How to motivate GTOP?

I posted a very similar question recently, but I think I was asking the wrong question. (Let me know if it would be better in the future to just post in the original thread. I wasn’t sure that it would garner attention if it was essentially filed as a reply under the “wrong” title).

What I asked before was for an example of a sentence with multiple verbs, because I thought this would motivate the purpose of the QEQ relationship between the TOP and the matrix verb. However, while the answer I was given (which I reworked to a simpler sentence) did help me get closer to understanding QEQs in general, it still did not convince me why there must be a QEQ between the TOP and the matrix verb.

The sentence I worked with was “The person believes a unicorn sneezes.” I drew out two scope-resolved trees for the two interpretations:

  1. “There exists a unicorn such that the person believes it sneezes.”

  2. “The person believes there to be a unicorn that sneezed, but the speaker doesn’t necessarily believe the unicorn exists” (Not 100% sure this is the interpretation, but the tree is still right).

Then what I tried to do was draw a scope tree that would violate the TOP =q believe constraint to motivate why it’s necessary:

However I got stuck because it appears to me that it would not be possible to draw such a tree without also violating believe =q sneeze. So why do we need the TOP=q constraint? Is there an example of a sentence that can motivate the need for the TOP=q constraint? That is, a sentence where, without it, you could draw more than one tree that fulfills all other QEQ constraints, but one of them would be for some unacceptable interpretation.

I have three guesses, both of which may be wrong, and I hope to be corrected:

  1. Perhaps the role of GTOP is as a pointer into the MRS, when someone manually or algorithmically picks it up to be used.
  2. It’s a kind of well-formedness constraint – just like a tree in a CFG has to have a single root (that matches the initial symbol), a well-formed MRS has to have a single GTOP.
  3. It has to do with the machinery of scoping. That is, without GTOP qeq ___, there would be no spot for the quantifiers to float into. (Though why they couldn’t just float above ___ is not clear to me).

Both of these are predicated on the idea that there are not in fact any cases where it’s possible to violate the qeq between GTOP and the relevant EP without alsoch ca violating some other qeq. If there are such cases, I’d be interested to know them!

We originally wanted GTOP as a location that could be the equivalence of the “top box” in DRT. I’m not sure whether or not there are potentially constraints that ought to be associated with GTOP given the various restrictions we’ve imposed on ourselves as to what gets done in the MRS, but at least in principle I think we should allow for richer semantics. In any case, having GTOP in the formalism is still a good idea even if there are no such constraints, because it allows us to use the single formal device of a qeq - otherwise we’d need a separate type of scoping relationship to express the idea that the quantifiers were floating “above” other things. All GTOP is doing is providing a way of talking about the root node in the tree so it doesn’t formally add significant complexity, it seems to me.

I just mechanically checked this example because I am also playing with quantifier scopes and Utool. Utool indeed presented three solutions for both versions of the input. One with the MRS without edition (with h0 qeq h1) and the other version with this constraint manually removed. So, yes, for Utool the extra constraint is redundant (in this example!)!