From the @guyemerson’s paper Linguists Who Use Probabilistic Models Love Them: Quantification in Functional Distributional Semantics - ACL Anthology, I thought that all scopes with quantifiers should have only one EP, the quantifier itself. But exploring the profiles under erg/tsdb/gold/ I found many cases a quantifier EP shares the same label with another non-quantifier EP.
Nearly every dog barked.
[ TOP: h0
INDEX: e2
RELS: < [ _nearly_x_deg<0:6> LBL: h4 ARG0: e5 ARG1: u6 ]
[ _every_q<7:12> LBL: h4 ARG0: x3 RSTR: h7 BODY: h8 ]
[ _dog_n_1<13:16> LBL: h9 ARG0: x3 ]
[ _bark_v_1<17:23> LBL: h1 ARG0: e2 ARG1: x3 ] >
HCONS: < h0 qeq h1 h7 qeq h9 > ]
[<EP object (h4:_nearly_x_deg(ARG0 e5, ARG1 u6)) at 4895099392>,
<EP object (h4:_every_q(ARG0 x3, RSTR h7, BODY h8)) at 4895099584>]
or
Not all those who wrote oppose the changes.
[ TOP: h0
INDEX: e2
RELS: < [ part_of<0:7> LBL: h4 ARG0: x3 ARG1: x5 ]
[ _all_q<0:7> LBL: h6 ARG0: x3 RSTR: h7 BODY: h8 ]
[ not_x_deg<0:7> LBL: h6 ARG0: e9 ARG1: u10 ]
[ generic_entity<8:13> LBL: h11 ARG0: x5 ]
[ _those_q_dem<8:13> LBL: h12 ARG0: x5 RSTR: h13 BODY: h14 ]
[ _write_v_to<18:23> LBL: h11 ARG0: e15 ARG1: x5 ARG2: i16 ]
[ _oppose_v_1<24:30> LBL: h1 ARG0: e2 ARG1: x3 ARG2: x17 ]
[ _the_q<31:34> LBL: h18 ARG0: x17 RSTR: h19 BODY: h20 ]
[ _change_n_of<35:42> LBL: h21 ARG0: x17 ARG1: i22 ] >
HCONS: < h0 qeq h1 h7 qeq h4 h13 qeq h11 h19 qeq h21 > ]
[<EP object (h6:_all_q(ARG0 x3, RSTR h7, BODY h8)) at 4895059680>,
<EP object (h6:not_x_deg(ARG0 e9, ARG1 u10)) at 4895059488>]
Are these MRS valid? If that is the case, the definition of scope tree given by @guyemerson does not apply to scope trees produced from MRS in general.
Each non-terminal node is a quantifier, with its bound variable in brackets. Its left child is its restriction, and its right child its body.
In both cases, the EP that shares the label with a quantifier is completely disconnected from the predicate-argument structure of the sentence. In my tentative to write an MRS to first-order logic translator, it seems that those cases need case-by-case analysis:
- not all = some
- nearly every = some
- about N = ?
- about a X = ?
- especially X = ?
…
I know… I know… we already hit the wall trying that in http://svn.delph-in.net/lkb/branches/fos/src/tproving/gq-to-fol.lisp! This is my first step towards a tentative of MRS to dependent types. Comments are welcome! From @AnnC I already expect the suggestion “forget logic… use DMRS…” 
In that paper, I didn’t discuss complex quantifiers of the kind you’ve mentioned. These have been discussed at length in the past, e.g.: https://github.com/delph-in/docs/wiki/TomarMrsWellformedness
Ann: The correct analysis of nearly every is that you have nearly taking scope over every, not over every dog
I think the consensus is that the current MRS representation isn’t completely satisfactory, but it’s good enough if you know what you want to do with those quantifiers. For example, EDS treats them as a special case:
To extend the account in my paper to deal with complex quantifiers, I would say that the predicates will compose to give an f_Q function (a function from [0,1] to [0,1]). I explain in the paper that every’s function maps [0,1) to 0 and {1} to 1. For not every, the function is simply flipped (which makes it truth-conditionally equivalent to some). For nearly every, the function should take a high value close to 1 (this gives a non-classical quantifer, with probabilistic behaviour).
Both not and nearly can also act as scopal modifiers. I don’t think this is a coincidence! But there is a different type of composition happening. I didn’t discuss scopal modifiers in my paper either, but I would say that they take a truth-conditional function and return a new truth-conditional function. Negation flips the probabilities, while nearly gives a function with high probabilities near the boundary of the input function.
When modifying a quantifier, they are doing the same thing, but applied to the f_Q function for the quantifier, rather than the truth-conditional function at that point in the scope tree.
So that’s my analysis – this is in agreement with the previous conclusion that we shouldn’t treat them as normal scopal modifiers… maybe we need some kind of new notation.