Processing Degree Operator Movement:Implications for the Semantics of Differentials

Micha Y. Breakstone*, Alexandre Cremers†, Danny Fox‡ and Martin Hackl‡

*HUJI and MIT; †SIGMA – ENS; ‡MIT

Contact: michab@mit.edu; alexandre.cremers@gmail.comExperimental Syntax and Semantics Lab: http://web.mit.edu/hackl/www/lab/

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Roadmap• Theoretical background: scope ambiguities with differentials• Two competing theories• A sentence processing study• Semantics of differentials in comparatives

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THEORETICAL BACKGROUNDScope Ambiguities

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Heim’s Ambiguity• exactly differentials give rise to scope ambiguities (Heim*):

John is required to be exactly 2" inches taller than Mary is.

• exactly reading: If Mary is 5 feet tall, the requirements will be met only if John is 5’2’’ tall. (“Narrow Scope”)

• at least reading: If Mary is 5 feet tall, the requirements will be met if John is at least 5’2’’ tall. (“Wide Scope”)

*Heim, Irene (2000): ‘Degree Operators and Scope’, Proceedings of SALT X, 40–64.

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Two Competing Theories

• Two competing theories of this ambiguity, that differ in what takes scope:

• [exactly 5 pages -er than…] Heim (2000)

• [exactly 5 pages] Oda (2008) 1, Beck (2010) 2

1 Oda, Toshiko (2008) ‘Degree constructions in Japanese’. Dissertations Collection for University of Connecticut.2 Beck, Sigrid (2010) ‘DegP Scope Revisited’. Ms., Universität Tübingen.

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Heim’s LFs

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- Basic Case- Narrow Scope

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Heim’s LFs - Wide Scope

[ [𝑇𝑃 ] ]𝑤 0=1 𝑖𝑓𝑓 max (𝜆𝑑 .𝑀𝑖𝑠𝑑 -𝑡𝑎𝑙𝑙 𝑖𝑛𝑤0 )+2′ ′

If Mary is 5’ tall, John must be at least 5’2’’ tall (at least reading)

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Oda / Beck LFs

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- Narrow Scope- Basic Case

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Oda / Beck LFs – 2 Wide Scopes

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High -er

Low -er

[ [𝑇𝑃 ] ]𝑤 0=1 𝑖𝑓𝑓 max (𝜆𝑑 .𝑀𝑖𝑠𝑑 -𝑡𝑎𝑙𝑙 𝑖𝑛𝑤0 )+2′ ′

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Distinguishing Oda/Beck from Heim• Under Heim only one LF for wide scope reading• Under Oda/Beck two possible LFs for wide scope reading• Assume we had a method of fixing wide scope for exactly

and asking what the syntactic position of the than-clause is:• If the location of DegP (er… than-clause) is always high Heim• If the location of DegP is ambiguous Oda/Beck

One might hope that the position of the DegP could have consequences for truth conditions. Based on Heim 2000 the differences would be subtle (pertaining to de dicto / de re). We will use a different strategy to distinguish the 2 theories but will return to the semantic distinction later 10

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Our Goals for this Talk• Present a real time sentence processing experiment designed

to tease apart these two structural analyses for exactly differentials• The results support Heim’s structure

• Propose a semantics for comparatives with exactly differentials that explains why Oda/Beck structures are not available

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REAL TIME PROCESSINGExperimental Design and Analysis of Results

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… in a Nutshell• Use real time sentence processing • Fix wide scope for exactly (by context) • Probe for location of than-clause (by ACE; following the logic

of HK&V*)

• The key result: exactly takes wide scope than clause is high

predicted by Heim’s LFs, left unexplained by Oda/Beck

*Hackl M., Koster-Moeller, J. & Varvoutis, J. (submitted): ‘Quantification and ACD: Evidence from Real Time Sentence Processing’. 13

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Forcing Wide Scope• How do we fix the scope for exactly? By context:

To become the all time champion, John was required to win exactly 3 more matches than Bill

• This context strongly disfavors an exactly reading• If you happen to win more than required, you’ll still be the all

time champion

• Wide scope is fixed for exactly14

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Fixing the Location of the than-clause

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John is required to be exactly 2" inches taller than Mary is d-tall / required to be d-tall

Oda/Beck: 2 options for wide scope Heim

• For Heim wide scope determines location of than-clause

• For Oda/Beck wide scope gives 2 options. One can fix the location of the than-clause in wide scope using non-local ACE

• For both theories, a than-clause containing non local ACE could only be in 1 place (to resolve antecedent containment)

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Probing for Location of [than…] in Real Time

To become the all time champion, John was required to win exactly 3 more matches than Bill was <required to win>• Heim: wide scope of exactly high attached than-clause• When ACE site is reached, there is only one available structure

(which also happens to license non-local ACE)• ACE site doesn’t trigger reanalysis no additional cost due to

reanalysis• Oda/Beck: wide scope doesn’t determine than-clause height• Two structures are still possible, only one of which licenses ACE• ACE site may trigger reanalysis possible increased processing

cost at ACE site

• Furthermore, HK&V observe that ACE will incur increased processing cost on the ACE site, unless a structure licensing ACE was previously established

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Experimental Paradigm

In order to become the all-time champion, the American athlete was required to win…

a. exactly 3 more matches than the British athlete didb. exactly 3 more matches than the British athlete was

c. a few more matches than the British athlete didd. a few more matches than the British athlete was

… and so practiced arduously for several months.

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The Base Line: A few• a few intuitively is not ambiguous (always ‘at least’ reading)

John was required to be a few inches taller than Mary• Assume non-local QR is more costly than local QR, then even with an

at least reading a few prefers narrow scope

• Formally:• [[a few –er]](P)(Q) = 1 iff max(Q) max(P)+ • John is required [[a few -er than Mary is <d-tall>]i [VP to be ti tall]]

• w[w Accw0 [max(d. J. d-tall in w) max(d. M. is d-tall) + ] 18

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PredictionsTo become the all-time champion, the American athlete was required to win…a. exactly 3 more matches than the British athlete did b. exactly 3 more matches than the British athlete wasc. a few more matches than the British athlete did d. a few more matches than the British athlete was

• Predictions for Heim: Resolution of non-local ACE in (b) should incur less of a processing cost than in (d)• exactly, in this context, triggers non-local movement facilitating ACE• a few yields semantically equivalent LFs so reanalysis is triggered only at ACE site

(local QR preferred)

• Prediction for Oda/Beck: exactly can move over require w/o pied-piping the than-clause, so there is no reason to predict a difference in processing cost

• Resolution of local ACE (a),(c) should incur the same difficulty (baseline)

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Predicted Interaction - Heim

Local ACE Non-Local ACE

Predicted Effect Associated with ACE Site

Ex-actly

RRTs

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Methods and Materials• 37 subjects from MIT BRL subject pool, native English speakers• Single word, self-paced reading, “moving window” paradigm• Response Times (RTs) measured between button presses for

each word• 4 conditions: • 2 equal-length differentials [“a few”/”exactly n”] x (non-)local ACE

• 44 target items (11 x 4 conditions) in a Latin square setup• 109 filler items• Each sentence followed by a comprehension question

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Analysis• Residual RTs (RRTs) calculated as difference between RT and a

linear model of average RT per subject per word length• Outliers trimmed: under 18 ms; over 2s; outside 2 SDs• Removed measurements for erroneous answers, and for last

word of each sentence• Removed subjects with total precision below 75% (only 1 of

the 37 subjects)• Effects analyzed using a contrast coded linear mixed effects

model

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Residual Reading Times

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Interaction after ACE site

Local ACE Non-Local ACE-50-45-40-35-30-25-20-15-10

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Mean Residual Reading Times(3rd slot following ACE site; interaction: p=0.001)

A FewExactly

RRTs

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Results• Mixed effects analysis (# of obs: 1336, groups: Item, 44; Subject, 36)

• Interaction of differential x ACE:• Mean over 3 words following ACE site: p=0.0279 (pMCMC)• 3rd word following ACE p=0.001 (pMCMC)• RRTs for [Exactly/Non-local] are lower than [A Few/Non-local]• RRTs for [Exactly/Local] are higher than [A Few/Local]

• We find that with local ACE, “a few” is easier than “exactly”, possibly due to the following preference (see: Romeo & Hardt*):

Scope should match size of ACE – wide scope is hard with small ACE

Ellipsis and the Structure of Discourse. Daniel Hardt and Maribel Romero. Proceedings of Sinn und Bedeutung VI, 2002.

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Conclusions• “exactly” differentials, but not “a few” differentials facilitate

wide scope ACE• This shows that the at least reading is not generated by the

following LF (otherwise we wouldn’t get facilitation):• [[exactly 2’’] … [require [… er-than]]]

• This is in line with Heim’s structure• Not predicted by Oda/Beck’s approach without some kind of

locality restriction, e.g.:• An exactly quantifier over degrees must receive narrowest

possible scope. The sister of exactly 2'' cannot dominate a distinct constituent of type t that dominates the trace of exactly 2''.

• Instead, we will pursue a compositional analysis of exactly differentials that predicts Heim’s structures

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COMPARATIVES WITH DIFFERENTIALSSemantics

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Theory of exactly differentials

• ] is a quantifier over degrees, if it raises alone out of a comparative it leaves behind a trace of type <dt,t>, hence the interpretation would still be narrow scope:

• [[exactly 2'']j [John is required [t<dt,t>,j -er than Mary is <d-tall>]i [VP to be td,i tall]]]

• Hence, the only way to receive wide scope under this spell-out of Heim’s theory is for the DegP+than-clause to pied-pipe

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Two Consequences• Exactly phrases are not scopally inert:

• Since all we did is change Oda/Beck’s semantics of the comparative from <d,<dt,<dt,t>>> to <dtt,<dt,<dt,t>>>, we predict there should be environments in which exactly phrases are not scopally inert. Specifically: any environment where the exactly phrase leaves a d-type trace and moves across an operator with which it is not scopally commutative. Evidence is mixed:

• Scopally active (at most reading): You’re allowed to miss exactly 3 classes• Scopally inert (unambiguous): In order to join the NBA, John is required

to be exactly 6 feet tall

• De dicto unavailable for wide scope readings• In our analysis the modal is always below than clause for at least reading de dicto for than-clause is ruled out

• Evidence seems to support this prediction but is quite involved

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De Dicto / De Re• For wide scope reading (ignoring ACE)

• Heim: modal always below than clause de dicto for than-clause is ruled out• Oda/Beck: modal may be above than clause optional de dicto for than-clause

• Forcing wide scope + de re (ok according to both)• Base line: forcing wide scope for 'exactly' without controlling for de-dicto/de-re

Speaker A: What do I need to do in order to pay no taxes at all? Do I have to earn less than 20K?Speaker B: Not quite. You are allowed to earn exactly $300 more than that.

• Now Speaker A: What do I need to do in order to pay no taxes at all? Do I have to earn less than last year's average?Speaker B: Not quite. You are allowed to earn exactly $300 more than last year's average.

• Forcing wide scope + de dicto (ruled out by Heim)• Base line: forcing de-dicto (without thinking about the scope of the degree phrase)

Speaker A: What do I need to do in order to pay no taxes at all? Do I have to earn less than average?Speaker B: Not quite. You are required to earn less than average plus $300.

• Speaker A: What do I need to do in order to pay no taxes at all. Do I have to earn less than average??? Speaker B: Not quite. You are allowed to earn exactly $300 more than average.

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Summary

• Can differentials take semantically detectible scope independently of the comparative? NO!• We investigated this question in a specific environment

with ‘at least’ contexts combined with ACE, using real time processing• Our results show that an at least reading is not

generated by the following LF:• [[exactly 2’’] … [require [… er-than]]]

• The question is why? • Our answer: we devised a compositional semantics

that does not give this LF an at least reading 31

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Thank You!

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Appendix

Main Effect on Differential

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Residual Reading Times

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Main Effect on diff: exactly harder

Local ACE Non-Local ACE-45-40-35-30-25-20-15-10

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Mean Residual Reading Times(2nd word following beginning of differential; main

effect for differential p=0.001; interaction com-pletely insignificant)

A FewExactly

RRTs

35• Main effect of differential (2 words after “a”/“exactly”):

• p =0.001 (pMCM) ; “Exactly” condition harder