It turns out that the solution is actually much simpler. First, we will introduce a very simple change to the syntax rule for a sentence to account for sentential Projectors (a part of speech which heads independent clauses in WSL):
S → P QP e AP
Next, we'll stop treating the specifier phrase separately, and account for it as a special case of an argument phrase:
S → P AP
AP → A AP | 0
A → QP R | QP e
We could also choose to treat the specifier clitic as a kind of Role, which would be slightly simpler, but this formulation better reflects my own psychological perception of what a specifier is (and thus presumably reflects the intuition of the fictional native speakers of WSL as well). Note that this allows a single clause to contain multiple specifiers, as well as putting them in arbitrary positions with respect to the other arguments; that situation is accounted for in the WSL Primer, which says that the semantics of multiple specifier phrases is the same as that of multiple conjoined specifiers, except that using multiple specifiers allows you to place them all in different quantifier scoping levels- the same as the interpretation for repeated roles.
The semantics for these bits of syntax is as follows:
[|S|] = [|P|]([|AP|])
[|AP|] = λx.[|A|](x) & [|AP|](x)
[|A: QP R|] = λx.[|QP|](λy. [|R|](x)(y))
[|A: QP e|] = λx.[|QP|](λy. y ⊆ x)
To summarize: the denotation of a sentence is the denotation of the argument phrase chain filtered through the denotation of the Projector; the denotation of an argument phrase is the denotation of the argument given an entity variable x conjoined with the denotation of the remaining argument phrase given x; and the denotation of an argument is the denotation of a relation on the shared entity variable x and the phrase-specifier entity variable z filtered through the denotation of the quantifier phrase (which for the moment is unchanged from last time).
The case of an argument containing a QP and specifier clitic instead of a QP and a Role just contains the explicit relation that the phrasal entity z is a subset of x. Note that this requires interpreting z and x not as representing single referents, but as sets of possible referents- an idea I introduced earlier in the series on semantics for a monocategorial language.
For now, we'll only deal with a single Projector: "ka", which indicates a simple declarative sentence. It's semantics are very simple:
[|P|] = G["ka"] = λy.∃x. y(x)
This just says that some x (which now refers to a set) exists, and its identity will be constrained by y (which is bound to the denotation of the argument phrase chain). We could build this into the interpretation of an S directly, but we will have to deal with the semantics of Projectors at some point, so we might as well start here.
We now have the ability to intersperse specifiers and other arguments in any order, with the members of the specifier set constrained by the specifier phrases, and the scope of all quantifiers corresponding exactly to their surface order!
Generalizing Quantifiers
The next step in building our model of WSL semantics is to improve the handling of quantifiers. As described in this article, the denotation of a Quantifier phrase will be represented not by a proposition in predicate logic, but by a set of sets of possible referents- all those sets of referents which contain the appropriate quantity of the type of referent identified by a given Noun phrase. This helps our model conform to the intuition that a bare noun or quantifier phrase does not correspond to a logical assertion- i.e., that a given entity exists- but merely to a possible entity itself. The semantics for any individual quantifier are given by a function which takes in the denotation of a Noun phrase and uses it to construct the appropriate set of sets. The new interpretation rule for QPs is as follows:[|QP|] = G[Q]([|N|])
And some examples of Quantifier semantics are as follows:
G["ves"] = λy. {x ⊆ U : y ⊆ x} ("every" or "all"; i.e, the set of all sets in the universe U that contain the entire set y)
G["jest"] = λy. {x ⊆ U : |y ∩ x| > 0} ("some"; i.e, the set of all sets in the universe that contain at least on element of y)
G["hiq"] = λy. {x ⊆ U : |y ∩ x| = 5} ("five"; i.e, the set of all sets in the universe that contain exactly some five elements of the set y)
This of course also requires a new formulation of the semantics for Noun phrases that produces the basis set of "referents of the right type". The new Noun rule is as follows:
[|N: n N|] = G[n] ∩ [|N|]
[|N: 0|] = U
Rather than binding a new entity variable which we assert to satisfy a given predicate, or to be a member of a given set (given by looking up the Noun n in the lexicon), we simply directly construct the intersection of all of the sets that are the denotations of individual Nouns.
Finally, we have to update the interpretation rules for arguments (again) to handle the new kind of denotation for QPs:
[|A: QP R|] = λx. {y : ∃z. z ∈ x & [|R|](z)(y)} ∈ [|QP|]
[|A: QP e|] = λx.∃y. y ∈ [|QP|] & y ⊆ x
This says that a normal argument containing a role marker asserts that the set of all entities y which satisfy a particular relation with some element of the specifier set x is in the denotation of the QP; or, that an argument containing the specifier clitic asserts that some element y of the the denotation of the QP is a subset of the specifier set x.
Unfortunately, we've just undone our progress in allowing for the correct quantifier scopes! I constructing a compositional semantics for quantifiers in terms of sets, we have thrown out the conventions for establishing variable scopes in predicate logic- because we eliminated the variables! In order to recover the proper quantifier scopes, we're going to have to find a way to take into account the constraints imposed in lower scopes while constructing the sets for higher-scoped quantifiers.
We'll take a look at that problem in the next post in the series.
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