Tuples, Discontinuity, and Gapping in Categorial Grammar* 
Glyn Morrill t & Teresa Solias t 
tDepartament de Llenguatges i Sistemes Informgtics 
Universitat Polit~cnica de Catalunya 
Edifici F I B, Pau Gargallo, 5 
08028 Barcelona 
e-maih morrill@lsi.upc.es 
tDepartamento de Filologfa Espafiola (Lingfiistica) 
Universidad de Valladolid 
Facultad de Filosoffa y Letras, Plaza de la Universidad, s/n 
47001 Valladolid 
e-mail: solias@cpd.uva.es 
Abstract 
This paper solves some puzzles in the for- 
malisation of logic for discontinuity in cat- 
egorial grammar. A 'tuple' operation intro- 
duced in \[Solias, 1992\] is defined as a mode 
of prosodic combination which has associ- 
ated projection functions, and consequently 
can support a property of unique prosodic 
decomposability. Discontinuity operators 
are defined model-theoretically by a resid- 
uation scheme which is particularly arn- 
menable proof-theoretically. This enables 
a formulation which both improves on the 
logic for wrapping and infixing of \[Moort- 
gat, 1988\] which is only partial, and resolves 
some problems of determinacy of insertion 
point in the application of these proposals 
to in-sits binding phenomena. A discontin- 
uons product is also defined by the residu- 
ation scheme, enabling formulation of rules 
of both use and proof for a 'substring' prod- 
uct that would have been similarly doomed 
to partial logic. 
We show how the apparatus enables char- 
acterisation of discontinous functors such 
as particle verbs and phrasal idioms, and 
binding phenomena such as quantifier rais- 
ing and pied piping. We conclude by show- 
ing how the apparatus enables a simple cat- 
egorial analysis of (SVO) gapping using the 
discontinuity product and the wrapping op- 
erator. 
*The work by Glyn Morrill was for the most part car- 
ried out under the support of the Ministerio de Edu. 
caciJn V Ciencia, Madrid, in the form of visting scholar- 
ship grant SB90 P413308C. 
1 Introduction 
In \[Lambek, 1958\] the suggestive recursive fractional 
categorial notations of \[Ajdukiewicz, 1935\] and \[Bar- 
Hillel, 1953\] were provided with a foundational set- 
ting in mathematical logic. This takes the form of a 
model theory interpreting category formulas in alge- 
braic structures. A Gentzen style sequent proof the- 
ory for which there is a Cut-elimination result means 
that a decision procedure is provided on the basis of 
sequent calculus. 
The category formulas are freely generated from 
atomic category formulas (e.g. N for referring nom- 
inals, S for sentences, CN for common nouns, ...) 
by binary operators \ ('under'), / ('over') and • 
('product'). The interpretation is in semigroups, 
i.e. algebras (L, +) where + is a binary operation 
satisfying the associativity axiom sl + (s~ + s3) = 
(sl + s2) + s3. (In the non-associative formulation 
of \[Lambek, 1961\], this condition is withdrawn.) We 
may in particular consider the algebra obtained by 
taking the set V* of strings over a vocabulary V; 
then L is V* - {t} where t is the empty string. Each 
category formula A is interpreted as a subset D(A) of 
L. Given such a mapping for atomic category formu- 
las it is extended to the compound category formulas 
thus: 
D(A\B) = {sirs' ~ D(A),s' + s E D(B)} (1) 
D(B/A) -- {slVd E D(A),s + s' E D(B)} 
D(A*B) = {sl + s213Sl e D(A),s2 e D(B)} 
In general we may define L in terms of a semigroup 
algebra (L*, +, t) where f E L is an identity element, 
i.e. an element such that s + t = t + s -- s; then 
L is L* - {t}. In the sequent calculus of \[Lambek, 
1958\] a sequent is of the form A1,..., An =~ A where 
n > 0,1 and is read as asserting that for any elements 
1The requirement n > 0 blocks the inference from 
287 
sl,..., Sn in A1,. • •, An respectively, sl + ... + Sn is 
in A. Thus the relevant prosodic operations are en- 
coded by the linear ordering of antecedents in the 
sequent, and structural rules of permutation, con- 
traction, and weakening are not valid. The calculus 
is as follows. The notation P(A) represents an an- 
tecedent containing a subpart A. 
a. ~.id r =~ A A(A) =~ B (2) 
A =~ A Cut a(r) B 
b. r =~ A A(B) =~ C A, r ==~ B 
\L ~R 
aft, A\B) :=~ C r ::~ A\B 
C. r =~ A A(B) ==~ C r, A =~ B 
• /L B/A/R A(B/A, r) =~ C r 
d. F(A,B) ~ C r ~ A A =~ B 
• L .R r(A.B) ~ C r, A =t~ A.B 
As is normal in sequent calculus, each operator has 
a L(eft) rule of use and a R(ight) rule of proof. Cut- 
free backward chaining proof search is terminating 
since in every proof step going from conclusion to 
premises, the total number of operator occurrences 
is reduced by one. 
The original development of categorial grammar 
grew from semantic concerns, and as is well known, 
the formalism embraces compositional type-logical 
semantics. In particular, division categories A\B 
and B/A can be seen as Fregean functors: incom- 
plete Bs the meanings of which are abstracted over 
A argument meanings. Complete (or: saturated) ex- 
pressions bearing primary meanings belong to atomic 
categories. Given some basic semantic domains (e.g. 
truth values {0, 1}, a set of entities E, ...) a hi- 
erarchy of spaces for a type-logicai semantics may 
be generated by such operations as function forma- 
tion (r\[2: the set of all functions from r2 into rl) 
and pair formation (rl x r~: the set of all ordered 
pairs comprising a rl followed by a r2). Each cat- 
egory formula A is associated with a semantic do- 
main T(A). Such a type map T for atomic category 
formulas (e.g. T(N) = E,T(S) = {0, 1},T(CN) = 
{0, 1} E) is extended to compound category formulas 
by T(A\B) = T(B/A) = T(B) T(A) and T(A*B) = 
T(A) x T(B). Each category formula A is now inter- 
preted as a set of two dimensional 'signs': a subset 
D(A) of L x T(A). Such an interpretation for atomic 
category formulas is extended to one for compound 
A =~ A to =~ A/A which as a theorem would assert that 
the identity element t is a member of each category of 
the form A/A (similarly for A\A). Since we have defined 
categories to be interpreted as subsets of a set L which 
does not necessarily contain an identity element, such a 
theorem would not be valid, and it is prevented by defin- 
ing sequents as having at least one antecedent formula. 
category formulas by: 2 
D(A\B) 
D(B/A) 
D(A.B) 
= {(s,m)IV(s',m') e D(A), (3) 
(s' + m(m')) e D(B)} 
-- {(s,m)lV(s',m') e D(A), 
(, + e, m(m')) e O(B)} 
= 
(st, ml) • D(A), (s2, ms) • D(B)} 
Proofs can be annotated to associate typed semantic 
lambda terms with each theorem \[Moortgat, 1988\]. 
A sequent has the form zi:A1,...,xn:An ::~ ¢:A 
where n > 0, no semantic variable is associated with 
more than one category formula, and ¢ is a typed 
lambda term over (free) variables {xl,..., x,}. It is 
to be read as stating that the result of applying the 
prosodic operation implicit in the ordering, and the 
semantic operation represented explicitly by ¢, to 
the prosodic and semantic components of any signs 
in At,..., An yields a sign in A. This system is un- 
derstood as observing the type map in the obvious 
way, and is an instance of the Curry-Howard corre- 
spondence between (intuitionistic) proofs and typed 
lambda terms. It was first employed in relation to 
categorial grammar in \[van Benthem~ 1983\]; for gen- 
eralisation to other connectives see \[Morrill, 1990b; 
Morrill, 1992a\] 
2 Prosodic Labelling 
As we shall see, the implicit coding of prosodic op- 
erations in the ordering of a sequent is not expres- 
sive enough to represent the logic of discontinuity 
connectives. In this connection, \[Moortgat, 1991b\] 
employs \[Gabbay, 1991\] notion of labelled deductive 
system (LDS). When we label for prosodics as well 
as semantics, a sequent has the form al - zi: At, ..., 
am - x, :Am ::~ a - ¢: A where n >_ 0, no prosodic or 
semantic variable is associated with more than one 
category formula, c~ i s a prosodic term over variables 
{al,..., an} and ¢ is a typed lambda term over (free) 
variables {zl,..., Zn}. The prosodically and seman- 
2A general preparation for such multidimensional 
characterisation is provided by \[Oehrle, 1988\] which ef- 
fectively refines Montague's program in order to pro- 
vide a more even-handed treatment of linguistic dimen- 
sions. But note that Oehrle anticipates only functions as 
prosodic and semantic objects. Here the prosodic alge- 
bra is not marie up of functions, and nor are functions 
the only kind of semantic object. The symmetric treat- 
ment of prosodies and semantics concurs with the con- 
temporary trend for 'sign-based' grammatical formalisms 
such as HPSG \[Pollard and Sag, 1902\], though this latter 
only goes so fax as recursively defining a relation between 
prosodic and semantic forms, i.e. representations. By in- 
terpreting categories in the way set out in \[Morrill, 1992a\] 
as pairings of prosodic and semantic objects we make di- 
rect reference to their properties as defined in terms of 
mathematical models, and use forms only in the meta- 
theory. 
288 
tically labelled calculus is as follows. 3 to the prosodic dimension. 
a. 
id 
a- z:A =~ a- x:A 
(4) 3 Residuation 
b. r ~. - ¢:A a - ~:A, a ~ ~(a) - ¢(~): Bent 
r, a ~ t~(~) - ¢(¢): B 
F =t. a - ¢: A b - y: B, A ~ 7(0 - X(Y): C \L 
F, d - w: A\B, A =V 3'(a + d) - X(w ¢): C 
C. 
d. F,a-x:A =*, a+3"-¢:B "\R 
r :0 3' - Axe: A\B 
e. r =~ a-¢:A b-v:B,A=v3"(b)-¢(v):C 
~, d"- w'.'~/A, -A ~-~d ~k-~ ---~(-w ¢-~'. C .\]L 
f. r,a-x:A~3"+a-¢:B ./a 
F =V 7 - Axe: B/A 
g. a-z:A,b-y:B,A=t, 3"(a+b)-x(x,y):C oL 
c - z:A..B, A ~ :'r(e) - x(lrlz, Ir2z):C 
h. r~a-¢:A A =t,/~ - ¢:B 
.,,It F,A =:*. a+/~- (¢,¢):A..B 
The pattern of prosodic interpretation and prosodic 
labelling given above is entirely general. The inter- 
pretation scheme is called residuation. Under the 
scheme we define in terms of any binary operation 
+n complementary (or: dual) division operators \n 
and/n and product operator ..n by the clauses given 
in (5). 
D(A\.B) 
D(B/nA) 
D(A.nB) 
= (,IVs' e D(A),s' +n s • D(B)} (5) 
= {sirs' • D(A),s +, s'• D(B)} 
-- {81 -~n 821381 • D(A), s2 • D(B)} 
As a consequence the following laws hold (see \[Lam- 
bek, 1958; Lambek, 1988; Dunn, 1991; Moortgat, 
1991a; Moortgat and Morrill, 1991\]: 4 
A ::t, C/riB HF- A.nB =~ C Hk B ::V A\nC (6) 
The LDS logic directly reflects this interpretation. It 
always has the following format, together with label 
equations in accordance with the axioms of the alge- 
bra of interpretation. 
a. (7) "id 
a:A :=~ a:A 
b. 
We are free to manipulate labels according to 
the equations they satisfy. In the case of asso- 
ciative Lambek calculus there is the assoeiativity c. 
law; in the case of non-associative Lambek calcu- 
lus there would be no equations on labels. Ob- 
serve that with prosodic labelling, the structural 
rules permutation, contraction, and weakening are d. 
valid. In our labelling, we maintain the convention 
that antecedent formulas are labelled with prosodic 
and semantic variables. As a result each theo- 
rem al -xvA1,...,an -xn:An =:" a-¢:A can be 
read as a Montagovian rule of formation with input 
categories AI,...,An and output category A and 
prosodic and semantic operations a and ¢. Other f. 
versions of labelling allow labelling antecedent for- 
mulas with prosodic and semantic terms in general. 
However such labelling constrains the value of the el- 
ements to which the theorems apply by reference to g. 
the terms that represent them. In relation to gram- 
mar, this would mean conditioning rules on the se- 
mantic and/or prosodic form of the input. For in- 
stance, with respect to semantics, this would consti- h. 
tute essential reference to semantic form in the way 
which Montague grammar deliberately avoids. We 
advocate exactly the same transparency in relation 
3In prosodic and semantic terms we allow omission of 
parenthesis under associativity, and under a convention 
that unary operators bind tighter than binary operators. 
F =~ a:A a:A,A ::t,/5(a): B 
Cut r, ~ ~ ~(a): B 
F::~a:A b:B,A:o7(b):C 
r,d:A\nB, a =~ 3"(a +. d):C \"L 
r,a:A ::*. a+n 3':B .\,It 
r =¢, 3": A\,B 
e. r~a:A b:B,A=,.3"(b):C 
r, d: B/hA, A :0 3'(d +n a): C/nL" 
r,a:A ::~ 3"+. a:B ~.It 
P ::~ 3': B\],A 
a:A,b:B,A =*. 7(a +n b):C 
an L c:A..nB, A ::~ 7(c):C 
F =V a:A A ::V B:B 
an R r, A :,. a +. ~: A*.B 
4In fact the residuation scheme is even more general 
than that which we need here: is applies to ternary 'ac- 
cessibility' relations in general, not just to binary func- 
tions, i.e. deterministic ternaxy relations. 
289 
The semantic interpretation with respect to func- 
tion and Cartesian product formation can also be ap- 
plied uniformly, with systematic labelling as in the 
previous section. 
4 Discontinuity 
Elegant as such categorial grammar is, it is more 
suggestive of an approach to computational linguis- 
tic grammar formalism, than actually representa- 
tive of such. Amongst the various enrichments that 
have been proposed (see e.g. \[van Benthem, 1989; 
Morrill et al., 1990; Barry et al., 1991; Morrill, 1990a; 
Morrill, 1990b; Moortgat and Morrill, 1991; Morrill, 
1992a; Morrill, 1992b\]), \[Moortgat, 1988\] advanced 
earlier discussion of discontinuity in e.g. \[Bach, 1981; 
Bach, 1984\] with a proposal for infixing and wrap- 
ping operators. The operators not only provide scope 
over these particular phenomena but also, as indi- 
cated in e.g. \[Moortgat, 1990\], seem to provide an 
underlying basis in terms of which operators for bind- 
ing phenomena such as quantification and reflexivisa- 
tion should be definable. The coverage of pied piping 
in \[Morrill, 1992b\] would also be definable in terms 
of these primitives, but all this depends on the reso- 
lution of certain technical issues which have been to 
date outstanding. 
Amongst the examples we shall be able to treat by 
means of our present proposals are the following. 
a. Mary rang John up. (8) 
b. Mary gave John the cold shoulder. 
c. John likes everything. 
d. for whom John works. 
e. John studies logic, and Charles, phonetics. 
In the particle-verb construction (8a) and discontin- 
uous idiom (8b), the object 'John' infixes in discon- 
tinuous expressions with unitary meanings. In (8c) 
the quantifier must receive sentential semantic scope, 
and in (Sd) the pied piping must be generated, with 
the semantics of 'whom John works for'. In (Be), the 
semantics of the verb gapped in the second conjunct 
must be recovered from the first conjunct. 
Binary operators T and ~ are proposed in \[Moort- 
gat, 1988\] such that BTA signifies functors that wrap 
around their A arguments to form Bs, and BIA sig- 
nifies functors that infix themselves in their A argu- 
ments to form Bs. Assuming the semigroup algebra 
of associative Lambek calculus, there are two possi- 
bilities in each case, depending on whether we are 
free to insert anywhere (universal), or whether the 
relevant insertion points are fixed (existential). We 
leave semantics aside for the moment. 
Existential (9) 
D(BT~A) = {s\]3sl, s2\[s = Sl + s2 A Vs' • D(A), 
sl + d + s2 • D(B)\]} 
Universal 
D(BTvA) = {s\]Vsl, s2\[s = sl + s2 --* Vs' • D(A), 
Sl + s' + s2 • D(B)\]} 
Existential (10) 
D(BIjA) = {s\]Vd e D(A), qSl, s: 
Is' = sl +s2 Asl +s' + s2 e D(B)\]} 
Universal 
D(SivA) = {sIVs' • D(A), Vsl, s2 
\[s' = sl + s2 ---* Sl + s' + s2 • D(S)\]} 
Inspecting the possibilities of ordered sequent pre- 
sentation, of the eight possible rules of inference (use 
and proof for each of four operators), only TjR and 
IvL are expressible: 
a. rl,A,r~ =~ B (11) 
rl, F2 =~ BT~A TJFt 
b. El,F2 =¢, A A1,B, A2 ::~ C 
IvL A1, El, BIvA, F2, A2 =¢, C 
This is the partiM logic of \[Moortgat, 1988\]. Note 
that the absence of a rule of use for existential wrap- 
ping means that we could not generate from discon- 
tinuous elements such as ring up and give the cold 
shoulder which we should like to assign lexical cat- 
egory (N\S)TsN. (Evidently Tv would permit incor- 
rect word order such as *'Mary gave the John cold 
shoulder'.) The problem with ordered sequents is 
that the implicit encoding of prosodic operations is of 
limited expressivity. Accordingly, \[Moortgat, 1991b\] 
seeks to improve the situation by means of explicit 
prosodic labelling. This does enable both rules for 
e.g. ~v but still does not enable the useful TjL: the 
remaining problem is, as noted by \[Versmissen, 1991\], 
that we need to have an insertion point somehow de- 
terminate from the prosodic label for an existential 
wrapper in order to perform a left inference. 
In \[Moortgat, 1991a\] a discontinuity product is 
proposed, again implicitly assuming just a semigroup 
algebra: 5 
D(A ® B) = {sl + s2 +dl \]Sl + st e D(A), (12) 
s2 E D(B)} 
As for the discontinuity divisions, ordered sequent 
presentation cannot express rules of both use and 
proof: only ®R can be represented: 
rl, F2 ::~ A A =~ B (13) 
'®R F1,A,F2 =~ A®B 
Even using labelling, the problem for ®L remains 
and is the same as that above: there is no proper 
management of separation points. 
In \[Moortgat, 1991a\] it is observed how the 
quantifying-in of infix binders such as quantifier 
SThe version given is actually just the existential case 
of two possibilities, existential and universal, as before. 
No rules for the universal version can be expressed in 
ordered sequent calculus, or labelled sequent calculus. 
290 
F,a- x:A =:~ 71 +a+72 -¢:B TR 
F =:~ (71,72) - (Xz¢): BTA 
P,a- z:A ==~ la+x+2a- ¢:B IR 
r x - (axe): BIA 
F ::~ (hi, a2) - ¢: A A ::~ fl - ¢: B ®L 
r,A ~ ,~1 +~+a2 - (¢,¢):A @ B 
A=~a-C:A 
F, A, c - z: BTA 
r, b - v: s ,(b) - D TL 
=~ 6(lc + a + 2c) - w((z ¢)): O 
F ==~ (as,a2) - ~b:A A,b- v:B ::~ 6(b) - w(y):D IL 
F, A, c - z: B~A =*, 6(hi + c + ~2) - w((z ¢)): D 
F, a - z: A, b - y: B :=~ 6(la + b + 2a) - X(z, Y): C @R 
r,c - z:A ® B =t, 6(c) - x(~rlz,~r2z):D 
Figure h Labelled rules for discontinuity operators 
phrases seems almost definable as SI(STN): they in- 
fix themselves at N positions in Ss (and take seman- 
tic scope at the S level - that is why they must be 
quantified in). And if this definability could be main- 
tained, it would enable these operators to simulate 
the account of pied piping in \[Morrill, 1992b\]. None 
of the interpretations above however enable the ex- 
pression of the requirement that the positions re- 
ferred to by the two operator occurrences are the 
same. Our proposals will facilitate this definability, 
and also admit of a full (labelled) logic. 
5 Tuple Control of Insertion Points 
The present innovation rests on extending the 
prosodic algebra (L*,+,t) as above to an algebra 
(L*, +, t, (., .), 1, 2) where (., .) is a binary operation 
of tuple formation (introduced in \[Solias, 1992\]), with 
respect to which 1 and 2 behave as projection func- 
tions. Thus the algebra satisfies the conditions: 
l(sl, s2) = sx 2(sl, 82) = 82 (14) 
(Is, 2s) -" s 
We may in particular think of the algebra of elements 
V* obtained from disjoint sets V and {\[, ;, \]} by clos- 
ing V under two binary operations: concatenation 
+, and pairing \[.; .\] where pairing can be defined as 
concatenation with delimitation and marking of in- 
sertion point. 
The proposal can be related to \[Moortgat and 
Morrill, 1991\] which also considers algebras with 
more than one adjunction operation (each either as- 
sociative or non-associative), and defines divisions 
and products with respect to each by residuation. 
Note however that firstly, our tuple prosodic oper- 
ation is not simply that of non-associative Lambek 
calculus which is characterised by the absence of any 
axiom (associative or otherwise), since the projec- 
tion axioms entail specific conditions not imposed 
in the non-associative case: we might describe the 
tuple system as unassociative. Tupling is bijective 
and a prosodic object s formed by tupling records 
a separation point between two objects ls and 2s 
whereas a prosodic object formed by non-associative 
adjunction has no such recoverable separation point. 
Secondly, we are not primarily interested here in di- 
visions and products based on tupling but in the 
combined use of the associative and unassociative 
operations to define discontinuity operators. (Note 
however that residuation with respect to tupling, as 
proposed in \[Solias, 1992\], would define operators 
suitable for verbs regarded as head-wrappers such 
as 'persuade'.) This brings us to the essence of the 
present proposals with respect to wrapping and infix- 
ing. The prosodic interpretation for the discontinuity 
operators is to be as follows: 
D(BTA) ={s\[Vs' e D(A), ls + s' + 2s e D(B)} (15) 
D(BIA) = {siVa' e D(A), ls' + s + 28' G D(B)} 
D(A ® B) = {181 + 82 + 282\[81 • D(A),82 • D(B)} 
It can be seen that the operators are the residuation 
divisions with respect to a binary prosodic opera- 
tion I defined by szIs2 = 181 + s2 + 281 just as the 
Lambek operators are the residuation divisions with 
respect to +. Use of the tuple operation collapses 
the former distinction between existential and uni- 
versal in (9) and (10). Because pairing is bijective 
and tuples express a unique insertion point, there is 
a unique decomposition of tupled elements. Exis- 
tential and universal wrappers collapse into a single 
wrapper and existential and universal infixers col- 
lapse into a single infixer. 
Turning to include the semantics, the type map 
is as is to be expected for functors and for product: 
T(BTA) = T(BIA) = T(B) T(A) and T(A @ B) = 
T(A) x T(B), and as usual a category formula A is 
interpreted as a subset of L x T(A). 
D(BTA) = {(s,m)\[V(s',m') • D(A), (16) 
(ls + 8' + 2s, m(m')) • D(S)} 
D(B~A) = {(s, rn)\]V(s',rn') • D(A), 
(Is' + s + 28', m(m')) • D(B)} 
D(A®B) = {(lSl+S2+2sl,(ml,m2))\[ 
(sl, rnl) • D(A), (s2, m2) • D(B)} 
The full prosodically and semantically labelled logic 
is given in Figure 1. In TL lc and 2c pick out the 
first and second projections of the prosodic object c 
in the same way that projections pick out the com- 
ponents of a semantic object in the eL rule of (4g); 
291 
likewise in ~l~ for the projections la and 2a. The 
resulting prosodic forms are only simplifiable when 
the relevant objects are tuples. 6 
6 Discontinuity Examples 
6.1 Phrasal Verbs 
As a first example of discontinuity consider the parti- 
cle verb case 'Mary rang John up' and the discontin- 
uous idiom case 'Mary gave John the cold shoulder'. 
The meaning of the particle verb and the phrasal id- 
iom resides with its elements together, which wrap 
around their object. The lexical assignments re- 
quired are: 
(rang, up) - ring-up (17) 
:= (N\S)TN 
(gave, the + cold + shoulder) - give-tes 
:= (N\S)TN 
A derivation is given in Figure 2. The lexical prosod- 
ies and semantics of the proper names may be as- 
sumed to be atoms. For 'Mary rang John up', substi- 
tution of the lexical prosodies thus yields (18) which 
simplifies as shown. 
Mary + 1(rang, up) + John + 2(rang, up) 
Mary -t- rang + John + up 
(18) 
Similarly, substitution of the lexical semantics gives (19). 
((ring-up john) mary) (19) 
For 'Mary gave John the cold shoulder', substitution 
of the lexical prosodies yields: 
Mary + l(gave, the + cold + shoulder) q- John 
+ 2(gave, the + cold + shoulder) .,z 
Mary + gave + John + the + cold + shoulder 
(20) 
The semantics is: 
((gave-tcs john) mary) (21) 
°Having the projection functions defined for all 
prosodic objects rather then just tuple objects allows 
us to consider the prosodic algebra to be untyped (or: 
unsorted). Consequently, there is no need to check for 
the data type of prosodic objects such as by pattern- 
matching on antecedent terms (see comment above on 
transparency of rules). It may be possible to develop the 
present proposals by adding sort structure to the prosodic 
algebra in a manner analogous to the typing of the seman- 
tic algebra. Such sorting could be essential to defining 
a model theory with respect to which the calculus can 
be shown to be complete. Recursive nesting of infixation 
points does not appear to be motivated linguistically, and 
the present calculus does not support it. A sorted model 
theory which excludes the recursion might provide an in- 
terpretation with respect to which the present calculus is 
both sound and complete. 
6.2 Quantifier Raising 
In Montague grammar quantifying-in is motivated 
by the necessity to achieve sentential scope for 
all quantifiers and quantifier-scope ambiguities. 
Quantifying-in allows a quantifier phrase to ap- 
ply as a semantic functor to its sentential context. 
Quantifying-in at different sentence levels enables 
a quantifier to take scope accordingly, and alterna- 
tive orderings of quantifying-in enable quantifiers to 
take different scopings relative to one another. In 
\[Moortgat, 1990\] a binary operator ~ is defined for 
which the rule of use is essentially quantifying-in, so 
that a Montagovian treatment of quantifier-scoping 
is achieved by assignment of a quantifier phrase like 
'something' to N~S, and assignment of determiners 
like 'every' to (N~S)/CN. In \[Moortgat, 1991a\] he 
suggests that a category such as A ~ B might be de- 
finable as B~(BTA), but notes that this definability 
does not hold for his definitions, for which, further- 
more, the logic is problematic. On the present formu- 
lation however, these intuitions are realised. The cat- 
egory S~(STN) is a suitable category for a quantifier 
phrase such as 'everything' or 'some man', achiev- 
ing sentential quantifier scope, and quantificational 
ambiguity. 
Assume the lexical entry (22). 
everything - XzVy(x y) := SI(SI"N) (22) 
For 'John likes everything' there is the derivation in 
Figure 3. In this derivation, and in general, lines are 
included showing explicit label manipulations under 
equality in the prosodic algebra, in such a way that 
all rule instances match the rule presentations. Sub- 
stitution of the lexical prosodies and semantics as- 
sociates John + likes + everything with (23) which 
simplifies as shown. 
(AzVy(x y) Ac((like c) john)) --* (23) 
Vy((like y) john) 
In this example the' quantifier is peripheral in the 
sentence and a category (S/N)\S could have been 
used in associative Lambek calculus. However, an- 
other category S/(N\S) would be needed to allow the 
quantifier phrase to appear in subject position, and 
further assignments still would be required for post- 
verbal position in a ditransitive verb phrase, and 
so on. Some generality can be achieved by assum- 
ing second-order polymorphie categories (see \[Emms, 
1990\]), but note that the single assignment we have 
given allows appearance in all N positions without 
further ado, and allows all the relative quantifier 
scopings at S nodes. 
6.3 Pied Piping 
In \[Moortgat, 1991a\] and and \[Morrill, 19925\] a 
three-place operator is considered which is like A 
B, except that quantifying-in changes the category of 
the context expression.. \[Morrill, 1992b\] shows that 
this enables capture of pied piping. It follows from 
292 
m-m:N :~m-m:N b-b:S ::~b-b:S \L 
j-j:N =~j-j:N m-m:N,a-a:N\S =~m+a-(am):S tL 
m - m: N, r - r: (N\S)TN, j - j: N =~ m+lr+j+2r - ((r j) m): S 
Figure 2: Derivation for 'Mary rang John up' and 'Mary gave John the cold shoulder' 
c-c:N =~ c- c:N 
j-j:N =~j-j:N f-f:S =~f-f.'S \L 
j-j:N,d-d:N\S :~j+d-(dj):S /i 
j - j: N, l - h (N\S)/N, c - c: N =~ j+l+c - ((1 c) j): S 
j -j:N, l-I:(N\S)/N =~ j+l+c+t - ((l c)j):S 
j -j:N, 1-h (N\S)/N =~ (j+l, t) - Ac((1 c)j):StN TR b-b:S =*. b - b:S 4L 
j -j: N, 1 - l: (N\S)/N, e -e: SI(STN) ::~ j+l+e+t - (e Ac((l c) j)): S 
j - j: N, 1 - h (N\S)/N, e - e: S~(STN) =~ j+l+e - (e Ac((l c) j)): S 
Figure 3: Derivation for 'John likes everything' 
the nature of the present proposals that A~(BTC) 
presents the desired complicity between the opera- 
tors. As a result, the treatment of \[Morrill, 1992b\] 
can be presented in these terms. 
Consider the example 'for whom John works'. The 
relative pronoun is lexically assigned as follows where 
R is the common noun modifier category CN\CN. 
whom - w) ^ (u (x (24) 
:-- (R/(STPP))~(PPTN) 
There is the derivation in Figure 4. The result of 
prosodic substitution is 
for + whom + 0'ohn + works, t) (25) 
The result of semantic substitution is 
((A~AYAzAwC(z w) ^ (Y (x w))\] (26) 
As(for a)) Ah((work h) john)) -,~ AzAw\[(z 
w) A ((work (for w)) john)\] 
As for the quantification, this example is potentially 
manageable in just Lambek calculus. But an exam- 
ple where the relative pronoun is not peripheral in 
the pied piped material, such as 'a man a brother of 
whom from Brazil appeared on television' would be 
problematic for the same reasons as quantification. 
The solution, in terms of infixing and wrapping, is 
the same in the two cases, but pied piping has been 
a more conspicuous problem for categorial grammar 
because while the scoping of quantifiers can be played 
down, the syntactic realisation of pied piping is only 
too evident. In the phrase structure tradition, pied 
piping has been taken as strong motivation for fea- 
ture percolation (see \[Pollard, 1988\]). We have seen 
here how discontinuity operators challenge this con- 
strual. 
Categorial grammar is well-known to provide 
OSSibilities for 'non-constituent' coordination (see 
teedman, 1985; Dowty, 1988J) less accessible in the 
phrase structure/feature percolation approach. We 
turn now to another example which is glaringly prob- 
lematic for all approaches, gapping. It is entirely 
unclear how feature percolation could engage such a 
construction; but as we shall see the discontinuity 
apparatus succeeds in doing so. 
7 Gapping 
The kind of examples we want to consider are: 
John studies logic, and Charles, phonetics. (27) 
The construction is characterised by the absence 
in the right hand conjunct of a verbal element, 
the understood semantics of which is provided by 
a corresponding verbal element in the left hand 
conjunct. Clearly, instanciations of a coordinator 
category schema (X\X)/X will not generate such 
cases of gapping. The phenomenon has attracted a 
fair amount of attention in categorial grammar (e.g. 
\[Steedman, 1990; Raaijmakers, 1991\]). 
The approach of \[Steedman, 1990\] aims to reduce 
gapping to constituent coordination; furthermore it 
aims to do this using just the standard division op- 
erators of categorial grammar. This involves special 
treatment of both the right and the left conjunct. We 
present our discussion in the context of the present 
minimal example of gapping a transitive verb TV. 
With respect to the right hand conjunct, the initial 
problem is to give a categorisation at all. Steedman 
does this by reference to a constituent formed by 
the subject and object with the coordinator. This 
constituent is essentially TV\S but with a feature 
293 
a-a:N =~a-a:N c-c:PP =~c-c:PP /L 
f-f:PP/N, a-a:N ~f+a-(fa):PP 
f-f:PP/N, a-a:N =~f+a+t-(fa):PP- Trt 
f-f:PP/N =~(f,t)-~a(fa):PPTN 
j-j:N =~j-j:N k-k:S =~k-k:S \L 
h-h:PP =~h-h:PP j-j:N, i-i:N\S =~j+i-(ij):S 
j-j:N, w-w:(N\S)/PP, h-h:PP =~j+w+h-((wh)j):S/L 
j-j:N, w-w:(N\S)/PP, h-h: PP =~j+w+h+t-((wh)j):STR 
j-j:N, w-w:(N\S)/PP =¢,(j+w, t)-),h((wh)j):STPP g-g:R =~g-g:R/L 
d-d:R/(STPP), j-j:N, w-w:(N\S)/PP =~d+(j+w,t)-(d),h((wh)j)):R IL 
f-f:PP/N, o-o:(R/(SI"PP))I(PPTN), j-j:N, w-w:(N\S)/PP =~f+o+t+(j+w,t)-((o~a(fa))~h((wh)j)):R 
m 
f-f:PP/N, o-o:(R/(STPP))I(PPTN), j-j:N, w-w:(N\S)/PP =~f+o+(j+w,t)-((o),a(fa)))~h((wh)j)):R 
Figure 4: Derivation for 'for whom John works' 
both blocking ordinary application, and licensing co- 
ordination with a left hand conjunct of the same 
category. The blocking is necessary because 'and 
Charles, phonetics' is clearly not of category TV\S: 
'Studies and Charles, phonetics' is not a sentence. 
Now, with respect to the left hand conjunct, Steed- 
man invokes a special decomposition of 'John stud- 
ies logic' analysed as S, into TV and TV\S. There 
is then constituent coordination between TV\S and 
TV\S. Finally the coordinate structure of category 
TV\S combines with TV on the left to give S. 
Although this treatment addresses the two prob- 
lems that any account of gapping must solve, cate- 
gorisation of the right hand conjunct and access of 
the verbal semantics in the left hand conjunct, it at- 
tempts to do so within a narrow conception of cate- 
gorial grammar (only division operators) that neces- 
siates invocation of distinctly contrived mechanisms. 
We believe that the radical reconstruals of grammar 
implicated by this analysis are not necessary given 
the general framework including discontinuity oper- 
ators we have set out. We address for the moment 
just our minimal example. 
Within the context of categorial grammar we have 
established, the right hand conjunct is characteris- 
able as STTV. It remains to access the understood 
verbal semantics from the sentence that is the left 
hand conjunct. In order to recover from the left 
hand side the information we miss on the right hand 
side, we would like to say that this information, 
the category and semantics of the verb, is made 
available to the coordinator when it combines with 
the left conjunct. In accordance with the spirit of 
Steedman, we can observe that the left hand con- 
junct contains a part with the category SI"TV of the 
right hand constituent, but it is discontinuous, be- 
ing interpolated by TV. But this is precisely what 
is expressed by the discontinuous product category 
(STTV)®TV. Furthermore, an element of such a cat- 
egory has as its semantics a pair the second pro- 
jection of which is the semantics of the TV. Conse- 
quently gapping is generated by assignment of 'and' 
to the category (((STTV)®TV)\S)/(STTV) with se- 
mantics ~x~y\[(rly lr2y) A (x 7r2y)\]. 
The complete derivation for (27) is as in Figure 5, 
where TV abbreviates (N\S)/N. When we substitute 
the lexical prosodics (here each just a prosodic con- 
stant) for the prosodic variables in the conclusion, 
we obtain the prosodic form (28). 
John + studies + logic + and (28) 
+ ( Charles, phonetics) 
Similarly substituting the lexical semantics (all se- 
mantic constants except for the coordinator seman- 
tics as above), we obtain the associated semantics 
(29) which evaluates as shown. 
^ (29) 
Aw((w phonetics) charles)) 
(As((s logic) john), studies)) -,~ 
\[((studies logic) john)A 
((studies phonetics) charles)\] 
Some generalisation to cover different categories 
of gapped element and different categories of coor- 
dination is given by straightforward schematisation. 
In general, gapping coordinator categories have the 
form ((Z ® Y)\X)/Z where Z is XTY. In this 
scheme, X is the category of the resulting coordinate 
structure and Y is the category of the gapped mate- 
riM. This allows interaction with other coordination 
phenomena such as node raising. For example, a 
referee pointed out that gapping can occur within 
incomplete sentences thus: 'John gave a book and 
Peter, a paper, to Mary'. Such a case would be cov- 
ered by the instanciation where Y is the ditransitive 
verb category and X is S/PP. 
For generalisation including multiple gapping (sev- 
eral discontinuous segments elided) see \[Solids, 1992\], 
which employs in addition operators formed by resid- 
uation with respect to tupling. That approach has 
certain affinities with \[Oehrle, 1987\], and makes it 
possible to begin to address examples of Oehrle's re- 
lating to scope and Boolean particles. The purpose 
of the present paper has been to lay the groundwork 
for empiricM inquiry into gapping and other notori- 
ous nonconcatenative phenomena, made possible in 
294 
j-j:N,s-s:TV,I-I: N=}j+s+l- ( (sl)j):S.1.R 
s-s:TV=}s-s:TV j-j:N,I-hN=~(j,I)-As((sl)j):SI"TV ®R 
j-j:N=}-j-j:N n-n:S=~n-n:S 
I-hN=M-I:N j-j:N,g-g:N\S =~j+g-(gj):S.fL L 
j-j:N,H:TV,l-hN=>j+s+l-(As((sl)j),s): (S}TV)®TV f-f:S=~f-f:S '\L 
j-j:N,s-s:TV,l-h N,e-e:((STTV)®TV)\S=~j+s+l+e-(e(As((sl)j),s)):S 
j-j:N,s-s:TV,l-h N,e-e:((STTV)(DTV)\S=>j+s+I+e-(e(As((sl)j),s)):S 
p--p:N=:>p-p:N 
c-c:N=~c-..c:N n-n:S=~n-n:S \L 
c-c:N,y-y:N\S=~c+y-(yc):S /L 
c-c:N,p-p:N,w-w:TV=~c+w+p-((wp)c):S TR 
c-c:N,p-p:N=>(c,p)-Aw((wp)c):STTV '/L 
j-j:N,s-s:TV, l-hN,a-a:((Sq)TV)\S)/(STTV),c-c:N,p-p:N=~j+s+l+a+(c,p)-((aAw((wp)c)) (As((sl)j),s)):S 
Figure 5: Derivation for 'John studies logic; and Charles, phonetics' 
categorial grammar by a proper treatment of discon- 
• tinuity. 
8 Conclusion 
When \[Moortgat, 1988\] introduced discontinuity op- 
erators for categorial grammar, he noted that or- 
dered sequent calculus was an inadequate medium 
for the representation of a full logic. In \[Moortgat, 
1991b\] the LDS formalism was invoked, but as we 
have seen, the LDS format alone is not enough. The 
present paper has argued that a different view is re- 
quired on the model theory of discontinuity than that 
suggested by interpretation in just a semigroup alge- 
bra. This view is provided by adding to the algebra 
of interpretation the tuple operation of \[Solias, 1992\]. 
Not only does this clear up some vagueness with re- 
spect to existential and universal formulations, it also 
admits of a full labelled logic. This has brought us to 
a stage where it is appropriate to address such issues 
as completeness and Cut-elimination. 
Acknowledgements 
We thank the following for comment on an earlier ab- 
stract: Juan Barba, Alain Lecomte, Michael Moort- 
gut, Koen Versmissen, and three anonymous EACL 
reviewers. Particular thanks go to Mark ttepple who 
happens to have been thinking along similar lines. 
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