Examples:

- A dog entered with every man
- for each man x, there is some dog y and event z, such that z involves y entering with x (i.e. the group of men is distributive; one dog/event per man)
- there is some dog x and some event y, such that for each man z, y involves x entering with z (i.e. the group of men is collective; just one dog/event)
- there is some dog x, such that for each man y, there is some event z such that z involves x entering with y (i.e. the group of men is distributive; just one dog, but one event per man)
- there is some event x, such that for each man y, there is some dog z such that x involves z entering with y (i.e. the group of men is collective; just one event, but one dog per man)

i.e. six possible scope orderings, but only four are distinct - where 'dog' and 'enter' are adjacent, there is no contrast. [Therefore, we need a representation format which allows these four fully specified structures but no others. There should also be five underspecified possibilities in the description language:

dog/man/enter > dog/man enter > dog man enter [3] > [1] > dog/enter man [2] > man/enter dog > [1] > enter man dog [4] > dog man/enter > [3] > [2] > man dog/enter [1] > enter dog/man > [2] > [4]Note: this is a well-formed type hierarchy.]

- We didn't see every dog
- for all dogs x, we didn't see x
- it is not true that for all dogs x, we saw x

- Everyone thought that Fido or Fifi would win
- For all x, x thought that [Fido would win or Fifi would win]
- Everyone thought that Fido would win or everyone thought that Fifi would win

- He will feed the hungriest dog tomorrow
- Take the hungriest dog x: go to tomorrow: he will feed x then
- Go to tomorrow: take the hungriest dog x: he will feed x then

- A fat dog always loses the race
- always: there is some fat dog x such that x loses
- there is some fat dog x such that x always loses the race

Three functions of NPs:

- definite reference - the addressee should be able to identify the object/set (e.g. the, possessive determiners)
- existential quantification/indefinite reference - introduce new objects/sets into the discourse (e.g. a, some, several, many, a few, two, seven, no; test: "There are _ men who like golf")
- universal quantification - all, each, every, most

Two interpretations for NPs denoting sets:

- collective - just one event in which all members participate as a group (e.g. The men met at the corner)
- distributive - a separate event per member (e.g. each man bought a suit)

- each man bought a suit -- distributive
- every man bought a suit
- all the men bought a suit
- the men bought a suit -- strongly collective

The collectiveness/distributiveness of an NP affects how it interacts with other NPs - in "X lifted a piano", if the subject has a collective interpretation, then the object cannot take wide scope (e.g. "Together, the men lifted a piano").

Quantifier scope ambiguity resolution - determining the correct linear ordering (i.e. the 'outscopes' relation) on the quantified NPs in a sentence.

NP1 and NP2 are **horizontally related** iff. every S and NP which dominates NP1 also dominates NP2 and vice versa (i.e. NP1 and NP2 are dependents of the same head?).

NP1 and NP2 are **vertically related** iff. NP1 dominates NP2 and every other S or NP which dominates NP2 also dominates NP1 (i.e. NP1 is the head of NP2?).

One possible way of disambiguating any pair of horizontally related QNPs is by referring to a hierarchy of "quantifier strength", e.g.

- each > who > every > all/some/several/a

- preposed dependent > surface subject > postposed adverbial > object

For disambiguating pairs of vertically related QNPs, we can note that full relative clauses are generally **scope islands**:

- Some man rewarded a boy who gave each dog a bone

- The dogs that won each race are hungry

-- MarkMcConville - 16 Sep 2008

This topic: TFlex > WebHome > Proposals > DeepGRs > QuantScope

Topic revision: r4 - 18 Sep 2008 - 08:51:40 - MarkMcConville

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