Nominalised APs: λ-calculus and Movement into Argument Position

Walking to from University into town, I usually pass a shop with a big sign, saying, “Your Local Needs you.” It made think for a while, and this is what I came up with.

Nominalised APs – that is the adjectives having to behave as nouns – seem intriguing, at least in an elementary syntax/semantics interface. The syntactic structure of the infamous saying on the sign would be something like (1)

(1)

Nominalised APs seem to λ-skip a step when it comes to functional application (FA) . Normal Adjuncts – that APs are esentially identified with – encounter a paradigmatic functional application. However, I am not confident in distinguishing the FA mode with (i) specifiers, (ii) adjuncts, and (iii) complements.

The elementary and general principle of FA is as follows:

If α is a branching node and {β, γ} is the set of α’s daughters, then α is in the domain of [[  ]] if both β and γ are and [[ β ]]  is a function whose domain contains [[ γ ]]. In this case,α = [[ β ]] ( [[ γ ]] ). (Heim & Kratzer 1998: 49)

Should the AP not having to bear the nominal burden, the sign might have said:

(2) “A Your [NP local community] needs you,”

and the semantic denotation of the NP in (2) would be something like (3). For the sake of my argument I will temporarily replace the intensional and deictic possessive determiner with an indefinite one.

(3) [[ local community ]] ≈ λx[LOCAL(x) & COMMUNITY(x)]

In case of the elided NP, the denotation in (4) would involve a predicate absence.

(4) [[ local community ]] ≈ λx[LOCAL(x) & COMMUNITY(x)]

I would postulate, however that the absent predicate [NP community] ≈ λx.∃x[COMMUNITY(x)] encounters an overt movement into argument position, as shown in (5).

(5) {[[ local community ]] ≈ λx.∃x[LOCAL(x) & COMMUNITY(x)]} → {◊[[ local (community) ]] ≈ λx.∃x[LOCAL(x)] (community) = LOCAL(community)}.

The entire compositional account of (1) would therefore be something like (6), still retaining an indefinite determiner.

(6)

a [[ NP1 ]] ≈ λx.∃x[LOCAL(x)] (community) = LOCAL(community)
b [[ D ]] = λPλQ. [#(P Λ Q) ≥ 1] → ◊[λPλq[#(P (x))≥ 1] (given the theory of (5))
c [[ DP ]] = ◊[#(LOCAL(community)≥ 1]

This helps raise questions like, what are the conditions for Nominalised APs? Given that they are an unmarked linguistic (if not necessarily syntactic) phenomenon and as APs are embedded in NPs (i.e. their being nominalised), what is the functional application making nominal sense? I propose the following.

Provided we categorically juxtapose D(eterminer)s and N(oun)s, the N0 (heading the NP which the AP is embedded into), as shown in (7),

(7)

is seen to be in specifier position in overt determiner mode.  Although the noun is absent, it seems to determine the adjectivity nominally. The latter is structurally well comparable to a conventional D-spec position shown in (8).

(8)

Another question is whether the noun in [NP [...] local [N ∅]] is overt or absent or do both instances point to the same syntactic state. I shall firstly assume its being overt (i.e. overtly present) and secondly, its being syntactically absent.

Overtness Hypothesis & Underspecification Theory

Provided it is overt, there seems to be a semi-vacant place in NP2 position, where the brackets denote possibility of anything pragmatically-related.

(9)

(10)  [[ ((NP)) ]] ≈ ◊∀x [x ∈ <w,t>]

Such overtness hypothesis would have to involve an underspecification property of the super-NP (i.e. NP1). Examples such as “Your local needs you” may thus semantically-equally refer to community, school, authority, etc.

In relation terms, let us assume the following specification movement.

(11) [NP local __x__ ]

a. community
b. school
c. authority
etc.

Examples (a – c) may all equally well correspond to x-slot in (11). The notion of movement of an out-of-worldly (para-linguistically-dependent) N into an overt position within the super-NP, would have be based upon the assumption that the entire super-NP is underspecified and the N0 is overt, leaving its position available for (out-of-worldly) fill-in.

Syntactic Absence & Game Over Theory

As the latter attempt involved out-of-worldly importing of the x-slot filling, an other option would be much more hermenautic. We could siply assume there not being anything in N0 position rather than allowing everything. Since predicated restrict all elements of the universe and since predicate local (L) refers to everything local, i.e. {x: x ∈ L}, there not being another (and coordinated) predicate (such as community), there is less set-restriction overall. (11) thus refers to everything local as opposed to importing a secondary restriction such as [NP local community] which narrows down the set of elements. In this respect, the N0 is an empty set.

N0 being either a function or argument of [AP local] (discusse below!) the functional application does not bear fruition.

(12) λx.∃x [LOCAL(x)] (∅) = LOCAL (∅) = ∅ (for further discussion on the Cartesian product and function-being of empty sets, see Heim & Kratzer)

The empty-set-ness of the FA resul makes this theory crash: a similar ‘game over’ gave rise to Quantifier Raising theory. What would have to raise or lower here to make scientific sense of Nominal APs?

λ-calculus and Function-Argument Interchangebility

The phrase [NP local community] may have only three options of semanctic denotation.

(13) [NP local community]

a. LOCAL (community)
b. COMMUNITY (local)
c. λx.∃x [LOCAL(x) & COMMUNITY(x)]

The latter (c) is the most optimal but it breaches an axiom where the two branching nodes have to assume a function and argument position. Since (c) assumes both are predicated and thus their both being functions, there can be no functional application with NP merge.

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Copyright © 2008 Moreno Mitrovic