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In Dependency Grammar we consider the meaning of a wordform either as a semantic predicate (:=predicate) or as a semantic name. Let us suppose we have a predicate, which has a predicate as argument (e.g. in 'He speaks Danish well' we have speaks=speaks(He,Danish) and well=well(speaks)). The questions are

a. How does this perspective agree with Predicate Calculus (where a predicate cannot be argument of another predicate)?

b. Has some suitable to this occasion Calculus been developed?

c. What about the negation operation in a Calculus as in (b)? What would happen in a case as P(~Q(x)), where P,Q predicates and x a term? Would it be equal to ~P?

ADDENDUM

At A. Polguere, I. Mel'cuk, Dependency in Linguistic Description, p.10 we read: "[...] the meaning of a sentence can be represented using the formalism of the predicate calculus. We say that an argument of a predicate semantically depends on this predicate; for P(a) we write P–sem→a. As I have said, an argument of a predicate P_1 can be another predicate P_2 with its own arguments [...]". (This book is just great as all Mel'cuk's works.) Here the author treats the idea of predicate-inside-predicate absolutely naturally. Furthermore he makes straight reference to predicate calulus!

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    @Lambie I am really trying to understand what do you mean by this comment.
    – SK_
    Commented Jul 3 at 16:20
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    @Lambie That's like saying "Probability is not medicine, it's from statistics" when someone asks what model best accounts for the risk factors of a certain disease. The question is about how to describe a certain linguistic phenomenon employing a mathematical tool.
    – Natalie Clarius
    Commented Jul 3 at 19:39
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    ... and not how predicate logic itself works, so it is not a question about logic.
    – Natalie Clarius
    Commented Jul 3 at 20:07
  • "an argument of a predicate P_1 can be another predicate P_2 with its own arguments" - Not in formal predicate calculus. In first-order calculus not anyway, but even with predicates of higher order, the argument would have to be a predicate P_1(P_2), and not an application of a predicate to arguments P_1(P_2(x)).
    – Natalie Clarius
    Commented Jul 5 at 11:06
  • @NatalieClarius This is exactly the root of my questioning; either a new Logic is rising or this fragment has false meaning (or no meaning at all).
    – SK_
    Commented Jul 5 at 11:16

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You are correct that predicates of predicates, in particular adverbs like "well", are not straightforwardly translated into (first-order) predicate calculus, as something like P(Q(x)) or P(Q) would simply not be a well-formed formula.

A way to solve this problem (and more) is so-called event calculus, where you amend the core predicate with an individual argument that stands for the event on which the action occurred, and the adverb can then be an additional property (predicate) of this event:

"John buttered the bread slowly"

= "There is an event which is a buttering event between John and the bread and which happend slowly"

∃e(Butter(j, b, e) ∧ Slow(e))

Your example could similarly be rephrased as

∃e(Speak(x,d,e) ∧ Well(e))

It's not great in this case because the speaking would be understood to be a permanent property of John rather than a temporarily occurring event, but you get the idea. One could apply a similar trick with a concept such as states instead of events that may be more suitable for this example.

With events added we're still in the realm of first order logic, so negation proceeds as usual (namely on formulas, not on predicates). It would be ~∃e(Speak(x,d,e) ∧ Well(e)) or ∃e(Speak(x,d,e) ∧ ~Well(e)), depending on which meaning is intended.

You can read more about it in chapter 11 here.

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  • Τhank you for your answer (+1). I would insist as such: do you think that the formalization of Dependency Semantics would be Event Calculus? No Higher Order like approach would formalize it? Additionally the negation of the last rephrase is as I can imagine (i.d. for all e (~Speak or ~Well))?
    – SK_
    Commented Jul 3 at 14:30
  • No, I don't see it particularly related to dependency grammar. If anything, this formalization maps more closely to a traditional tree structure, as in the the formalization syntactically the adverb predication is a sister to the verb predication rather than in each other's scope. I have to admit I haven't seen a formal semantics on the basis of dependency grammar yet.
    – Natalie Clarius
    Commented Jul 3 at 18:37
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    May well be that there is a first-order logic based compositional semantics for dependency grammar, I'm just not aware of one, and saying that introducing the notion of events on top of FOL isn't really something specifically relating to DG.
    – Natalie Clarius
    Commented Jul 3 at 21:08
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    My preferred DG analysis of the example in the question, though, would be to view both speaks and well as predicates: well (speaks). But then the two words together form a single complex predicate such that we have: speaks well (he, Danish). In the dependency structure, speaks and well form a catena,(i.e. a subtree) and as such they qualify as a concrete unit of dependency structures. Thus, I would advocate a formal semantics that combines catenae as opposed to constituents.
    – Tim Osborne
    Commented Jul 4 at 7:06
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    @TimOsborne About the first comment; I think I haven' t understood why the n-arity causes problem in development of a strict DG formalization. What do you mean by "Function...time"? About the second; I strongly prefer the view of maximal division of elements inside a sentence (in any Structure, so in SemS too and mainly) -even interventing the periphrastic wordforms. The reason is that this way the passage from meaning to sentence is more radical and complete. Given this approach we can choose then any accepted combination of our Structure's elements that helps our respective purposes.
    – SK_
    Commented Jul 4 at 20:39

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