A Progressive Model of WSL Syntax & Interpretation: Part 3

Last time we saw how to introduce generalized quantifiers and arbitrary specifier positions into our model of WSL; but, in the process, we lost any recognition of the scoping effects between quantifiers. Figuring out how scoping effects work for generalized quantifiers represented by sets-of-sets can get pretty complicated and confusing, so we're gonna really slow and step-by-step.

First, let's revise and review our model of the syntax so far. The complete syntactic model that I'll be using for the rest of this post is given by the following grammar:

S  → P AP
AP → RP AP | 0
RP → QP R | QP e
QP → Q NP
NP → N NP | 0

Where an S is a Sentence, an AP is a Argument Phrase, an RP is a Role Phrase, an R is a Role, a QP is a Quantifier Phrase, a Q is a Quantifier, an NP is a Noun Phrase, and N is a Noun.

Next, let's look at some examples to get a better grasp on how scope should work. Consider the English sentence

"Everybody loves somebody."

This has several possible translations into WSL; two of them are:

1) Ka ves anz i siru jest anz jo.

and

2) Ka jest anz jo siru ves anz i.

Sentence (1) states that, for every person, there is someone whom that person loves- but not every lover necessarily loves the same lov-ee. Sentence (2), on the other hand, states that there is some single person whom everybody else loves, because the existentially-quantified patient is now outside the scope of the universally quantified agent.

Additionally, sentence (1) allows that every agent might be participating in a totally separate instance of "loving", while sentence (2) indicates that there is only one instance of "loving" going on, and every person is a simultaneous agent in it. This is because of differences in the scope of the existentially-quantified specifier ("siru", with a phonologically-null quantifier). Moving "siru" to different positions produces more subtly-different interpretations; e.g., taking (2) and moving the specifier to the end would indicate that there is one person who is separately and independently loved by everyone- possibly at different times. And adding multiple conjoined specifiers would make things even more complicated.

Now let's take a look at how Roles get assigned to QPs inside Role Phrases, as of last time:

[|RP: QP R|] = λx. {y : ∃z. z ∈ x & [|R|](z)(y)} ∈ [|QP|]

What we're doing here is constructing the set of all things that bear a particular relation to some element of the specifier set, and then asserting that that is the same as one of the potential referent sets from the Quantifier Phrase. This implicitly imposes some constraints on the identity of the specifier set as well. A Role Phrase containing a specifier works a little differently:

[|RP: QP e|] = λx.∃Y. Y ∈ [|QP|] & Y ⊆ x

This just imposes an explicit constraint on the identity of the specifier set. (Note that I have chosen here to use an upper-case Y for the referent set in the rule for specifiers, to distinguish it from the lowercase y used for an element of the referent set for normal Role Phrases.)

In order to respect quantifier scoping, we need to arrange things so that we can reconstruct all lower-scope quantifier sets and select the relevant constraints independently for every element y of the referent set that we're constructing for the current RP.

In order to do that, we first have to make the denotations of all lower-scoped RPs available during the interpretation of any given RP. That means modifying our interpretation rules for APs as follows:

[|AP|] = λx.[|RP|](x)([|AP|])

such that the denotation of next-lower-scope Argument Phrase (which contains all the remaining Role Phrases) is filtered through the current Role Phrase as a parameter. That will, of course, require updating the rules for RPs to take multiple arguments, and doing the right thing with them:

[|RP: QP R|] =
        λx.λa. {y : ∃z. z ∈ x & [|R|](z)(y) & a(x)} ∈ [|QP|]
[|RP: QP e|] =
        λx.λa. ∃Y. Y ∈ [|QP|] & Y ⊆ x & a(x)

This places the evaluation of each Argument Phrase inside the scope of the variable y bound by the next higher Argument Phrase. Although y itself is not accessible in the lower scopes (we only pass along the shared specifier set as a parameter to a), this means that lower quantifiers are re-evaluated for every y. Thus, in "Ka ves anz i siru jest anz jo.", it is possible in the evaluation of "jest anz jo" to select a different existentially-quantified referent to correspond to every member of the universally-quantified referent set of "ves anz i". This is obvious in the semantics for specifiers, where we're still cheating a little bit by using the "∃" symbol to bind the variable Y and borrowing its scoping behavior.

The interpretation for the internal structure of a QP, containing a Q and an NP, remain unchanged. If we round things out with an explicit rule for null APs, we get the following completed model for the syntax-semantics interface:

[|S|]  = [|P|]([|AP|])

[|AP: RP AP|] = λx.[|RP|](x)([|AP|])
[|AP: 0|]     = λx. true

[|RP: QP R|] =
        λx.λa. {y : ∃z. z ∈ x & [|R|](z)(y) & a(x)} ∈ [|QP|]
[|RP: QP e|] =
        λx.λa. ∃Y. Y ∈ [|QP|] & Y ⊆ x & a(x)

[|QP|] = [|Q|]([|NP|])

[|NP: N NP|] = [|N|] ∩ [|NP|]
[|NP: 0|]    = U

[|P|] = G[P]
[|R|] = G[R]
[|Q|] = G[Q]
[|N|] = G[N]

This gets us the ability to model a pretty big chunk of all WSL declarative sentences. Still to come: non-intersective Nouns, modality, alternative Projectors, subordinate clauses, and controlling semantic projection.