ dom(c)
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# Sequential Update with SIs
A scalar implicature is computed sequentially after the assertion, c[A][SI] (contrary to what we assumed earlier)
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(27) There's a studentx who solved most of the problems
(28) There's a studentx who solved all of the problems
Utterance of (27) in context c ➠ c[(27)][¬(28)]
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Suppose that the Novelty Condition is about the use of an indefinite (cf. Heim 1982) and it doesn't apply to alternatives.
- In (27), x needs to be a new variable.
- The Novelty Condition doesn't apply to x in the SI, ¬(28), so it can be anaphoric.
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# Example Computation
(29) There's a studentx who solved most of the problems
student(x) ∧ solvedMost(x)
(30) There's a studentx who solved all of the problems
student(x) ∧ solvedAll(x)
In c[(29)][¬(30)], only the first occurrence of x is new.
(See Sudo 2016 for a concrete implementation of dynamic generalized quantifiers)
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# Non-Sequential Reasoning
But sequential update is not necessary.
(31) There's a studentx who solved most of the problems
c[(31)] = {⟨f[x↦d], w⟩ | ⟨f,w⟩∈c, d is a student in w and d solved most or all of the problems in w}
(32) There's a studentx who solved all of the problems
c[(32)] = {⟨f[x↦d], w⟩ | ⟨f,w⟩∈c, d is a student in w and d solved all of the problems in w}
c[(31)]−c[(32)] = {⟨f[x↦d], w⟩ | ⟨f,w⟩∈c, d is a student in w and solved most but not all of the problems in w}
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# Two Ways to Achieve the Same Thing
1. Sequential update Process the negation of ψ after φ; Variables in ψ can have antecedents in φ.
c[φ][¬ψ]
2. Reasoning about hypothetical utterance: The effects that ψ could have brought about are excluded.
c[φ] − c[ψ]
For multiplicity inference and SIs under indefinites, either way is fine.
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class: center, middle
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 $1000. Hearing the contest is over, John says:
(1) Either someone got the question wrong, or someone is $1,000 richer.
(2) #Someone either got the question wrong, or is $1,000 richer.
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# Why This is a Puzzle
(33) Someone married John or someone married Sue.
(34) 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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Dekker (2001)
[...] indefinite noun phrases generally are or ought to be used with referential intentions, mediated by a speaker’s belief objects.
Dekker's idea: To utter 'Someone married John or Sue', the speaker has to have a particular referential intention about the witness of 'someone'.
But this fails to account for the fact that disjunction is a necessary ingredient here.
Cf. 'Someone became my in-law', in a context where John and Sue are my siblings.
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# Standard Reasoning
The standard pragmatic reasoning doesn't yield the results we want.
(34) 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 the same person to be candidate partners for John and Sue.
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# Sequential Reasoning
The only way out we can think of now is sequential reasoning about the same value of x.
(34) 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:
(33) 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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class: center, middle
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Conclusion
We need discourse referents for anaphora.
Then they should participate in pragmatic inferences (scalar implicatures, ignorance implicatures).
For some reason, this has been neglected, although both Gricean Pragmatics and discourse referents have been with us since the 1960s.
Multiplicity inferences and SIs under indefinites are given a straightforward account under this view.
Ignorance inferences under indefinites seem to require sequential reasoning for the same value of the variable.