(Chierchia 1995, Yoon 1996; Krifka 1996, Champollion et al. 2018)
---
# Uniqueness
For some speakers (1) seems to presuppose that every farmer who has a donkey has only one.
(1)
This can be avoided by using a mass noun, but we simply this complication today.
(Evans 1977, Kadmon 1989, Champollion et al. 2018)
---
# Determiners
In addition to contexts, determiners also affect the preference between the two readings.
--
- Universal quantifiers generally prefer the ∀-reading.
(5) Every/each farmer who owns a donkey loves it.
(6) All (the) farmers who own a donkey love it.
-
No and
some prefer the ∃-reading.
(7) No/Some farmers who own a donkey love it.
No/Some farmers who own a donkey/donkeys love ∃ of their donkeys.
These general patterns have been experimentally supported
(Yoon 1996, Geurts 2002, Foppolo 2008)
---
# Theories
Two theories about which determiners prefer which readings:
- Kanazawa (1994)
- Champollion, Bumford & Henderson (2018)
(and related theories, Champollion 2016, Yoon 1996, Krifka 1996)
We'll focus on Kanazawa today
(Appendix for Champollion et al.).
--
Kanazawa's Generalization (for monotonic determiners)
- ↑MON↑ and ↓MON↓ determiners only have ∃-readings (e.g. some, no)
- Default readings for ↑MON↓ and ↓MON↑ determiners (e.g. all, not all) are the ∀-reading.
- Pragmatic and other contextual effects may override the deafult reading.
---
# Kanazawa's Monotonicity Principle
Kanazawa further argues that the default reading is the one that preserves the (left) mononoticity of the determiner ('
Monotonicity Principle').
E.g.
every and
no are both left downward monotonic.
--
- Under the ∀-reading but not under the ∃-reading, (8)⟹(9).
(8) Every farmer who owns a donkey loves it.
(9) Every farmer who owns a female doneky loves it.
- Under the ∃-reading but not under the ∀-reading, (10)⟹(11).
(10) No farmer who owns a donkey loves it.
(11) No farmer who owns a female doneky loves it.
---
class: middle, center
# Donkeys in Non-Monotonic Contexts
---
# Kanazawa on Non-Monotonic Determiners
Kanazawa (1994) remarks that the Monotonicity Principle says nothing about determiners that are (left) non-monotonic, so no preferences are predicted for them.
E.g.
most and other proportional determiners are left non-monotonic. Kanazawa takes this prediction to be desirable, but empirical facts are not so clear.
--
However, Kanazawa concedes that 'existential non-monotonic determiners' like
exactly three are problematic, because they seem to favor the ∃-reading.
(12)
Exactly three farmers who own a donkey love it.
- Three donkey-owning farmers love ∃ of their donkeys;
- No other donkey-owning farmers love ∃ of their donkeys
---
# Continuity and Symmetry
Kanazawa suggests the preference fo the ∃-reading with
exactly three might be due to some other logical property, e.g.
-
Continuity
Q is left continuous iff for any A⊆B⊆C and for any D, whenever Q(A)(D) and Q(C)(D), Q(B)(D).
-
Symmetry
Q is symmetric iff for any A and B, Q(A)(B) ⟺ Q(B)(A)
The left continuity and symmetry of
exactly three are preserved under the ∃-reading, but not under the ∀-reading.
Kanazawa's proposal: At least one of these properties matters for donkey anaphora.
---
# Left Continuity
(13) |
a. |
Exactly three people who own a female cat love it
|
|
b. |
Exactly three people who own a pet love it |
|
c. |
Exactly three people who own a cat love it |
Under the ∀-reading:
- A, B, C, D own at least one cat, and love all their cats.
⟹ (13c) is false
- A, B, C own at least one female cat, D only owns male cats.
⟹ (13a) is true
- D also have some dogs, but doesn't love them.
⟹ (13b) is true
---
# Symmetry
Conservativity + Symmetry = Intersection
$$Q(A)(B) \text{ iff } Q(A\cap B)(\top)$$
(12)
Exactly three farmers who own a donkey love it.
Under the ∀-reading, (12) ≠ (12')
(12')
There are exactly three farmers who own and love a donkey.
---
# Motivations for Our Study
We can tease apart the predictions of the two possibities that Kanazawa (1994) discusses by comparing two non-monotonic determiners,
exactly three and
all but one.
|
L-Cont. |
Symm. |
exactly 3 |
Yes |
Yes |
all but one |
Yes |
No |
--
Predictions
- Both of them are left continuous, so if left continuity needs to be preserved, they should both prefer the ∃-reading.
-
Exactly three is symmetric but
all but one is not, so if symmetry needs to be preserved in donkey sentence,
exactly three should prefer the ∃-reading, while there should not be preferences for
all but one.
No previous experiments tested non-monotonic determiners.
---
# Main Findings
Findings
- ∃-reading available for
exactly three and
all but one
- Evidence for ∀-reading of
all but one but not of
exactly three
➥ Monotonicity is not the whole story (as Kanazawa says)
➥ Symmetry might be another factor
➥ Left continuity does not seem to matter
Further directions
Appendix 1: Challenges for Champollion et al. (2018)
Appendix 2: Experiment 3 on
all
- Difference between
all vs.
all but one
- No correlation with ∀-readings and subjective monotonicity/symmetry
---
class: middle, center
# Experiment 1
Exactly Three
---
# Experiment 1
- Truth-value judgment task |
- 6 Conditions x 6 items |
- Continuous scale |
- 65 participants on MTurk |

---
# Conditions
A non-monotonic quantifier has a downward- and upward-monotonic component:
E.g.
exactly three =
at most three (↓) +
at least three (↑)
Each of the ↓ and ↑ components could receive a ∃- or ∀-reading, giving rise to four logically possible readings:
(13)
Exactly three of the triangles that are above a star are connected to it
- ↑: Three △s above ☆s are connected to
∃/∀ of the ☆s
- ↓: no other △s above ☆s are connected to
∃/∀ of the ☆s
---
# Conditions (cont.)
These four readings stand in entailment relations:

At least one of these readings is true in the following scenarios:
1.
DEstrongUEstrong validates all four readings
2.
DEstrongUEweak validates only ↓∃↑∃ and ↓∀↑∃
3.
DEweakUEstrong validates only ↓∀↑∀ and ↓∀↑∃
4.
DEweakUEweak validates only ↓∀↑∃
---
Exactly 3 triangles/squares that are above a star/heart are connected to it.
1. DEstrongUEstrong |
2. DEweak-UEstrong |
 |
 |
3. DEstrong-UEweak |
4. DEweak-UEweak |
 |
 |
---
# False Conditions
Exactly 3 triangles/squares that are above a star/heart are connected to it.
5.a DEstrong-UEfalse |
5.b DEfalse-UEstrong |
 |
 |
---
# Detecting the Four Readings
|
↓∃↑∀ |
↓∃↑∃ |
↓∀↑∀ |
↓∀↑∃ |
1. DEstrongUEstrong |
T |
T |
T |
T |
2. DEstrongUEweak | F | T | F | T |
3. DEweakUEstrong | F | F | T | T |
4. DEweakUEweak | F | F | F | T |
5. FalseConditions | F | F | F | F |
--
Therefore:
- 4. DEweakUEweak > 5. FalseConditions ➠ evidence for ↓∀↑∃
- 3. DEweakUEstrong > 4. DEweakUEweak ➠ evidence for ↓∀↑∀
- 2. DEstrongUEweak > 4. DEweakUEweak ➠ evidence for ↓∃↑∃
---
count: false
# Detecting the Four Readings
|
↓∃↑∀ |
↓∃↑∃ |
↓∀↑∀ |
↓∀↑∃ |
1. DEstrongUEstrong |
T |
T |
T |
T |
2. DEstrongUEweak | F | T | F | T |
3. DEweakUEstrong | F | F | T | T |
4. DEweakUEweak | F | F | F | T |
5. FalseConditions | F | F | F | F |
Detecting the strongest reading ↓∃↑∀ is more complicated.
- 1. DEstrongUEstrong > 2. DEstrongUEweak, 3. DEstrongUEstrong > 4. DEweakUEstrong might obtain without ↓∃↑∀.
- Similarly, 1. DEstrongUEstrong > higher-of(2. DEstrongUEweak, 3. DEstrongUEstrong) might obtain without ↓∃↑∀.
---
count: false
# Detecting the Four Readings
|
↓∃↑∀ |
↓∃↑∃ |
↓∀↑∀ |
↓∀↑∃ |
1. DEstrongUEstrong |
T |
T |
T |
T |
2. DEstrongUEweak | F | T | F | T |
3. DEweakUEstrong | F | F | T | T |
4. DEweakUEweak | F | F | F | T |
5. FalseConditions | F | F | F | F |
Detecting the strongest reading ↓∃↑∀ is more complicated.
- For the participants who don't access ↓∃↑∃ or ↓∀↑∀: i.e.
2. DEstrongUEweak ≤ 4. DEweakUEweak; or
3. DEweakUEstrong ≤ 4. DEweakUEweak
- DEstrongUEstrong > the other of 2. and 3. ➠ evidence for ↓∃↑∀
---
# Results

- No evidence for ↓∀↑∃
(χ2(1)=0.06, p=.8)
- Evidence for ↓∃↑∃
(χ2(1)=100, p<.001)
- No evidence for ↓∀↑∀
(χ2(1)=0.02, p=.9)
- No evidence for ↓∃↑∀
(χ2(1)=0.01, p=.9)
---
count: false
# Results

-
No evidence for ↓∀↑∃ (χ2(1)=0.06, p=.8)
- Evidence for ↓∃↑∃
(χ2(1)=100, p<.001)
- No evidence for ↓∀↑∀
(χ2(1)=0.02, p=.9)
- No evidence for ↓∃↑∀
(χ2(1)=0.01, p=.9)
---
count: false
# Results

- No evidence for ↓∀↑∃
(χ2(1)=0.06, p=.8)
-
Evidence for ↓∃↑∃ (χ2(1)=100, p<.001)
- No evidence for ↓∀↑∀
(χ2(1)=0.02, p=.9)
- No evidence for ↓∃↑∀
(χ2(1)=0.01, p=.9)
---
count: false
# Results

- No evidence for ↓∀↑∃
(χ2(1)=0.06, p=.8)
- Evidence for ↓∃↑∃
(χ2(1)=100, p<.001)
-
No evidence for ↓∀↑∀ (χ2(1)=0.02, p=.9)
- No evidence for ↓∃↑∀
(χ2(1)=0.01, p=.9)
---
# Summary

- No evidence for the weakest reading ↓∀↑∃
- ↓∃↑∃ is available but no evidence for ↓∀↑∀,
- No evidence for the strongest reading, ↓∃↑∀, either. Even if it is accessible, it wouldn't be detected as ↓∃↑∃ is already at the ceiling.
This is consistent with Kanazawa's view that
exactly three favors the ∃-reading (↓∃↑∃).
---
class: center, middle
# Experiment 2:
All but One
---
# Experiment 2:
All but One
- Exactly the same visual stimuli as Experiment 1
- This was possible because there were always exactly 4 big figures (△ or ☐) that are above at least 1 small figure (☆ or ♡)

---
# Results

- No evidence for ↓∀↑∃
(χ2(1)=2.35, p=.12)
- Evidence for ↓∃↑∃
(χ2(1)=64.8, p<.001)
- Evidence for ↓∀↑∀
(χ2(1)=10.5, p=.001)
- Some evidence for ↓∃↑∀
(χ2(1)=3.62, p=.057)
---
count: false
# Results

-
No evidence for ↓∀↑∃ (χ2(1)=2.35, p=.12)
- Evidence for ↓∃↑∃
(χ2(1)=64.8, p<.001)
- Evidence for ↓∀↑∀
(χ2(1)=10.5, p=.001)
- Some evidence for ↓∃↑∀
(χ2(1)=3.62, p=.057)
---
count: false
# Results

- No evidence for ↓∀↑∃
(χ2(1)=2.35, p=.12)
-
Evidence for ↓∃↑∃ (χ2(1)=64.8, p<.001)
- Evidence for ↓∀↑∀
(χ2(1)=10.5, p=.001)
- Some evidence for ↓∃↑∀
(χ2(1)=3.62, p=.057)
---
count: false
# Results

- No evidence for ↓∀↑∃
(χ2(1)=2.35, p=.12)
- Evidence for ↓∃↑∃
(χ2(1)=64.8, p<.001)
-
Evidence for ↓∀↑∀ (χ2(1)=10.5, p=.001)
- Some evidence for ↓∃↑∀
(χ2(1)=3.62, p=.057)
---
# Summary

- No evidence for the weakest reading, ↓∀↑∃
- Evidence for both intermediate readings, ↓∃↑∃ and ↓∀↑∀
- ↓∃↑∃ is preferred over ↓∀↑∀
(χ2(1)=26.8, p<.001)
- Unclear about ↓∃↑∀. If it exists, nearly everyone who accessed it also assessed ↓∃↑∃ or ↓∀↑∀.
---
class: center, middle
Discussion
---
# Summary of the Results
Experiment 1 on exactly three
- No evidence for ↓∀↑∃
- Evidence for ↓∃↑∃, but not for ↓∀↑∀
- No evidence for ↓∃↑∀
Experiment 2 on all but one
- No evidence for ↓∀↑∃
- Evidence for both ↓∃↑∃ and ↓∀↑∀
- ↓∃↑∃ is preferred over ↓∀↑∀
- Unclear about ↓∃↑∀
---
# Conclusions
|
L-Cont. |
Symm. |
exactly 3 |
Yes |
Yes |
all but one |
Yes |
No |
The ∀-reading (↓∀↑∀) is available for
all but one but does not seem to be for
exactly three
- Monotonicity is not the only relevant logical property
- The difference can be explained by
symmetry.
The symmetry of
exactly three is preserved under the ∃-reading, not under the ∀-reading.
(14)
Exactly three farmers who own a donkey love it
-
Left continuity need not be preserved.
---
# Further Directions
The ∀-reading was not the preferred reading for
all but one.
This could be due to the task. But Experiment 3 shows that
all receives the ∀-reading more often.
Is it logical notions or subjective notions that matter?
(Chemla, Homer & Rothschild 2011)
Experiment 3 finds no correlation between the ∀-reading and Subject Monotonicity/Symmetry.
---
---
count: false
class: middle center
# Appendix: Champollion et al.
---
count: false
# The Conjunctive Reading
Champollion, Bumford & Henderson (2018) liken donkey anaphora to homogeneity
(earlier related works: Krifka 1996, Yoon 1994, 1996, Champolion 2016).
Putting the details aside, they claim that the default reading in each case of donkey anaphora is the reading that makes both ∀- and
∃-readings true.
(A1)
Every farmer who owns a donkey loves it
- Every donkey-owning farmer loves at least some of their donkeys; and
- Every donkey-owning farmer loves all of their donkeys
= ∀-reading
---
count: false
# The Conjunctive Reading
(A2)
No farmer who owns a donkey loves it
- No donkey-owning farmer loves at least any of their donkeys; and
- No donkey-owning farmer loves all of their donkeys
= ∃-reading
Indefinites are not quantifiers and always receive the ∃-reading.
Non-default readings are due to a QUD that collapses all non-false cases.
---
count: false
# Predictions for NM Quantifiers
For NM quantifiers, they claim that the default reading is ↓∃↑∀.
(A3)
Exactly three farmers who own a donkey love it
- Exactly three donkey-owning farmers love some of their donkeys; and
- Exactly three donkey-owning farmers love all of their donkeys
= ↓∃↑∀ reading
---
count: false
# Challenges
One lesson from our experiments is that it is not easy to detect ↓∃↑∀, because of the entailment patterns.
In our results:
- No evidence for ↓∃↑∀ of
exactly three
- Mild evidence for ↓∃↑∀ of
all but one
Challenges for Champollion et al.
- Provide evidence for this reading.
- Why is ↓∀↑∀ accessible for
all but one, but not for
exactly three
- Why is ↓∃↑∃ so easy to access with NM quantifiers (cf.
all in Experiment 3). Note that the QUD is constant in our experiment.
---
count: false
class: middle, center
# Appendix 2: Experiment 3
---
count: false
# Ideas for Experiment 3
Chemla, Homer & Rothschild (2011) argue that Subjective Monotonicity better predicts the distribution of NPIs than Logical Monotonicity.
Inspired by their work, we checked if the following two notions correlated with the ∀-reading for
all.
- Subjective monotonicity
- Subjective symmetry
We also compared the results for
all but one and the results for
all. Both of them are non-symmetric, but they have different monotonicity profiles.
---
count: false
# Two Tasks
Task 1 |
Task 2 |
Truth-Value Judgments |
Inference Judgments |
 |
 |
6 conditions x 6 items |
9 conditions x 6 items |
---
count: false
# Conditions
Since
all is monotonic, we only need to consider the ∃- and ∀-reading.
Since ∀-readig ⟹ ∃-reading, we have two types of conditions:
-
Strong Condition: both readings are true
-
Weak condition: only the ∃-reading is true
We also have False Condition
---
count: false
# Results: Truth-Value Judgments

∀-readig more robustly associated with
all (χ2(1)=13.8, p<.001)
---
count: false
# Results: Inferential Judgments
Subjective Monotonicity |
Subjective Symmetry |
 |
 |
No correlation
(χ2(1)=0.13, p=.7; χ2(1)=0.02, p=.9)