Research on Language and Computation (2005) 3:411–428
DOI 10.1007/s11168-006-0005-9
© Springer 2006
n-ary Quantifiers and the Expressive Power
of DP-Compositions
SEUNGHO NAM
Department of Linguistics, Seoul National University, Sillim-dong, Kwanak-gu,
Seoul 151-742, Republic of Korea (E-mail: [email protected])
Abstract. This paper, extending Keenan’s (1987, 1988) Case-extension of generalized
quantifiers, proposes a natural algebraic semantics of DP-coordination and DP-composition. DPs in subject and non-subject positions are uniformly identified as a case-extension of a usual generalized quantifier, and DPs with different semantic cases combine with
each other to yield polyadic quantifiers. The paper proves that each set of case-extensions
forms a complete atomic Boolean algebra consisting of a full set of unary quantifiers,
and it further shows that natural language requires the full power of binary quantification, i.e., the full set of (Fregean) reducible and unreducible binary quantifiers. This result,
Type < 2 > Effability, is derived from the fact that the set of all binary quantifiers can
be constructed by taking the meet/join closure of the set of (composed) reducible type
< 2 > quantifiers. This fact is illustrated with non-constituent coordination constructions
(e.g., gapping in English and Korean), whose interpretation requires arbitrary meets and
joins of reducible binary quantifiers.
Key words: expressive power, generalized quantifier, non-constituent coordination, polyadic
quantifier, reducibility.
0. Introduction
This paper addresses the following two related issues: (i) How to give a
unified semantics of non-subject DPs (determiner phrases) as well as subject DPs in the spirit of Generalized Quantifier Theory; and (ii) Can we
determine the expressive range of n-ary quantifiers (functions from n-ary
relations into truth-values) in natural language?
Regarding (i), this paper, based on type-polymorphism, generalizes the
case-extension approach of Keenan (1987, 1988). This forces us to a
somewhat more general than usual notion of n-ary relation – one motivated on grounds of linguistic analysis, specifically, the need to represent all
possible scope ambiguities without inducing argument ambiguities. On the
other hand, our polymorphic DP-type makes it possible to directly compose two DPs without invoking Type-lifting operations, as in Dowty (1988)
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and Steedman (1990). Equally, the case-extension approach leads us to certain observations concerning the uniform interpretation of DPs in English
(cf. (5)) as well as a natural way of representing quantifier scope ambiguities (cf. Proposition 1).
The second issue (ii) has been discussed by van Benthem (1989), Keenan
and Moss (1985), Keenan (1992) and Keenan and Westerståhl (1997).
Illustrating various types of unreducible binary quantifiers, they conclude
that the expressive power of English goes well beyond (Fregean) reducible
binary quantifiers. But they do not assess how far beyond. Here we argue
that the possibility of forming non-constituent coordinations requires, in
principle, all binary quantifiers (type 2 functions) as possible denotations
of natural language expressions over a finite universe.
The sort of non-constituent coordinations of concern here are those
illustrated by (1) from English and (2) from Korean, where we get a
conjunction (or disjunction) of binary quantifiers. In (1), (EVERY-BOY◦
A-NOVEL) and (EVERY-GIRL◦A-PLAY) are conjoined (or disjoined), and in (2),
(JOHN◦APPLE) and (MARY◦PEAR) are conjoined.
(1)
Every boy read a novel, and/or every girl a play.
(2)
John-i sakwa-lul, (kuliko) Mary-ka pay-lul Bill-hantey cwuessta
J.-Nom apple-Acc (and) M.-Nom pear-Acc B.-Dat gave
‘John gave an apple to Bill, and Mary gave a pear to Bill.’
We prove that such boolean compounds of n-ary quantifiers are not
(Fregean) reducible to iterated applications of unary generalized quantifiers. Specifically, limiting ourselves to the functions from binary relations to
truth values [R → 2], we show that conjunction and disjunction of binary
quantifiers provide English with the full expressive power on [R → 2] over
finite universe. (This paper uses the notation [A → B] to refer to the whole
set of functions with domain A and range included in B.)
1. Case-extensions of Generalized Quantifiers
Generalized quantifiers (GQs) are functions from properties into truth values [P → 2], i.e., sets of properties. In a simple intransitive sentence, a subject DP takes a one place predicate as an argument to give a truth value.
The theory of GQs can be extended to non-subject DPs in a sentence. For
example, in a transitive sentence, the object DP can be seen as a function
taking a two place predicate (P2 ) to give a one place predicate (P1 ), then
the subject takes the derived P1 to give a truth value (zero place predicate, P0 ). Fully generalizing this view, we can treat DPs polymorphically as
denoting functions from (n + 1)-ary predicates into n-ary predicates, that is,
we extend GQs to take (n + 1)-ary relations to give n-ary relations for all
non-negative integers n.1
EXPRESSIVE POWER OF DP-COMPOSITIONS
413
Adopting the notation of Keenan and Timberlake (1988), we might
sketch the interpretation of everyone loves someone as in (3). Here, we
assign DPs the “polymorphic” type Pn /Pn+1 , which reveals the idea that
a DP saturates one argument of a predicate, reducing its arity by one. An
expression of this type will denote a single function whose domain is the
set of all (n + 1)-ary relations for n ≥ 0.2
(3)
everyone
loves
someone
Pn/Pn+1
P2
Pn/Pn+1
P1: SOMEONE(LOVE)
P0: EVERYONE(SOMEONE(LOVE))
This representation, however, forces the first generalized quantifier that
applies to an n-place predicate to saturate its n-th argument. Thus SOMEONE in (3) saturates the second argument of LOVE and takes narrow
scope relative to EVERYONE. We need, obviously, a notation for allowing
scope assignment of DPs and the arguments they saturate to be independent. And while such representation is relatively unproblematic for binary
relations, the linguistic issues loom larger for n ≥ 3-ary relations as in (4),
containing a ternary relation:
(4) Every teacher sents a letter to exactly three students.
Traditionally, n-ary relations are treated as sets of ordered n-tuples of
elements of the universe E. Interpreting send in (4) as a set of triples
α, β, γ we see that there are three ways to derive a binary relation from
it according to which argument is saturated first: (i) saturating the Recipient argument first we derive {α, β| α sent β to exactly three students}.
(ii) Saturating the Theme argument first we derive {α, γ | α sent a letter
to γ }, and (iii) saturating the Agent argument first we derive {β, γ | every
teacher sent β to γ . The traditional treatment of relations which interprets
any binary relation as a set of ordered pairs, however, does not distinguish
such different types of binary relations derived from ternary ones.
Now, let us suppose we interpret (4) in such a way that EVERY TEACHER
has narrowest scope (cf. (iii)). Then, A LETTER is looking at a binary relation, but which argument is it to saturate? Intuitively, the first member
of the set of pairs. But that is not the argument it would saturate if
EXACTLY THREE STUDENTS had been interpreted with narrowest scope – then,
the remaining argument for A LETTER to saturate would be the second of
the remaining pairs (cf. (i)). Thus, intuitively what we want to be able to
do here, as linguists, is to remember which arguments of an n-ary predicate remain when some have been saturated. This is what motivates our
enriched notion of n-ary relations to be proposed shortly.
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Note that (i)–(iii) reflect different scope relations among the three argument DPs of send, i.e., (i) exactly three students gets interpreted as having
narrow scope, and in (ii) a letter does, and in (iii) every teacher does. This
suggests that if we allow the three DP functions to apply to SEND in any
possible order, we can interpret scope ambiguities in a straightforward way.
But we must be able to do this without altering the argument a DP saturates. This problem arises less dramatically if attention is restricted to just
one and two-place predicates, since saturation of one argument leaves little
choice for the remainder.
So to capture our linguistic intuitions concerning argument choice and
scope ambiguities, we shall in the first instance represent an n-ary sequence
of elements of a universe E as a function s from {1, 2, . . . , n} into E. Each
such s corresponds to a unique n-tuple s(1), s(2), . . . , s(n) and the map
sending each such s to that n-tuple is easily seen to be a bijection. So
n
[{1, 2, . . . , n} → E ] is isomorphic to E . And we shall represent an n-ary
n
relation over E not directly as a subset of E but rather as a subset of
[{1, 2, . . . , n} → E ], a set of functions with domain {1, 2, . . . , n} and range
included in E. This approach to n-ary relations is just a notational variant
of the standard representation. But it generalizes slightly in a way that is
useful to us. Namely, for each finite subset K of positive integers we define
a K-ary relation to be a set of functions from K into E. We write RK for
the set of K-ary relations over E. Formally,
DEFINITION 1. For all non-empty universes E, all finite sets K of
positive integers,
R ∈ RK iff R ⊆ [K → E].
If |K| = n we call R an n-ary relation over E.
We note that when |K| = |K | then RK and RK are isomorphic. Our interest in the (slight) generalization is that we will treat R{2,5} , for example, as
the set of binary relations derived from relations of arity 5 or greater by
saturating all but the second and fifth arguments. We will indulge in one
traditional notational simplification however. A unary relation (property)
p over E should be a set of functions from e.g. {1} into E. So for example the set {a, b, c} on this representation is {1, a, 1, b, 1, c}. But this
representation is uniquely recoverable from the set {a, b, c} by the function
that sends any subset X of E to {1, x|x ∈ X}. So we shall continue to note
subsets of E as usual, and P, the set of unary relations, is just the set of
subsets of E, as usual.
We think of determiner phrases initially as denoting functions
from properties to truth values, and we extend them in various ways to
take (n + 1)-ary relations to yield n-ary ones, according to the arguments
they saturate. Following Keenan (1987, 1988), we call this way of extension
EXPRESSIVE POWER OF DP-COMPOSITIONS
415
“case-extension,” since the Case of a DP (e.g., Nominative or Accusative,
etc.) determines the argument it affects semantically.
DEFINITION 2. For all f ∈ [P → 2], fi (i ≥ 1), or i-th case-extension of f ,
is defined as follows:
For all finite sets K of positive integers, ∀R ∈ RK ,
R,
if i ∈
/ K and
fi (R) =
{h ∈ [K − {i} → E]|f {α|h ∪ {i, α} ∈ R} = True}, otherwise.
DEFINITION 3. The set CXi of i-th case-extensions of generalized
quantifiers is defined by
CXi =df . {fi | f ∈ [P → 2]}.
As stated in Definition 2, given a generalized quantifier f and an (n + 1)-ary
relation R ∈ RK , if i ∈ K, fi (the i-th case-extension of f ) saturates the i-indexed
argument of R, to yield an n-ary relation which is a set of functions from
K −{i} into E, and thus a relation where the i-indexed argument has been
saturated. Thus suppose SEND is a set of functions from K = {1, 2, 3} into E,
and let F, G, H be DP denotations. Then, applying G2 to SEND, we get a binary
relation G2 (SEND) = {f ∈ [{1, 3} → E]|f (1) sent G to f (3)}.3 In this way, we
can express all the six possible scope readings without altering the argument
saturated by F1 , G2 , or H3 : F1 (G2 (H3 (SEND))), F1 (H3 (G2 (SEND))), G2 (F1 (H3
(SEND))), G2 (H3 (F1 (SEND))), H3 (F1 (G2 (SEND))), H3 (G2 (F1 (SEND))).
The set CXi of i-th case-extensions has a straightforward (if tediously
shown) but important property: it is a complete, atomic boolean algebra
(see the Case-extension Theorem in (Theorem 1)).4 Linguistically, in what
follows, we need this because we want to interpret conjunctions, disjunctions, and negations of DPs as boolean meets, joins and complements, as in
John saw [every teacher and exactly two students], and the intended denotation sets must be rich enough to guarantee this. We will see later that
the simple composition classes we use to interpret non-constituent coordinations in (1) and (2) fail to satisfy all the boolean conditions, and it is
the additional requirement of boolean closure that forces FULL expressive
power.
THEOREM 1. Case-extension
For all i, (CXi , ≤) is a complete atomic boolean algebra, where
for all f, g ∈ CXi , f ≤ g iff f (R) ⊆ g(R).
We include the proof of this theorem in the appendix, since while basically
straightforward, it is important to realize that the enriched notion of n-ary
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relation we are using does not induce any booleanly unexpected properties
(like being undefined at certain points, etc.).
The proof of Theorem 1 essentially shows that the case-extension
algebra CXi is closed under pointwise meets, joins, and complements. For
example, saw [every boy and some girl ] gets the same interpretation as saw
every boy and saw some girl, and love [not [every teacher]] gets interpreted
as not [love every teacher]. Further, we have the following:
COROLLARY 1. For all i = j, CXi and CXj are disjoint, i.e., CXi ∩ CXj =
φ, and CXi is isomorphic to CXj .
So, each CXi is isomorphic to [P → 2], the set of generalized quantifiers. This shows the fact that the i-th extensions of the set of GQs just
extends them to a larger domain, but does not really introduce any “new”
functions. The algebraic characterization of case-extensions, as shown in
Theorem 1, accounts for one basic universal:
(5) Universal: In natural language, two DPs interpreted by different Caseextensions cannot be conjoined.
Since every boolean algebra is closed under meets and joins, we can
interpret a conjunction of DPs with the same case as denoting their greatest lower bound in their case-extension algebra. However, we cannot interpret conjunctions of DPs which are interpreted in different case-extensions
CXi and CXj (i = j ), since boolean meets and joins of two elements lying
in different algebras do not make sense.
This accounts for the semantic unnaturalness of coordinations like he
and him, and I and her. Note that this semantic unnaturalness is not to
be confused with syntactic ill-formedness. A variety of languages – Nandi
(Nilo-Saharan), Malagasy (Malayo-Polynesian), Irish (Celtic) – present
surface coordinations of DPs in different morphological cases. Essentially
the first DP conjunct goes in the case required by the syntactic construction, and the remaining conjuncts fall into a default case (Accusative in
Irish and Nandi, Nominative in Malagasy). (6) below from Irish (McCloskey 1986:265) illustrates a case of default-case assignment in coordination.
(6) Chuaigh se-isean
agus
Went
he(Nom)-contr and
‘He and he went home.’
e-isean
‘na bhaile
him(Acc)-contr home
Thus whether he and him sounds odd in a language depends in part on the
mechanism assigning default morphological case. Our claim in (5) above,
however, is a claim about how expressions may be semantically interpreted,
regardless of their surface form. One is tempted to generalize (5) as follows:
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EXPRESSIVE POWER OF DP-COMPOSITIONS
(7) Speculation: For all type-preserving binary operators
natural language, [DP∗ DP ]i = DP∗i DPi .5
∗
on DPs in
That is, case-extension preserves the binary operation ∗ , so the speculation
does hold when ∗ is a Boolean operator: [DP and DP ]i = DPi and DPi .
2. Functional Composition and n-ary Quantifiers
We now consider the increase in expressive power forced by non-constituent
coordinations (e.g., gapping and right/left-node raising) illustrated by English
(8) and Korean (9) (or equivalently (1) and (2) mentioned earlier). It is
basically this increase which forces full finite type 2 expressive power.
(8) a. The teacher showed [every girl two plays], and [every boy three novels]
b. John loves Mary, and Harry Sue.
(9) a. John-i Mary-lul, (kuliko) Harry-ka Sue-lul salanghanta
J-Nom M-Acc (and) H-Nom S-Acc love
‘John loves Mary, and Harry Sue.’
b. John-i Mary-lul, (kuliko) Harry-ka Sue-lul Bill-hantey sokayhayssta
J-Nom M-Acc (and)
H-Nom S-Acc B-Dat
introduced
‘John introduced Mary to Bill, and Harry introduced Sue to Bill.’
Having interpreted DPs as functions with all (n + 1)-ary relations in
their domain, pairs of DPs like [every-girl two plays] or [John-i Mary-lul ]
are naturally interpreted as the composition of the elements of the pair,
yielding a function from (n + 2)-ary relations into the n-ary ones. Formally,
for example, JOHN1 ◦ MARY2 will be an element of the composition class
CX1 ◦ CX2 , defined in general by
DEFINITION 4. For all case-extension algebras CXi and CXj , CXi ◦CXj =df .
{f◦g|f ∈ CXi , g ∈ CXj and f◦g(R) = f (g(R))}.
Elements of CXi ◦ CXj are called basic composite functions of an i-th caseextension and a j -th case-extension.
One advantage of our polymorphic DP-type (Pn /Pn+1 ) is that we can
directly compose two DPs without Type-changing operations like Lifting
which is necessary for Dowty (1988) and Steedman’s (1990) account of
DP-compositions. For example, they lift two DPs in different ways to allow
them to be composed: thus, DP can be lifted to either (S/DP)/((S/DP)/DP)
or ((S/DP)/DP)/(((S/DP)/DP)/DP) or higher types (directionality of slash
ignored). We also note here that our DP-composition approach based on
case-extension is more linguistically motivated than Hendriks’ (1988) type
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raising mechanism. Hendriks applies type changing operations (i.e., Argument Raising or Lowering) to “verbs” so as to derive different scope relations
among DPs. For example, ditransitive verbs with three arguments like send
can be assigned six different functional types, each of which derives a unique
scope reading according to its order of the semantic saturation of the argument DPs. However, verbs in natural language do not indicate its possible
saturation orders or the scope-relations among arguments. Instead, DPs usually carry a mark to indicate their CASE, focal/topical features, etc. Further,
the semantic characteristics of DPs – e.g., monotonicity, (in)definiteness, etc.
– constrain their scope interaction with other scopal elements.
Let us also briefly compare our semantics with Oehrle’s (1987). In order
to interpret gapping structures in general, Oehrle first takes a transitive
verb like eat as a function f mapping the Cartesian product NP×NP to
truth values, i.e., eat denotes a function EAT such that EAT(np1 , np2 ) →
EAT(np2 )(np1 ). Then he extends this function f to f ∗ , which acts on
conjoined/disjoined/negated pairs of arguments, so the domain of f ∗ is the
boolean closure of [NP×NP], which he labels L[NP×NP]. Oehrle, without proof, takes the domain of f ∗ as a distributive lattice, and claims that
each f ∗ is a unique homomorphism preserving meets (∧), joins (∨), and
complements (–) in L[NP × NP]. That is, f ∗ (x ∧ y) = f ∗ (x) ∧ f ∗ (y), and
f ∗ (x ∨ y) = f ∗ (x) ∨ f ∗ (y), and f ∗ (−x) = −(f ∗ (x)). Oehrle (1987) only deals
with the gapping structures which contain individual-denoting NPs, and
he interprets the conjoined/disjoined NP-pairs as an argument of the verb.
Our semantics proposed here, however, deals with full range of boolean
compounds of pairs of generalized quantifiers, and each pair of generalized
quantifiers denotes a composed function, i.e., a binary quantifier.
Now we note two basic properties of the CXi ◦ CXj defined under
Definition 4: namely, Propositions 1 and 2.
PROPOSITION 1. Scope Dependency
For all i = j ,
a. CXi ◦CXj = CXj ◦CXi and
b. F ∈ CXi ◦ CXj ∩ CXj ◦CXi if ∃f, g such that F = fi ◦gj = gj ◦fi .
It is this which accounts in general for the possibility of generalized quantifier scope ambiguities. The proof of Proposition 1a given in
Appendix shows that the different composition orders of the universal and
existential quantifiers determine different type 2 functions (binary quantifiers): (EVERY-E)i ◦(SOME-E)j = (SOME-E)j ◦(EVERY-E)i , thus scope dependency is
just failure of commutativity. However, (EVERY-E)i ◦(SOME-E)j ∈ CXi ◦CXj −
CXj ◦CXi , whence the two composition classes are distinct.
EXPRESSIVE POWER OF DP-COMPOSITIONS
419
As stated in Proposition 2b, if F = fi ◦gj = gj ◦fi , there would not
arise a scope ambiguity between fi and gj , and F is in both CXi ◦ CXj
and CXj ◦ CXi . For instance, as proved by Zimmerman (1987), proper
names are scopeless, whence [JOHN1 ◦NO-STUDENT2 ] is the same function as
[NO-STUDENT2 ◦ JOHN1 ], and so shared by CX1 ◦ CX2 and CX2 ◦ CX1 .
PROPOSITION 2.
a. CXi ◦ CXj is closed under complements (since −(fi ◦ gj ) = −fi ◦ −gj ),
and
b. CXi ◦ CXj is not closed under pointwise meets or joins.
This last property is of our special interest, since the form of the
sentences in (8) and (9) suggests precisely that we must take meets and
joins of basic composite functions. Keenan (1992) points this out in
terms of Fregean reducibility of Type 2 functions. In what follows,
using this method of testing reducibility (Theorem 2), we prove that
such boolean compounds of DP-compositions are not interpretable by
basic composite functions in CXi ◦CXj (see also van Benthem’s (1989:446)
theorem). Further, we argue that non-constituent coordinations in natural language force FULL finite type 2 expressive power, i.e., Type 2
Effability.
Type 2 functions are ones from binary relations into truth-values, and
type 1 functions are ones from properties into truth values. We will use
these terms in broader sense to refer to functions from (n + 2)-ary relations to n-ary ones, and functions from (n + 1)-ary relations to n-ary ones,
respectively. Following Keenan (1992), we define,
DEFINITION 5. A type 2 function F is Fregean (= reducible) iff there
are type 1 functions f and g such that F = f ◦g.
In effect, with the generalized notion of ‘type 2’ functions, the set of
(Fregean) reducible functions from (n + 2)-ary relations into n-ary relations
is ∪i=j CXi ◦ CXj . Keenan (1992) uses the following theorem as a basis for
testing (Fregean) reducibility of type 2 functions.
THEOREM 2. Reducibility Equivalence (RE )
Let F, G be reducible functions of type 2. Then,
F = G iff for all subsets P , Q of E, F (P ×Q) = G(P ×Q).
This theorem says that the values of a type 2 reducible function on the
cross product relations (P ×Q) determine its values at any binary relation.6
Thus, to show a function F to be unreducible, we only have to find a
reducible function which gives the same values to all the cross product
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relations but which is not identical to F . We refer Keenan (1992) for a
variety of linguistic contexts where unreducible type 2 functions derive.
Specifically, he shows that the compositions of a universal quantifier and
an anaphoric DP-function (e.g., (EVERY-STUDENT)◦HIMSELF) are not (Fregean)
reducible.
As illustrated earlier in (8) and (9), non-constituent coordinations
involve conjunctions or disjunctions of DP-compositions, but such conjunction or disjunction is not guaranteed to be interpreted as a reducible type
2 function, i.e., a function in CXi ◦ CXj . For example, each of the composite functions in (8b) and (9), (JOHNnom ◦MARYacc ) and (HARRYnom ◦SUEacc ),
is reducible, i.e., they are in CXnom ◦CXacc , but we prove that their
conjunction F = (JOHNnom ◦MARYacc )∧(HARRYnom ◦SUEacc ) is not (Fregean)
reducible:
To prove this, we find a reducible function G = F such that for all product relations R, G(R) = F (R). Let G = (JOHNnom ∧ HARRYnom )◦(MARYacc ∧
SUEacc ) ∈ CXnom ◦CXacc . Then, G takes the same values at all cross product
relations R as F does. We see that F = G, since, for all binary relations
R, F (R) = True iff John and Harry bear the relation to Mary and Sue,
respectively, but G(R) = True iff each of John and Harry bears the relation to each of Mary and Sue. Therefore, by Theorem 2, F = (JOHNnom ◦
MARYacc ) ∧ (HARRYnom ◦ SUEacc ) is not reducible, i.e., F ∈
/ CXnom ◦ CXacc .
Equally, for suitable non-trivial choice of GIRL and PLAY etc., the function [EVERY-GIRLdat ◦ TWO-PLAYSacc ] ∧ [EVERY-BOYdat ◦ THREE-NOVELSacc ] needed
to interpret (8a) is unreducible.
Essentially, this proves Proposition (2), that is, CXi ◦ CXj is not
closed under meets and thus under joins since meets can be defined in
terms of joins and complements, and so forces us to extend the set of
possible denotations of DP-compositions. In other words, the nonconstituent coordinations force expressive power beyond the (Fregean)
reducible functions in CXi ◦ CXj . The following gapping examples reveal
that boolean compounds of composite functions can be obtained in
various ways:
(10) a. Neither does John love Mary, nor Fred Susan.
= John doesn’t love Mary, or Fred Susan.
b. Either John loves Mary, or Fred Susan.
For example, (10a) can be paraphrased as ‘John doesn’t love Mary, and
Fred doesn’t love Susan,’ and so we get a type 2 function, -(JOHN1 ◦
MARY2 ) ∧ -(FRED1 ◦ SUSAN2 ) which takes a binary relation to give a truth
value. To prove this function to be unreducible, we observe first from
Keenan (1992):
EXPRESSIVE POWER OF DP-COMPOSITIONS
421
(11) If F of type 2 is unreducible then so is its (boolean) complement
-F , its post-complement F -, and its dual -F -, where these notions are
defined as follows:7
For all functions F of type n, n ≥ 1, and R ⊆ E n , i.e., an n-ary relation,
a. -(F )(R) = -(F (R))
b. (F -)(R) = F (-R)
c. -F - = -(F -) = (-F )Observe also: (i) the post-complement operation distributes over meets:
((F ∧G)-)(R) = (F ∧G)(-R) = F (-R) ∧ G(-R) = (F -)(R) ∧ (G-)(R) =
(F - ∧ G-)(R), thus (F ∧ G)- = (F - ∧ G-). Dually (F ∨ G)- = (F -) ∨ (G-). And
(ii) individual denoting functions like JOHN1 and MARY2 are self-dual, i.e.,
john1 = -JOHN1 -. That is, John laughed if and only if it is not the case
that John didn’t laugh. Then the type 2 function derived from (10a) can
be reduced as the post-complement of (JOHN1 ◦ MARY2 ) ∧ (FRED1 ◦ SUSAN2 ) as
shown below:8
(12)
-(JOHN1 ◦ MARY2 ) ∧ -(FRED1 ◦ SUSAN2 )
= -(-JOHN1 - ◦ -MARY2 -) ∧ -(-FRED1 - ◦ -SUSAN2 -)
= (JOHN1 ◦ MARY2 )- ∧ (FRED1 ◦ SUSAN2 )= [(JOHN1 ◦ MARY2 ) ∧ (FRED1 ◦ SUSAN2 )]-
We have already shown that (JOHN1 ◦ MARY2 ) ∧ (FRED1 ◦ SUSAN2 ) is unreducible. Thus, due to (11), its post-complement we get from (10a) is also
unreducible, i.e., the function in (12) is not in CXi ◦ CXj . (10b) above gives
a disjunction of two composite functions, that is, (JOHN1 ◦ MARY2 ) ∨ (FRED1 ◦
SUSAN2 ). This is type 2 function and (13) shows it to be the dual of
(JOHN1 ◦ MARY2 ) ∧ (FRED1 ◦ SUSAN2 ), and thus unreducible.
(13)
(JOHN1 ◦ MARY2 ) ∨ (FRED1 ◦ SUSAN2 )
= -[-(JOHN1 ◦ MARY2 ) ∧ -(FRED1 ◦ SUSAN2 )] De Morgan Law
= -[(JOHN1 ◦ MARY2 ) ∧ (FRED1 ◦ SUSAN2 )] from (11) above
This completes the proof of Proposition 2: CXi ◦CXj is not closed under
meets or joins. The following examples further illustrate that we need a
much extended set of type 2 functions to give proper interpretations to
English expressions.
(14) a. Either John interviewed Mary, and Fred Susan, or else John Susan,
and Fred Mary.
b. John interviewed Mary, and Fred Susan, but neither John Susan,
nor Fred Mary.
Let α = (JOHN1 ◦ MARY2 ), β = (FRED1 ◦ SUSAN2 ), γ = (JOHN1 ◦ SUSAN2 ), and
δ = (FRED1 ◦ MARY2 ), then from (14a and b) above, we get the following
422
S. NAM
boolean compounds of type 2 functions: (a) (α ∧ β) ∨ (γ ∧ δ), (b) (α ∨ β) ∧
(γ ∨ δ), (c) (α ∧ β) ∧ (-γ ∧ -δ).
These complex type 2 functions suggest that we need to extend the
possible denotations of DP-compositions to the closure of basic composite
functions CXi ◦CXj under pointwise meets and joins, noted CF(CXi ◦CXj )
in Definition 6. Further, from Proposition 3, we see that CF(CXi ◦CXj ) is
closed under complements, too.
DEFINITION 6. For all CXi and CXj , CF(CXi ◦CXj ), or the set of
composite functions of CXi and CXj , is the closure of CXi ◦CXj under
arbitrary meets and joins pointwise.
PROPOSITION 3. CF(CXi ◦ CXj ) is closed under complements.
It follows then that for all F, G ∈ CF(CXi ◦ CXj ), (F ∧ G)(R) = F (R) ∩
G(R), and (F ∨ G)(R) = F (R) ∪ G(R), and (-F )(R) = -(F (R)). And one
shows:
THEOREM 3. Composite Algebra
(CF(CXi ◦ CXj ), ≤) is a complete atomic boolean algebra, where for all
F, G ∈ CF(CXi ◦ CXj ), F ≤ G iff for all R ∈ RK (|K| ≥ 1), F (R) ⊆ G(R).
For example, the zero and the unit elements of the algebra are given as
follows:
∀R ∈ RK ,
R,
if i ∈
/ K or j ∈
/ K and
(a) 0CF (R) =
Ø,
otherwise.
R,
if i ∈
/ K or j ∈
/ K and
(b) 1CF (R) =
[K−{i, j } → E],
otherwise.
In effect, 0CF = 0CXi ◦ 0CXj , and 1CF = 1CXi ◦ 1CXj . Thus they are in
CF(CXi ◦CXj ), and ∀F ∈ CF(CXi ◦ CXj ), 0CF (R) ⊆ F (R), and F (R) ⊆
1CF (R).
From Definition 6, we have that, for all F, G ∈ CF(CXi ◦ CXj ), the greatest lower bound and the least upper bound of {F, G} are (F ∧ G) and
(F ∨ G), respectively. And, ∀F ∈ CF(CXi ◦ CXj ), hF , the complement of F ,
is given by: for all R ∈ RK ,
R,
if i ∈
/ K or j ∈
/ Kand
hF (R) =
[K−{i, j }→E] − F (R),
otherwise.
Then (hF ∧)(R) = 0CF (R), and (hF ∨ F )(R) = 1CF (R).
EXPRESSIVE POWER OF DP-COMPOSITIONS
423
To establish the major result of this section – Type 2 effability
(Theorem 4), first, we restrict the domain of type 2 functions F ∈
CF(CX1 ◦ CX2 ) to the set of binary relations, R{1,2} , and characterize the
atoms of CF(CX1 ◦ CX2 ) with the restricted domain.
As we saw above, CF(CX1 ◦CX2 ) is a boolean algebra, and the atoms of
the algebra can be given as follows: define, for all R ∈ R{1,2} ,
FR = df . (∧f ∈R (If (1) ) ◦ (If (2) )) ∧ -(∨g∈R
/ (Ig(1) ) ◦ (Ig(2) ))
= (∧f ∈R (If (1) ) ◦ (If (2) )) ∧ (∧g∈R
/ -(Ig(1) ) ◦ (Ig(2) )),
where for all b ∈ E, Ib is an individual (ultrafilter) in (℘ (E), ≤) generated by
the property {b} ∈ ℘ (E). Thus, for all S ∈ R{1,2} , FR (S) = True iff S = R.
One sees easily that FR is a minimal non-zero element, that is, an atom:
It is not zero since FR (R) = True; and since it holds of just one relation,
no other functions but zero can be strictly less than it. And if G is a
non-zero function in CF(CX1 ◦ CX2 ) – so G holds of some R, then FR ≤ G.
Therefore, every such G dominates an atom, and so the algebra is atomic.
Finally, we have:
THEOREM 4. Type 2 Effability
Let R{1,2} be the set of binary relations, i.e., ℘[{1, 2}→E].
Then, F ∈ [R{1,2} →2] iff F ∈ CF(CX1 ◦ CX2 ) with the restricted domain R{1,2} .
Proof (i) Trivially, from the definition of CF(CX1 ◦CX2 ), for all F ∈
CF(CX1 ◦CX2 ) with the domain restricted to R{1,2} , F ∈ [R{1,2} →2].
(ii) Now, for R∈R{1,2} , define FR as follows: FR (S) = True iff S = R.
Thus, FR = (∧f ∈R (If (1) )◦(If (2) )) ∧ -(∨g∈R
/ (Ig(1) )◦(Ig(2) ) = (∧f ∈R (If (1) )◦(If (2) )) ∧
(∧g∈R
/ -(Ig(1) ) ◦ (Ig(2) )). (The same definition of FR is given on the previous page.) Since f and g above are functions in [{1, 2} → E], both (If (1) ) ◦
(If (2) ) and -(Ig(1) ) ◦ (Ig(2) ) are in CX1 ◦CX2 . Thus, each R ∈ R{1,2} , FR is
in CF(CX1 ◦CX2 ), since it is a boolean function of basic composite functions of the form (Ia ◦Ib ). Now, for all F ∈ [R{1,2} →2], F = ∨R,F (R)=1 FR ,
thus any type 2 function F can be represented as joins of meets of the
reducible functions in CX1 ◦ CX2 . Therefore, for all type 2 functions
F ∈ [R{1,2} →2], F ∈CF(CX1 ◦CX2 ).
This proves that CF(CX1 ◦CX2 ), the closure of (CX1 ◦CX2 ) under meets
and joins, contains all the functions from binary relations into truth values. In other words, conjunction and disjunction in gapping sentences in
English provide the FULL expressive power on (un)reducible type 2 functions from binary relations into truth values over finite universe E, since
joins of atoms can be denoted by finite disjunctions.
424
S. NAM
3. Concluding Remarks
The first half of this paper argues that case-extensions of GQs are
motivated by linguistic intuitions concerning grammatical relations and
scope ambiguities between quantifiers. Further, proving that each set of
case-extensions form a complete atomic boolean algebra, we saw that
the algebraic semantics proposed here accounts for the universal (5):
Two DPs interpreted by different Case-extensions cannot be conjoined.
Type-theoretically, our polymorphic DP-type Pn /Pn+1 allows direct composition of DPs without invoking Type-lifting operations.
In section 2, we built up Composition algebras (Definition 6, Proposition 3, and Theorem 3) from the Case-extension algebras (Definition 2 and
3, Theorem 1) by taking the meet/join closure of the set of reducible type 2
functions. This is shown to include all type 2 functions and is needed as
the denotation set for the non-constituent coordinations exhibited in the text.
Thus such boolean compounds tell us that all type 2 functions over a finite
universe are needed as possible denotations of natural language expressions.
Acknowledgements
I would like to thank Edward Keenan for comments on earlier versions of
this article. I also thank anonymous reviewers for their critiques. All errors
are my own.
Appendix
Proof of Theorem 1. Case-extension Theorem.
For all i, (CXi , ≤) is a complete atomic algebra, where
for all f, g ∈ CXi , f ≤ g iff f (R) ⊆ g(R).
Proof.
(i) Since the relation (≤) is defined in terms of subset relation (⊆), it is a
partial order, i.e., reflexive, antisymmetric, and transitive.
(ii) The zero (0CXi ) and the unit (1CXi ) elements: Define, for all K, and
∀R ∈RK ,
Ø,
if i ∈ K and
(a) 0CXi (R) =
R,
otherwise.
[K−{i}→E],
if i ∈ K and
(b) 1CXi (R) =
R,
otherwise.
We show the functions, 0 and 1 given as the above, are the zero and the
unit element of CXi : ∀f ∈ CXi , ∀R ∈ RK , if i ∈
/ K, f (R) = R, thus f (R) =
EXPRESSIVE POWER OF DP-COMPOSITIONS
425
0(R) = R; and if i ∈ K, 0(R) = Ø ⊆ f (R). Therefore, ∀f ∈ CXi , 0 ≤ f . Dually,
∀f ∈ CXi , f ≤ 1.
(iii) We show for all F, G ∈ CXi , {F, G} has a greatest lower bound and a
least upper bound: Let F = fi , and G = gi for f, g ∈ [P→2]. Define hf,g as
follows: for all R ∈ RK ,
fi (R) ∩ gi (R),
if i ∈ K and
hf,g (R) =
R,
otherwise.
Then, we show hf,g is the greatest lower bound of {fi , gi }:
First, we show hf,g ∈ CXi . If i ∈
/ K, hf,g (R) = R = (f ∧ g)i (R); and
if i ∈ K, (f ∧ g)i (R) = {h ∈ [K−{i}→E]|(f ∧ g){α|h ∪ {i, α} ∈ R} = 1}
(by Definition 2)
= {h ∈ [K−{i}→E]|f {α|h ∪ {i, α} ∈ R} ∧ g{α|h ∪ {i, α} ∈ R} = 1}
= {h ∈ [K−{i}→E]|f {α|h ∪ {i, α} ∈ R} = 1} ∩ {h ∈ [K−{i}→E]|g{α|h ∪
{i, α} ∈ R} = 1}
= fi (R) ∩ gi (R). Thus, hf,g = (f ∧ g)i ∈ CXi .
Obviously, for all R ∈ RK , if i ∈ K, hf,g (R) ⊆ fi (R) and hf,g (R) ⊆ gi (R);
and if i ∈
/ K, hf,g (R) = fi (R) = R. Thus hf,g is a lower bound of {fi , gi }.
Now, for all lower bounds w for {fi , gi }, w(R) ⊆ fi (R), and w(R) ⊆ gi (R),
thus w(R) ⊆ fi (R) ∩ gi (R) = hf,g (R). Therefore hf,g is the greatest lower
bound of {fi , gi }.
Dually, we prove that (f ∨ g)i is the least upper bound of {fi , gi }.
(iv) Now we show CXi is complemented: For all fi ∈ CXi , and R ∈ RK ,
define hf i as follows:
R,
if i ∈
/ K and
hf i (R) =
[K−{i}→E] − fi (R),
otherwise.
To show hf i is the complement of fi , we show hf i ∈ CXi , and (hf i ∧ fi ) =
0CXi , and (hf i ∨ fi ) = 1CXi . To show hf i ∈ CXi , we show hf i = (-f )i : From
the definition of Case-extension in Definition 2, for all R ∈ RK ,
if i ∈
/ K, (-f )i (R) = R = hf i (R); and
if i ∈ K, (-f )i (R) = {h ∈ [K−{i}→E]|(-f ){α|h ∪ {i, α} ∈ R} = True}
= {h ∈ [K−{i}→E]|f {α|h ∪ {i, α} ∈ R} = F alse}
= [K−{i}→E] − fi (R) = hf i (R).
Thus, hf i = (-f )i ∈ CXi .
Now, for all R ∈ RK ,
if i ∈ K, (hf i ∧ fi )(R) = hf i (R) ∩ fi (R)
= ([K−{i}→E] − fi (R)) ∩ fi (R) = φ = 0CXi (R); and
if i ∈
/ K, (hf i ∧ fi )(R) = hf i (R) ∩ fi (R) = R ∩ R = R = 0CXi (R).
426
S. NAM
Thus, (hf i ∧ fi ) = 0CXi . Dually, (hf i ∨ fi ) = 1CXi . Therefore, CXi is
complemented.
From (i) to (iv) above, CXi is a boolean algebra.
To show CXi is atomic and complete, first we prove that the
Case-extension map f ⇒ fi defined in Definition 2 is an isomorphism
from [P→2] to CXi : we have already shown in (iii) and (iv) above
that Case-extension preserves meets, joins, and complements, i.e., ∀f, g ∈
[P→2], (f ∧ g)i = fi ∧ gi , (f ∨ g)i = fi ∨ gi , and (-f )i = -(fi ), thus Caseextension is a homomorphism from [P→2] to CXi . And from the definition
of Case-extension, we have that it is a surjection from [P→2] to CXi : that is,
for all g ∈ CXi , there is a function f ∈ [P→2] such that fi = g. Now we show
that Case-extension is an injection from [P→2] into CXi : For all f = g ∈
[P→2], ∃p ∈ P, such that, without loss of generality, f (p) = 1 and g(p) = 0.
Set ψ ∈ RK where K = {i}, i.e., ψ ⊆ [{i}→K], such that ψ = {i, α|α ∈ p}.
Then, from (Definition 2), fi (ψ) = {h ∈ [Ø→E]|f {α|h ∪ {i, α} ∈ ψ} =
True} = {Ø}, but gi (ψ) = {h ∈ [Ø→E]| g{α|h ∪ {i, α} ∈ ψ} = True} = Ø. Thus,
for all f = g ∈ [P→2], fi = gi ∈ CXi , and so Case-extension is an injection.
Now by the following Lemma, we see that CXi is a complete and
atomic boolean algebra.
LEMMA. Let B = B, ≤B be isomorphic to D = D, ≤D , for B,D boolean
algebras. Then,
(a) if B is atomic then so is D, and
(b) if B is complete then so is D.
Here, the atoms of CXi are i-th case-extensions of the atoms in [P→2],
thus, for all atoms Fp in [P→2], the isomorphic images (Fp )i in CXi are
given by
Fp = (∧a∈p Ia ) ∧ -(∨b∈p
/ Ib ); and (Fp )i = (∧a∈p (Ia )i ) ∧ -(∨b∈p
/ (Ib )i ).
Where for all b ∈ E, Ib is an individual (ultrafilter) in (℘ (E), ≤) generated by
the property {b} ∈ ℘ (E). Further, for all F ⊆ CXi . the greatest lower bound
(∧F ) and the least upper bound (∨F ) of F are given by:
∧F (R) = ∩{f (R)|f ∈ F }; and ∨F (R) = ∪{f (R)|f ∈ F }.
Proof of Proposition (1a): Scope Dependency
For all i = j, CXi ◦ CXj = CXj ◦ CXi .
Proof. To prove this, we find a function F such that F ∈ CXi ◦ CXj but
F∈
/ CXj ◦ CXi . Let F = (EVERY-E)i ◦ (SOME-E)j , where the universe E has at
EXPRESSIVE POWER OF DP-COMPOSITIONS
427
least two elements. We represent this type 2 function as the following:
F = (∧a∈E Ia )i ◦ (∨b∈E Ib )j ∈ CXi ◦ CXj , where for all b ∈ E, Ib is an individual
(ultrafilter) in (℘ (E), ≤) generated by the property {b} ∈ ℘ (E).
Then, F = (∧a∈E ∨b∈E (Ia )i ◦ (Ib )j ) = (∧a∈E ∨b∈E (Ib )j ◦ (Ia )i ).
To show F ∈
/ CXj ◦ CXi , that is, F is not a reducible type 2 function
in CXj ◦ CXi , from the Reducibility equivalence theorem (Theorem 2), we
find a distinct reducible type 2 function ϕ ∈ CXj ◦ CXi such that for all
product relations R ∈ RK over E (where |K| = 2), F (R) = ϕ(R). Observe that
R ∈ R{i,j } is a product relation iff ∃P , Q ⊆ E such that R = {f ∈ [{i, j }→P ∪
Q]|f (i) ∈ P and f (j ) ∈ Q}.
Set ϕ = (SOME-E)j ◦ (EVERY-E)i = (∨b∈E Ib )j ◦ (∧a∈E Ia )i ∈ CXj ◦CXi .
Then for all product relations R = P ×Q ∈ RK ,
(i) if K = {i, j }, F (R) = (EVERY-E)i ◦(SOME-E)j (R) = True iff P = E
and Q = Ø; and
ϕ(R) = (SOME-E)j ◦(EVERY-E)i (R) = T rueiff P = E and Q = Ø.
And if K = {i, j }, by the definition of case-extension in Definition 2, we
have:
(ii) if i, j ∈
/ K, F (R) = (EVERY-E)i ((SOME-E)j (R)) = (EVERY-E)i (R) = R,
ϕ(R) = (SOME-E)j ((EVERY-E)i (R)) = (SOME-E)j (R) = R;
(iii) if i ∈ K, and j ∈
/ K, F (R) = ϕ(R) = (EVERY-E)i (R); and
(iv) if i ∈
/ K, and j ∈ K, F (R) = ϕ(R) = (SOME-E)j (R).
Thus, from (i)–(iv), for all product relations R ∈ RK , |K| = 2, F (R) = ϕ(R).
Now we see F = ϕ from the following: Suppose R is the identity
relation, i.e., R = {f ∈ [{i, j }→E]|f (i) = f (j )}, then F (R) = (EVERY-E)i ◦
(SOME-E)j (R) =True, but ϕ(R) = (SOME-E)j ◦ (EVERY-E)i (R) = False.
Therefore, by the Reducibility Equivalence Theorem, F is not reducible
to a basic composite function in CXj ◦ CXi , i.e., F ∈
/ CXj ◦ CXi .
Notes
1
Composing two DP-functions, we would get a function from (n + 2)-ary relations to
n-ary relations, which saturates two arguments. In section 2, the category name Pn /Pn+2
is used for the compositions of two DPs.
2
In this paper, the polymorphic types are used exactly in the same spirit as Keenan
and Timberlake’s (1988) use of “n-tuple categories.” Moortgat (1988) takes polymorphic
types as ambiguous ones brought in either by Type-changing operations (e.g., lifting) or
by assigning incompletely instantiated categories to lexical items like and and or.
3
Sometimes we write fnom(inative) for f1 , and facc(usative) for f2 , and so forth.
4
What we have actually defined in Definition 2 and 3 is a boolean lattice, but as they
are interdefinable with boolean algebras we shall use this latter term for its greater familiarity. See Keenan and Faltz (1985) for the definitions.
5
This equality fails if * is replaced by ◦ (composition), but the latter is not typepreserving. The domain of two composed DP functions is a proper subset of their common domain.
428
S. NAM
R ∈ R{i,j } is a product relation iff ∃P , Q ⊆ E such that R = {f ∈ [{i, j } → P ∪ Q]|f (i) ∈ P
and f (j ) ∈ Q} and we write standardly R = P × Q.
7
This proposition says that the set of unreducible type 2 functions is closed under postcomplement and dual as well as complement. Further, it follows that the set of reducible
type 2 functions, CXi ◦CXj , is also closed under these operations.
8
We use the following facts: for all functions f and g,
(i) (f - ◦ -g)(R) = (f -)(-g(R)) = (f -)(-(g(R))) = f (g(R)) = f ◦ g(R), and
(ii) -(-f ◦ g)(R) = -(-f (g(R))) = -(-(f (g(R)))) = f (g(R)) = f ◦ g(R), and
(iii) (f ◦ g-)(R) = f ((g-)(R)) = f (g(-R)) = (f ◦ g)(-R) = ((f ◦ g)-)(R).
6
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