THE CATEGORY OF THE CONJUNCTION IN CATEGORIAL GRAMMAR
M.
Te esa Solias i A is
Uni e si a de Ba celona
In his wo k a ca ego ial ype o conjunc ions (and, o , e c) is p oposed wi hin he
Ca ego ial G amma o malism. Fi s o all,
I
p esen h ee main cha ac e is ics ha
ha e o
be
accoun ed o in any analysis o conjunc ion. Secondly,
I
explain he
di e en con ibu ions ha ha e been made wi hin his o malism o md a ca ego y
o conjunc ion ha allows us o accoun o
na u al
language phenomena. All hose
p oposals a e commen ed on wi h ega d o he h ee p ope ies o be explained.
Nex , a ca ego ial ype o conjunc ions is p oposed which can accoun o hose
cha ac e is ics. This ca ego y in oduces a new n- uple ope a o which is also use ul
o analysing o he na u al language phenomena.
INTRODUCTION
This pape is o ganized in wo pa s ha can
be
ead sepa a ely. The i s one is an in oduc ion
o he Ca ego ia1 G amma o malism, so people who a e amilia wi h hs linguis ic pa adigm
can di ec ely pass o he second pa o he pape . In he second pa ,
I
discuss wha is he
ca ego ial ype ha should
be
assigned o conjunc ions (and, o ).
I.
A
GENERAL INTRODUCTION TO CATEGORIAL GRAMMAR
In his sec ion I shall expose he main ea u es o Ca ego ial G a nma o allow people who do
no know any hing o his o malism o unde s and he objec i e o he second pa o his
pape .
Ca alan Wo king Pape s in Linguis ics (CWPL)
1991:
301-341
Uni e si a Au anoma de Ba celona
A Ca ego ia1 G amma is o med by a se o Ca ego ies,
a
se o Ope a o s and a se o
Ope a ions.
1.
Ca ego ies
The ca ego ies a e he o mulae assigned o lexical en ies. E e y lexical i em in he lexicon has
a ca ego y. Ca ego ies can
be
basic o unc o (slashed). The basic ones a e S (Sen ence) and N
(Noun) and all he o he ca ego ies
in
he g amma a e o med by combining hese basic
ca ego ies by means o ope a o s. The ca ego ies ha ha e ope a o s in hei composi ion a e
unc o ca ego ies. A unc o ca ego y is a unc ion (a ma hema ical unc ion) and is o med by
a unc o , an a gumen and a di ec ion whe e he unc o is looking o he a gumen . Fo
ins ance: A ca ego y which looks o a Noun on i s igh o gi e a Noun Ph ase is a
De e mine , a ca ego y which looks o an
NP
on i s igh o gi e a ca ego y which looks o
ano he NP on i s le o gi e a Sen ence is a ansi i e e b.
The o mal de ini ion o he se o ca ego ies is:
(i) I A is a basic ca ego y, hen A belongs o CAT ( he se o ca ego ies).
(ii) I A and
B
belong o CAT, hen A/B belongs o CAT, whe e
'1'
is any membe o he se o
ope a o s.
The clause (i) speci ies he se o basic ca ego ies and he clause (ii) allows he g amma o
gene a e he se o unc o ca ego ies by means o he se o ope a o s.
In his way, om
S
and N we ha e o be able o gene a e all he ca ego ies ( o mulae) o
na u al language g amma ical ca ego ies. The ca ego ies which bea ope a o s a e unc o
ca ego ies which seek a gumen s o
be
cancelled.
2.
Ope a o s
The se o ope a o s ha a e used o o n new unc o ca ego ies has g adually been made
bigge as Ca ego ia1 G a nma has been applied o na u al language phenomena. The i s
ope a o is he slash o Fon a d Applica ion, which is he ope a o ha ma ks ha he a gumen
has o
be
sough on a d, on he igh side o he unc o ca ego y:
(
1)
Fo wa d Applica ion Ope a o :
I
X YY-->X
Ex. De
=
NPIN
The second ope a o is he Backwa d Applica ion. This is simila o he Fon a d one bu he
di ec ion o he sea ch changes. In his case he a gumen
has
o
be
looked o on he le :
(2)
Backwa d Applica ion Ope a o :
Y X Y
-->
X
Ex. In ansi i e Ve b
=
SWP
The hi d ope a o is he Bidi ec ional one. Then, he a gumen can be sough ei he on he igh
o on he le o he unc o ca ego y.
(3)
Bidi ec ional Ope a o :
I
XIY
Y
-->
X
,
Y XIY
-->
X
Ex. An ex ended class o sen encial modi ie s: SIS
The ou h ope a o in pu e Ca ego ial G amma is he p oduc . This ope a o joins wo
adjacen ca ego ies.
This ope a o could be use ul in some languages whe e some lexical i ems always appea
conca ena ed, o ins ance, he wo objec s o di ansi i e e bs in English, al hough i would
be
a ca ego y speci ically p oposed o such languages, since in o he languages hese wo
objec s do no always appea adjacen
.
Mo eo e , in he second pa o his wo k we shall see
a new use o his ope a o .
This is he se o ope a o s ha we e de ined
by
Lam
bek
(1958) in his Calculus, and has been
widely adop ed by mos people who wo k
in
Ca ego ial G amma . The
La nbek
Calculus is
a
Fo mal Sys em ha allows us o deduce he ope a ions o he g amma
as
heo ems o he
Sys em. This Fo mal Sys em has been s udied in dep h by Michael Moo ga (1988) and is he
basis o he Gene alized Ca ego ia1 G a nma .
3.
Ope a ions
Ope a ions apply o ca ego ies. The e a e wo ypes o ope a ions o ules: One place
Ope a ions and Two Place Ope a ions. One place ope a ions apply o a ca ego y o gi e ano he
one. Two place ope a ions apply o wo ca ego ies o gi e one.
3.1.
One
Place Ope a ions
They apply o a ca ego y o any ype (in (5), T is any o he ca ego y):
(5)
Fon a d Type
Raising:
X
--q,
T/(TK)
I means ha i you ha e a ca ego y NP, o ins ance, you can change i o ge S/(SWP). This
lund o ope a ion, in any o i s possibili ies, is e y use ul. In ac , i pe mi s us o change an
a gumen ca ego y in o
a
unc o ca ego y. In he example gi en we ha e changed he ca ego y
o an NP in o a ca ego y which needs o ind on i s igh ano he ca ego y which needs o ind
on i
s
le ha NP. Wha is his ca ego y? This ca ego y is he ca ego y o a NP subjec .
Recall
304
ha a VP will be a ca ego y ha needs o ind a NP on i s le , (SWP), hen he ca ego y
esul ing om he ype aising is he one ha needs o ind a VP on i s igh , namely a NP
subjec .
(6)
Backwa d Type Raising:
X
---<
T (T/X)
An in e es ing example could
be
he aised ca ego y o an objec NP. This ime T
=
SWP, hen
he esul ing ca ego y is (SWP) (SWP)/NP. This ca ego y says ha in o de o o m a VP
-
(SWP)- we need o ind on he le a ansi i e e b, which has he ca ego y (SWP)INP
as
i
has al eady been explained.
3.2.
Two
Place
Ope a ions
They apply o wo ca ego ies
i
hei o m ma ches wi h he a iables.
(7)
Fon a d Func ional Applica ion:
XIY
Y
-->>
X
(>)
Thls ope a ion co esponds o he Fo wa d Applica ion Ope a o . I we ha e a unc o ca ego y
whose a gumen is Y and we ind his a gumen o he igh , hen we can de i e he unc o .
Fo ins
ance:
(8)
he boy
NPIN N
>1
NP
(9)
Backwa d Func ional Applica ion:
Y
WY
--><
X
(<)

This ule co esponds o he Backwa d Applica ion Ope a o . This ime he a gumen is ound
o he le o he unc o ca ego y:
(10) Mike uns
NP S NP
The in ansi i e e b seeks a
NP
on i s le o gi e a Sen ence.
(1 1)
Fo wa d
Func ional
Composi ion:
NY
Y
/z
--y.,
NZ
This is he ypical composi ion o ma hema ical unc ions. This is use ul in linguis ics, among
o he easons, because i allows an analysis om le o igh . Le us compa e wo de i a ions
wi h and wi hou Func ional Composi ion, espec i ely:
(12)
Ma y
ea s
apples
NP (SWP)/NP NP
.......................
>
SWP
-------------------------e-------
<
S
(13)
Ma y
ea s
apples
NP (SWP)INP NP
------T>
S/(SWP)
....................
C>
S/NP
In
(12),
he analysis begins in he middle o he sen ence while in
(13)
we s ic ly go om le
o igh .
The e is also he backwa d e sion o composi ion:
(
14)
Backwa d Func ional Composi ion:
YE X Y
-->C,
X Z
The ule o in oduc ion o he p oduc ope a o :
(
1
5)
P oduc Fo mulion:
X
Y
-->a
X*Y
This ule is used o conca ena e adjacen elemen s.
I
akes wo adjacen ca ego ies and gi es a
new ca ego y o med by he conca ena ion o bo h. I is an ope a ion wi h a long adi ion in
Ma hema ics and Logic.
(
16)
P oduc Decons uc ion:
X*Y
-->
X
Y
This is he in e se o
(15).
I is used o b eak a p oduc ca ego y in o wo adjacen ca ego ies.
These a e some o he ope a ions ha can be de i ed om he Lambek Calculus. Mo eo e
o he au ho s, mainly S eedman, add new ope a ions
in
o de o gi e mo e desc ip i e and
explana o y powe o he g amma . Some o hese a e Fo wa d and Backwa d Subs i u ion and
C ossed o Dishannonic ope a ions:
(
17)
Fon a d Subs i u ion:
(XN)IZ Y IZ
--+S
XIZ
(
18)
Backwa d Susb i u ion:
YE (X Z) Z
--><s
X Z
All he ope a ions seen un i1 now a e hannonic
in
he sense ha he di ec ion o hei ope a o is
he same. Dishannonic ope a ions ha e he di ec ion o ope a o s c ossed. One ca ego y is
o wa d and he o he is backwa d. These ope a ions a e p oposed by S eedman (Combina o y
Ca ego ia1 G amma ) in o de o accoun o some linguis ic phenomena ela ed wi h c ossed
dependencies:
(19)
Fon a d c ossed Composi ion:
X Y
Y Z
-->>Bx
X Z
(20)
Backwa d c ossed Composi ion:
Y/z x Y --><Bx
X/z
(21)
Ex:
I
shail buy oday
and
cook omo ow he mush ooms
NP (SWP)/VP VPINP VRVP CONJ VPINP VP VP NP
-----------
<Bx
-----------------
<Bx
VPINP VPINP
..............................................
&
VPINP
(22)
Fon a d c ossed Subs i u ion:
(XN)IZ YE
--+sx
X Z
(23)
Backwa d c ossed Subs i u ion:
YIZ (X Y)IZ --><sx
X/z
(24) Example
o
anaiysis o a Pa asi ic gap:
(.. he a icles) ha he boss kep wi hou eadlng
NINI(S1NP) NPlN N (SWP)/NP (SWP) SWPIGER GERINP
NP (SWP) (S WP) INP
........................................................................
Q
SlNP
NIN
(55)
John and
Ma
Y
NP (XK)/X NP
>
NP NP
.........................
<
NP
This analysis ob iously has he p eposi ional p ope y since he conjunc ion i s combines
wi h he cons i uen on i s igh . On he o he hand, i seems o
be
in ix because i seeks one
a gumen on i s igh and ano he on i s le bu he e is a p oblem. The p oblem is he e is
no hing ha o bids us om o ming he i s cons i uen
NPWP
and, a e ha , combining i
wi h he le cons i uen by means o a di e en ope a ion om applica ion, in a way ha
iola es he hi d cha ac e is ic, he conjunc ion o like ca ego ies.
Fo ins ance, an ung amma ical sen ence like
(56)
could
be
w ongly p edic ed by his ca ego ial
ype in combina ion wi h backwa ds composi ion. Ajdukie icz's ca ego ia1 ype has his
p oblem oo.
(56)
*
(A
man) who walks and he alks
NWI(SWP)~
SWP (XK)IX S
------------------
>
S S
...........................
<C
S NP
Thus, as we can combine he ca ego y esul ing om he conjunc ion plus he igh conjunc
wi h he le conjunc by means o backwa d composi ion, we ha e a w ong p edic ion.
In ui i ely, wha we lea n is ha we ha e o compel an analysis whe e he ope a ions ha apply
a e o wa d and backwa d applica ion on ca ego ies o he same ype.
3
16

2.3.
S eedman
's
Ca eg o y
S eedman has ea ed in dep h he p oblem o coo dina ion in many pape s and his ca ego ia1
ype o conjunc ion has been changing h oughou . I will only commen on he las one bu he
c i icisms a e alid o his pas accoun s.
S eedman
(1990)
p oposes
a
syn agma ic ule a ached o conjunc ion. This ule has wo pa s:
(57) Fon a d Coo dina ion
Rule
(>&)
conj X
-->
[X]&
By his ule he combines he conjunc ion wi h he ca ego y o he second conjunc o ge a new
ca ego y which is he same as he second conjunc bu he has added a
I&'
ea u e. This ule
accoun s o he p eposi ional p ope y.
The second ule o coo dina ion is:
(58)
Backwa d Coo dina ion
Rule
(<
&)
X [X]&
-->
X
ibid. pg.
223.
The second ule wipes he cons i uen ma ked wi h he ea u e
I&'
by combining i wi h ano he
cons i uen wi h he same ca ego y. The esul ing ca ego y is o he same ype as he o he wo.
I allows him o accoun o he in ix p ope y and conjunc ion o like ca ego ies. This ule
allows him o a oid he p oblem o Ajdulue icz's and Lambek's ca ego y which was shown in
In my opinion hese ules ha e a leas wo p oblems. The i s one is in a heo e ic. As we
ha e seen, a Ca ego ial G amma does no ha e any syn agma ic ule. I s de i a ions a e
p oduced by lexical ca ego ies and gene al ope a ions o e ca ego ies. Adding syn agma ic
ules would
be
ad-hoc wi hin he ca ego ial g amma mechanism.
The second one conce ns he na u e o he ules hemsel es. Le us assume ha we accep hese
syn agma ic ules, al hough we should no by he easons alluded o. Then, we canno admi
such ules which in oduce ad-hoc ea u es in hei applica ion. Recall ha S eedman needs a
ea u e o ma k he second cons i uen o he coo dina ion. This ma k allows him o ecognize
his second cons i uen o combine wi h he i s one by means o he second ule. I o bids he
applica ion o ano he ule like backwa d combina ion in
(56).
Fo hese easons I hink a lexical ca ego y mus be ound o coo dina ion wi hin a
ca ego ia11 y-based app oach o coo dina ion.
2.4.
Geach
and
Wood's
Ca ego y
I
will commen on hese wo p oposals oge he because hey a e e y simila . Geach (1971)
sugges ed he ca ego y (59):
(59)
:x(2x) (in his e minology)
This ca ego y says ha in o de o o m a ca ego y o ype
X,
we ha e o ind wo ca ego ies o
he sa ne
ype
X.
Mo o e i says ha he connec i e akes bo h a gumen s a he sa ne. ime,
as
Geach has no con empla ed he linguis ic da a exposed abo e o de end he p eposi ional
cha ac e o conjunc ions.
"...'and , 'o ', he connec i e is el o be joining wo clauses, no going wi h one
a he han ano he "
Geach (1971), pg 131.
To exp ess his idea mo e p ope ly Wood
(1988)
in oduces a new ope a o . This is he in ix
ope a o 'I' which indica es ha a unc o appea s be ween he elemen s o i s a gumen . This
ope a o can only be used when he a gumen is onned by a p oduc :
"as only will i ha e wo elemen s o he unc o o appea be ween, a he posi ion
in he s ing pa allel o ha o he
*
connec i e in he p oduc ca ego y
.
"
Wood
(1988)
pg
90.
The in ix ope a o allows us o cancel one a gumen on he le and ano he on he igh . The
p oduc ope a o indica es ha wo a gumen s o m a single uni , and d aws he place whe e
he lexical conjunc ion appea s in a eal sen ence. Then he ca ego ia1 ype o conjunc ions
sugges ed by
Wood
is:
whe e he supe index
+
means one o mo e occu ences o he ca ego y
X
(i is he as e isk o
Kleene). I is use ul o cases o mul iple coo dina ion like:
(61)
Ma y,
Be h and Pe e
The
Kleene
+
allows us o conjoin all hese coo dina ed ca ego ies and he p oduc symbol
(*)
poin s ou he place whe e he conjunc ion lexical i em appea s, in he las bu one posi ion.
This ca ego y has he in ix p ope y since i cancels bo h ca ego ies oge he , one on i s le and
he o he on i s igh and i conjoins only like ca ego ies
.
One p oblem is ha i does no
accoun o he p eposi ional na u e as i ea s he wo conjunc s
as
o ming one single
cons i uen wi h he conjunc ion.
An addi ional p oblem o his p oposa1 is he use ha Wood makes o he p oduc ope a o .
This ope a o is an old one wi hin he ca ego ial g amma li e a u e and has a e y well de ined
syn ax and seman ics. I was i s in oduced by Lambek (1958) as an ope a o o his Calculus.
A e ha i was esumed by Moo ga (1988) and i is de ined in he ollowing ems:
"An exp ession belongs o a p oduc ca ego y (X*Y) i i is he conca ena ion o an
exp ession o ca ego y X and an exp ession o ca ego y
Y,
in ha o de ."
Moo ga
(
1988)
Wi h his, i seems o be clea ha he in e p e a ion ha should be gi en o a ca ego y o ype
X*Y is a conca ena ion o wo adjacen ca ego ies.
Ne e heless, when Wood p oposes he ca ego y XI(X+*X) as associa ed wi h conjunc ions,
i seems ha she does no use he p oduc ope a o in his sense bu ins ead I*' poin s o he
place whe e he conjunc ion appea s in he sequence o ca ego ies. Then:
"
...
he ca ego y assignmen as XI(X+*X).
*
may
be
any conjunc ion is in ixed
be o e he inal conjunc .
"
Wood (1988), pg. 171.
I will be clea ha i we in e p e he sign I*' as he p oduc ope a o de ined by Lambek, we
should in e p e he ca ego y XI(X+*X) as one ha looks o wo o mo e i ems o he same
ca ego y which a e adjacen . Bu he in e p e a ion gi en in Wood (1988) is ha he p oduc is
pa allel o he conjunc ion, in ac i is a d aw o he conjunc ion. We can gloss his by saying
ha he conjunc ion would be an ope a o which appea s be ween wo ca ego ies ha would
o hen ise appea conca ena ed. Bu , in ac , hey a e no conca ened and he p oduc ope a o
does no desc ibe he si ua ion ai h ully.
Wood's use o p oduc ope a o seems o con use he algeb aic ope a o wi h he na u al
language one. Whene e we use an ope a o o Ca ego ia1 G amma , we a e no e e ing o
any ope a o o na u al language. This is clea in he use o he p oduc ope a o which also
appea s in Wood
(1988).
She says ha a di ansi i e e b is a ca ego y which has o combine
wi h a p oduc o wo NP o ge a VP (SWP): (S NP)/NP*NP. He e, he p oduc ope a o
only means adjacency, i is no d awing he place whe e
a
lexical ope a o o na u al language
appea s.
The e a e wo di e en uses o he p oduc ope a o in Wood's wo k and
I
hink ha i should
be
clea ha wha co esponds o he classical de ini ion o his ope a o does no gi e lexical
eali y o an algeb aic ope a o . The p oduc ope a o means adjacency; he e o e, as we canno
iden i y i wi h a na u al language ope a o , i seems ha i is no he ope a o ha joins he wo
(o mo e) cons i uen s ha en e coo dina ion and ha a e cancelled by he in ix ope a o .
Then, i p oduc canno join he coo dina ed cons i uen s, which ca ego ia1 ope a o does join
he cons i uen s coo dina ed by conjunc ion?.
2.5.
The
P oposa1
As we ha e seen, he p oduc ope a o has some p oblems as a ca ego ial ope a o joining he
cons i uen s coo dina ed by conjunc ion. Mo eo e , Wood's ca ego y has he p oblem o no
accoun ing o he p eposi ional p ope y. On he o he hand, S eedman's ca ego y does so bu
is no a lexical ca ego y as i is he claim o Ca ego ial G amma .
I
will y o gi e a lexical
ca ego y which mee s he h ee p ope ies desc ibed abo e o conjunc ions.
I
shall adop he in ix ope a o . The in ix ope a o is a unc ion o wo a gumen s, namely:
F(x,y). This o mula sugges s ha he wo a gumen s a e cancelled a he sa ne ime. Bu a
unc ion o wo a gumen s can also cancel i s a gumen s in wo s eps: F y(x). I will adop his
second way o cancelling a gumen s o enable his ope a o o nee he p eposi ional p ope y
o he conjunc ion. This mo e is linguis ically mo i a ed, since bo h a e o mally equi alen .
To exp ess his ope a o in Ca ego ia1 G amma o malism
I
will de ine i
as
an
ope a o whch
seeks each a gumen in a di e en di ec ion. I is an ope a o o double di ec ionali y. The i s

a gumen is looked o on he igh and he o he a gumen on he le . In Ca ego ial G amma
e minology i is a conca ena ion o ope a o s o o wa d and backwa d applica ion. Then:
(62)
I
=
I,
Bu I no only wan o exp ess ha a gumen s ha e o
be
sough on he igh and on he le ,
sucessi ely. I also wan o exp ess ha hey a e o
be
cacelled by means o backwa d and
o dwa d applica ion ules. No ope a ion in Ca ego ial g amma has been de ined o ope a o s
o wo a gumen s. So we ha e o build an ope a ion o cancella ion o he in ix ope a o , in he
same way ha he e is one (o mo e) o using p oduc o unc ional applica ion ope a o s. The
ope a ion whlch cancels he in ix ope a o is:
(63)
Rule
o
in@
ope a o :
[ZI(X,Y) Y -->ZLX] [X ZCX
-->
Z]
So he use o he in ix ope a o is equi alen o he conca ena ion o o dwa d and backwa d
applica ion. The sign
"V
is no a new ope a o . I only exp esses he in e media e s ep o he
applica ion o he in ix ope a o . I eminds us ha he second a gumen has o be looked o on
e le and cancelled by backwa d applica ion.
I is di e en om Wood's in ix ope a o , since (63) is
an
ope a ion o wo s eps. I does no
join i s wo a gumen a he sa ne ime: a he i i s ly joins he igh one and, immedia ely a e
ha , he le one.
Bu he ques ion is how o o mally ep esen he se o a gumen s ha he in ix ope a o has o
cancel. In ac , he wo cons i uen s ha he conjunc ion joins o n an o de ed sequence. Fi s
comes he le cons i uen and hen he igh one, bu hey a e no adjacen . Then, hey o m an
o de ed sequence, no a p oduc .
In Ma hema ics and Logic, he idea o o de ed pai and, gene alizing, o n- uple has been used
ui ully. I hink i would
be
e y use ul o in oduce a uple ope a o in o de o sol e many
empi ical linguis ic p oblems. In ui i ely i ep esen s an o de ed sequence o elemen s.
Fo mally, i can be de ined induc i ely:
(64)
<xl
,...
xn>=<<xl. ..xn- l>,xn>, i.e., he n- uple <xl
...
xn> is he o de ed pai whose
i s elemen is he (n-1)- uple cxl
...
xn-l> and whose second elemen is xn.
By in oducing i as
an
ope a o o ca ego ia1 g amma , we need a ule o cons uc ion:
(65)
N-Tuple Fo ma ion:
X
,...,
Y
-->
...
,-&,Y>
This means ha wo ca ego ies which a e in some o de in a sen ence can
be
g ouped in
a
sequence o n- uple o ca ego ies.
A
n- uple is an o de ed se o a conc e e nu nbe o elemen s.
Co espondingl y:
(66)
Tuple decons uc ion:
...
,<X,Y>
-->
X
,...,
Y
Then, ha ing his ope a o we can de ine he se o a gumen s ha he in ix ope a o needs o
ind as
<X,X>.
So he ca ego y ha is p oposed o he conjunc ion is:
The in ix ope a o will
be
aking i s wo a gumen s om he uple om igh o le un i1 i is
emp y by means o o wa d and backwa d applica ion.
323
coo dina ed ph ase would be:
(69)
John and
ma ^
NP XI<X,X> NP
....................
>
NPk<NP>
..........................
<
NP
The ca ego y p oposed,
XI<X,X>,
is equi alen o Lambek's (XK)/X ca ego y, bu excluding
he p oblem men ioned abo e as he in ix ope a o o ces he ca ego ial engine o cancel bo h
a gumen consecu i ely by means o on a d and backwa d applica ion. E en his ca ego ial
ype is e y simila o Ajdukiewicz's, al hough i speci ies he p ocedu e o combina ion in
mo e de ail.
This ca ego y obse es he p eposi ional p ope y since i i s ly joins he igh conjunc .
Mo eo e , i mee s he in ix p ope y and i obse es he like ca ego ies p ope y. Finally
,
i is
a una y ca ego y
as
i always
asks
o one ca ego y.
To allow his ca ego y o accoun o mul iple coo dina ion, we ha e only o add a Kleene
as e isc?. Then:
Recall ha wi h his ca ego y we need no s ipula e he place whe e he conjunc ion appea s as i
uns di ec ly om de ini ion. In addi ion, wi h his o mula ion we main ain in ix ope a o 's
cha ac e is ic o being diadic.
Recall ha he only ope a o ha can ake elemen s om a uple is he in ix ope a o since he
o he ope a o s only ake one a gumen . Then i a n- uple appea s in a ca ego y wi h
a
slash
ope a o
{[/,I),
he n- uple mus be cancelled as a whole. This kind o ope a ion will
be
use ul
in p o iding analysis o some in e es ing phenomena o na u al languages bu
I
shall no ea
hese applica ions he e.
The main di e ence be ween Wood's p oposa1 and wha has been ou lined abo e e e s o he
applica ion o he in ix ope a o . Wood's in ix ope a o akes bo h a gumen s in a single s ep as
she uni ies he p oduc ope a o wi h he conjunc ion and i s wo a gumen s wi h he wo
cons i uen s ha conjunc ion joins. Wi h he ca ego y p oposed he e he applica ion o he in ix
ca ego y uns s ep by s ep, aking i s he igh a gumen and a e ha he le one as has been
shown in
(63).
3.
Some
Examples
In his sec ion
I
would like o show some analysis using he ca ego ia1 ype p oposed in sec ion
2.5
o p o e i s co ec ness.
I
will dis inguish be ween "cons i uen coo dina ion" and "non-
cons i uen coo dina ion".
3.1.
Cons i uen Coo dina ion
To begin wi h,
I
would like o no e he easiness o analysing he coo dina ion o e e y
ca ego ia1 ype
by
means o his ca ego y. The e is no p oblem in analysing a coo dina ion o
NP,
VP,
o any ca ego y.
Then, wo in ansi i e e bs coo dina eci:
(82)
Geo ge is
an
idio
NP (SWP)ININ (NIN)IN N
--T>
S/(SWP)
......................
C>
-
-
-
--
-
-
--
-
--
-
-
-
>
S/(NIN) NIN
Thus, he de i a ion o
(44)
is:
(83)
Geo ge is ei he silly o an idio
NP (SW)/N/N XIX NIN XI&+,X> (NIN)IN N
--T>
SI(SWP)
-------e---------
C>
Sl(N1N)
----------------
>
NIN
..............................
>
(NIN)
C
(NIN)
............................................
<
NIN
......................................................
>
NIN
..............................................................................
>
S
The de i a ion o
(45):

(84)
Hany
is
cle e and ecei ing
a
good educa ion
NP (SWP)/N/N NIN
XI<X+,X>
(N1N)INP NPIN NIN N
--T>
S/(SWP)
------------
C>
Sl(N1N)
..........................................
>
NIN
...............................................................
<
NIN
(85)
Bill is ne ous and unde p essu e
NP (SWP)lN/N NIN
XI<X+,X>
(N1N)INP NP
--T>
S(S NP)
------------
C>
-------------------
>
S/(NIN) NIN
.................................
>
(NIN)k(NIN)
................................................
<
NIN
.......................................................................
>
S
Wi h ega d o he o he lund o sen ences ha allow non-cons i uen coo dina ion om
(47)
o
(49),
I
shall show ha hey ha e he same ca ego ial ype
as
hey occupy he same a gumen al
place.
So,
in Ca ego ia1 G amma
an
Ad e b and a
PP
modi iing a
VP
bo h ha e he same
ca ego ia1 ype
'(SWP) (SWP)'.
Then he de i a ion o sen ence
(48)
is s aigh on a d:
(86)
Geo ge comes omo ow and on Monday
NP S NP (SWP) (SWP)
XI<X+,X>
(SWP) (SWP)
---T>
S/(SWP)
.............................................................
<
SWP
And
(49):
Ca ego ial G amma allows all hese de i a ions in a simple way, by s udying he a gumen a1
possibili ies o cons i uen s. In his way, he e is no non-cons i uen coo dina ion. As shown,
all coo dna ion is cons i uen coo dina ion.9
4.
Conclusions
In his pape ,
I
ha e p oposed a ca ego ial ype o conjunc ion. This ca ego y mee s h ee
impo an obse a ional cha ac e is ics o na u al language conjunchons, p eposi ional na u e,
in ix loca ion and like
ype
coo dina ion.
We ha e conside ed o he possibili ies such as Ajdukiewicz's, Lambek's, S eedman's,
Geach's and Wood's. Theca ego ial ype p oposed he e is lexically based, which a unes wi h
he gene al basis o Ca ego ial G amma . I joins i s a gumen s by means o applica ions,
a oiding he empi ical p oblems ha Lambek's p oposa1 has. Mo eo e , i uses an adap a ion
o he in ix ope a o i s ly poin ed ou by Wood (1988). This new e sion allows us o
-
accoun o he p eposi ional p ope y o he conjunc ion, i s indica ed by Ross (1967).
Finally his ca ego ial ype in oduces a new ope a o wi hin Ca ego ial G amma o malism.
The uple ope a o will pe mi us o sol e o he in ica e p oblems which na u al language
p esen s ha will be ea ed in he u u e. In his wo k,
I
use i o indica e ha he se o
a gumen s ha a conjunc ion mus cancel a e no adjacen bu consecu i e. In his sense, i
leads o be e esul s han he p oduc ope a o , which has been used in he li e a u e, and picks
up he ue spi i o Ajdukiewicz's p oposal.
*The au ho is indeb ed o suppo om he Gene al Depa men o Educa ion o Ca alonia.
No es
1
E e y s ep o a de i a ion has a symbol which says which ope a ion has been applied.
2
The symbol
'B'
o ma k he composi ion is usually used by S eedman in all his wo ks
because his is he symbol o he Combina o o Combina o y Logics equi alen o Lambek's
Composi ion.
3
This ca ego y co esponds o a ela i e p onoun o subjec . I is a ca ego y which needs o
ind a VP on i s igh o gi e a Nominal modi ie .
4
No ice ha he Kleene as e isk could be ew i en
as
an
n- uple o like ype a gumen s.
5
This can be w i en in his way by he equi alence shown abo e. The Kleene as e isk should
also be w i en, as in de i a ion
(80)
below, bu we will no w i e i o he sake o simplici y
when he coo dina ion is no mul iple.
So he o mally s ic ca ego y should be: (SWP)@(SWP)+>.
6
This ca ego ial ype is associa ed wi h e b ph ase modi ie s. I only says ha in o de o
o m a ca ego y o ype e b ph ase (SWP), i needs o ind a e b ph ase on i s le . E e y VP
modi ie will ha e his ca ego ial ype. So,
a
p eposi ion which would modi y a VP will ha e
he ca ego ial ype: '(SWP) (SWP)/NP1.
7
O cou se, i we apply his ca ego ial ype o languages whe e wo d o de is di e en we
should change he di ec ion o ope a o s in he syn ax o in ix ope a o and, pe haps, he
posi ion o he Kleene as e isk.
8
The di ec ion o he ope a o o a modi ie is e y a iable. As
I
am no gi ing
a
heo y o
modi ie s
I
will no ea his poin
,
al hough. he di ec ion will change pa ame ically be ween
languages and occasionally in a language. These a ia ions should be s udied in dep h.

9
Fo u he de ails o gapping phenomena ea ed
as
cons i uen coo dina ion, see S eedman
(1990).
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