From Allen (1995)

Examples:

• A dog entered with every man
  1. 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)
  2. 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)
  3. 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)
  4. 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]

• We didn't see every dog
  1. for all dogs x, we didn't see x
  2. it is not true that for all dogs x, we saw x
• Everyone thought that Fido or Fifi would win
  1. For all x, x thought that [Fido would win or Fifi would win]
  2. Everyone thought that Fido would win or everyone thought that Fifi would win
• He will feed the hungriest dog tomorrow
  1. Take the hungriest dog x: go to tomorrow: he will feed x then
  2. Go to tomorrow: take the hungriest dog x: he will feed x then
• A fat dog always loses the race
  1. always: there is some fat dog x such that x loses
  2. there is some fat dog x such that x always loses the race

Three functions of NPs:

1. definite reference - the addressee should be able to identify the object/set (e.g. the, possessive determiners)
2. 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")
3. universal quantification - all, each, every, most

Two interpretations for NPs denoting sets:

1. collective - just one event in which all members participate as a group (e.g. The men met at the corner)
2. 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").

-- MarkMcConville - 16 Sep 2008