From Allen (1995)
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)
Note: a continuum of implication:
- 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
Another involves using a hierarchy of grammatical relations:
- preposed dependent > surface subject > postposed adverbial > object
Proposed algorithms for quantifier scope disambiguation usually involve some combination of these two hierarchies.
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
But not always:
- The dogs that won each race are hungry
-- MarkMcConville - 16 Sep 2008