Translation of Negative Quantifiers from a Scope-Resolved MRS to First-Order Logic Expression

I am currently exploring the translation of MRS logical form (scope-specified) to first-order logic expression. In particular, I am not sure how to translate negative quantifiers. For example, consider the sentence

No dogs chase no cats.

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Let’s say we pick this scope:

The translation of no to first-order expression should be:
no(x, R(x), B(x)) → ∀x R(x) ⇒ ¬B(x)

So, the first-order translation of the sentence would be:
∀d dog(d) ⇒ ¬[∀c cat(c) ⇒ ¬chase (e, d, c)], which simplifies to:
∀d dog(d) ⇒ ∃c cat(c) ∧ chase (e, d, c)

This translation makes sense to me. However, a similar analysis does not seem to make sense when I consider the sentence

No dogs chase no cats and bark.

image

Let’s say we pick this scope:

So, the first-order translation of the sentence would be:
∀d dog(d) ⇒ ¬[∀c cat(c) ⇒ ¬(chase (e, d, c) ∧ bark(d))], which simplifies to:
∀d dog(d) ⇒ [∃c cat(c) ∧ chase (e, d, c) ∧ bark(d)]

This first-order expression says that the dog barks anyway, which is not what the original sentence asserts.

I also notice a similar thread but it does not seem to answer the question here. I think the sentences I provided here are ‘simple’ enough for a complete translation to first-order logic expression. Does anyone know where did things go wrong? Many thanks!

it’s not translatable

This is correct, for the scope tree you’ve given.

What the original sentence asserts depends on which scope reading you think the sentence should have.

I suspect you want “bark” to be outside the scope of “no(c)”. To allow this, “and” needs its own position in the scope tree, with its two children being QEQ to “chase” and “bark”. Then “no(c)” can appear between “and” and “chase”.

Thanks, this is an interesting example! Covering all kinds of coordination is actually very difficult, and you shouldn’t consider it a solved problem. There is plenty of discussion in the wiki, if you dare to look, e.g. VirtualCoordinationComposition · delph-in/docs Wiki · GitHub

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Should we consider the scope tree I provided for No dogs chase no cats and bark. an extra scope reading that the MRS allows but in fact an invalid one, or is it actually that the translation itself from the MRS meta-language to first-order logic as the object language is problematic? Personally I could not understand such a reading which asserts all dogs do bark from the sentence.

Trying to understand all the possible scopes of a sentence can be a fun but time-consuming game. Having played this game many times, I can suggest doing the following:

  1. Simplify the sentence to isolate the confusing part.
  2. Modify the sentence to make the intended scope more reasonable (or make other scopes less reasonable).
  3. Come up with a discourse which supports the intended scope.

So, firstly, the confusing part is having a quantifier in one part of the coordination which scopes over the whole coordination. We can get rid of the first “no”, as in the example below. In the reading where the remaining “no” scopes over “bark”, Fido did not bark.

Fido chased no cats and barked.

Secondly, how do we make this reading more reasonable? Well, if “no” takes wide scope, the chasing and barking events are coordinated together at the bottom of the scope tree, so we have a kind of combined event which “no” takes scope over. So we could suggest this, as in the example below.

Fido chased no cats and barked at the same time.

But if Fido isn’t chasing cats, why do we care about barking? So, thirdly, we can build a discourse context:

My dog Fido likes to chase cats, but I really wish Fido would bark at the same time, so that I know what’s happening. Just this morning, Fido chased three cats, but I had no idea, because Fido chased no cats and barked at the same time.

You might still think it sounds a bit unnatural, but hopefully you can see what the scope is supposed to be. If you want to find a better example, you could also go back to step 2 and try with other words where there is a more obvious connection between the two events, e.g.:

Kim passed no exams and got a certificate to show it.