Anaphora and IntensionalUy^ in Classical Logic
Jørgen ViUadsen
Technical University of Danmark
Abstract
The dynam ic per^)ective on natural language suggests that texts change the value o f contextual param eteis in m uch the way com puter program s change the value o f program variables. O ne phenomenon w here dynam ic aspects em erge is that o f anaphora (introduction o f referents for later a n ^ h o ric links), but intensionality and tense are other candidate phenomena. How ever, m ost logics used in com puter science fo r the study o f the semantics o f program are often rather com plicated and d iffo e n t from classical logic. W e describe a rqnesentation directly in classical logic using a categorial grammar. In the logic a theory is defined (via axiom s) to describe how in a text a pronoun manages to pick up a referent that was introduced by a determ iner. Besides (nominal) anaphora we discuss how to handle intensionality presem in (belief and knowledge) attitude reports.
Introduction
W e p re se n t a th e o re tic a l stu d y o f c e n tral se m a n tic a l p ro b le m s o f a n a p h o ra a n d in te n s io n a lity in n a tu ra l la n g u a g e te x ts (E n g lish ). A b a sic k n o w le d g e o f a n a p h o ra , in te n s io n a lity a n d lo g ic is assu m ed .
A s te x ts w e ta k e se q u e n c e s o f d e c la ra tiv e an d d is a m b ig u o u s se n te n c e s; h e n c e e a c h te x t h a s a u n iq u e m e a n in g a n d th e o b je c tiv e is to d e fin e a se m a n tic a l th e o ry s p e c ify in g , in s o m e w a y a n d to s o m e e x te n t, ttie m e a n in g o f e a c h t e x t
W e th in k th a t in o rd e r to u s e th e te x ts to c o m m u n ic a te o u r in fo rm a tio n a b o u t fa c ts ( o r a s su m p tio n s ) c o n c e rn in g “ th e u n iv e rs e a ro u n d u s ” th e th e o ry m u s t h a n d le a n a p h o ra a n d in te n s io n a lity a s a m p ly p re s e n t in n a tu ra l la n g u a g e . F o r in sta n c e , c o n s id e r th e fo llo w in g L o ve S to ry te x t:
A m a n k is s e s a w o n u in . S h e b e lie v e s t h a t h e io v e s h e r .
T h e p ro n o u n s (h e , sh e ,...) im p ly a n a p h o ric lin k s a c ro ss se n te n c e s a n d a ttitu d e re p o rts ( b e lie v e t h a t,...) im p ly in te n sio n a l p ro p e rtie s fo r e m b e d d e d se n te n c es.^
F o r in sta n c e th e fo llo w in g s im i le e n ta ilm e n t d o e s n o t h o ld :
H o w e v e r, a re stric te d v e rs io n h o ld s (w h e n n o o v e rla p b e tw e e n a c tiv e q u a n tifie r v a ria b le s a n d free v a ria b le s in th e fo rm u la ). S e e [3] f o r d e ta ils a b o u t d y n a m ic p re d ic a te lo g ic a s w ell a s re fe re n c e s to th e e x te n d e d lo g ic a l sy ste m c a lle d d y n a m ic M o n ta g u e g ra m m a r.
Overview
F irs t d ie g e n e ra l id e a o f lo g ic a l se m a n tic s is d e sc rib e d , s u b s u m in g b o th s ta tic a n d d y n a m ic se m a n tic s. T h e n sta tic s e m a n tic s is d e fin e d (b a se d o n d ie n o tio n o f tru th v a lu e s ) a n d h e re a fte r d y n a m ic se m a n tic s (b a se d o n th e n o tio n o f in fo rm a tio n sta te c h a n g e s).
In fo rm a tio n sta te s c a n b e s tru c tu re d in m a n y w a y s - w e p ro p o se to le t a n in fo rm a tio n s ta te b e a s e t o f a lte rn a tiv e s.^ E v e n w ith o u t s a y in g m o re a b o u t w h a t a n a lte rn a tiv e is w e o b ta in in te re s tin g d e fin itio n s o f v a c u o u s, a b su rd an d d e fin ite in fo n n a tio n sta te s. M o re o v e r, re d u c tio n s a re p o s s ib le i f th e in fo rm a tio n sta te c h a n g e s a re d istrib u tiv e a n d e lim in a tiv e .
W e a rg u e th a t b y a p ro p e r c o n s tru c tio n o f th e s e t o f a lte rn a tiv e s aruqihora c a n b e h a n d le d in a n e le g a n t w a y . W e p ro p o se a re p re s e n ta tio n in lo g ic a l ty p e th e o ry u s in g a c a te g o ria l g ra m m a r. A tin y e x a m p le is p ro v id e d . F in a lly w e d isc u s s h o w to in c o rp o ra te in te n s io n a lity in th is fra m e w o rk .
A fe w w o rd s o n n o tatio n . In sc h e m a s w e ta k e <=> to m e a n ; i f a n d o n ly i f ( f o r a ll a f ^ r o p r ia te in sta n tia tio n s w ith e x p re ssio n s o f th e s h o w n s c h e m a v a ria b le s). W e h a v e tw o e q u a lity s y m b o ls : = m e a n s e q u a lity b y d e fin itio n an d = m e a n s e q u a lity b y c a lc u la tio n (h e n c e = im p lie s = ).
Logical Semantics fo r Natural Language
T h e m a in p u rp o se o f lo g ic a l se m a n tic s fo r n a tu ra l la n g u a g e is , w e th in k , to p ro v id e a p re c is e d e sc rip tio n o f th e in tu itiv e n o tio n o f a v a lid a rg u m en t.^
A rg u m e n ts c a n b e b u ilt in se v e ra l w ay s. W e ta k e a n a rg u m e n t to c o n s is t o f a se q u e n c e o f tex ts t ] , ... . tn (th e p re m ise s ) and a s in g le te x t U (th e c o n c lu sio n ). W e a rra n g e th e a rg u m e n t a s in th e fo llo w in g s c h e m a w ith th e sy m b o l in fro n t o f th e c o n c lu sio n :
W e tra n s la te e a c h te x t t in n a tu ra l la n g u a g e to a fo rm u la in a s u ita b le lo g ic a l la n g u a g e - a lo g ic .^ B y a lo g ic w e m e a n a fo rm a l la n g u a g e e n d o w e d w ith a n o tio n o f e n ta ilm e n t K w h ic h w e re q u ire m u s t m a tc h tiie n o tio n o f a v a lid a rg u m e n t in th e fo llo w in g se n se :
i n
A fo im a l la n g u a g e is a la n g u a g e w ith a p re c is e sy n ta x an d s e m a n tic s ^ T h e s y n ta x d e fin e s th e set o f fo rm u la s ( fo r in sta n c e v ia fo rm a tio n ru le s). T h e se m a n tic s d e fin e s f o r e a c h fo rm u la its m ea n in g ( fo r in sta n c e a m a th e m a tic a l o b j e a ) . In g e n e ra l w e w rite th e m e a n in g o f th e fo rm u la i* a s [r]. T h is h id e s th e tra n s la tio n , b u t no c o n fu s io n c a n arise . W e sh a ll a lso re fe r to [r] a s th e m e a n in g o f th e tex t
t.
W e re q u ire th a t a lth o u g h e n ta ilm e n t is in tro d u c e d a s re la tio n ^ b e tw e e n fo rm u la s (sy n tactical o b je c ts ) i t m u s t o o tie s p o n d to a re la tio n Ti b e tw e e n th e m e a n in g s o f th e fo rm u la s (se m a n tica l o b je c ts ):
N ^ • • • > [^n]> [f*])
S in c e w e a im a t u s in g m a th e m a tic a l o b je c ts a s th e m e a n in g s , o u r g o a l is to fin d o b je c ts w ith e n o u g h stru c tu re to m a k e th e d e fin itio n o f s u c h a re la tio n 71 p o ssib le . In th e s ta tic se m a n tic s th e o b je c ts a re ra th e r c o a rs e -g ra in e d , b u t in th e d y n a m ic se m a n tic s to b e d e s c rib e d la te r, th e o b jects a re m o re fin e -g ra in e d .
Static Semantics
In s ta tic se m a n tic s th e m e a n in g o f a te x t is a tru th v a lu e (0/1 fo r fa lse an d tru e ) w ith re sp e c t to a s o -c a lle d m o d e l. It w o u ld a c tu a lly b e m o te a p p ro p ria te to sa y d ia t th e m e a n in g th e n is a fu n ctio n fro m m o d e ls to tru th v a lu e s , b u t sin c e it tu rn s o u t th a t th e m o d e l p a ra m e te r is c o m m o n to all m e a n in g c o m p o s itio n s (b ase d o n th e fo rm a tio n ru le s) it m a k e s se n se to u s e th e “ w ith resprect to ” p h rase.
W e w rite [r]M f o r th e m e a n in g o f th e te x t r w ith re s p e c t to th e m o d e l Af = (D , d ), w h e re D is the d o m a in (a s e t o f o b je c ts) a n d d is th e d e n o ta tio n (a fu n ctio n fro m c o n sta n ts to m ath e m a tic a l s tru c tu re s o f o b je c ts ). T h e re c a n b e m a n y k in d s o f c o n sta n ts: in d iv id u a l c o n sta n ts (d en o tin g o b je c ts ), firs t o r d e r p re d ic a te c o n sta n ts (d e n o tin g re la tio n s o v e ro b je c ts ), se c o n d o r d e r i n d i c a t e c o n s ta n ts (d e n o tin g re la tio n s o v e r o b je c ts a s w ell a s o v e r re la tio n s o v e r o b je c ts ), etc . T h e d e n o ta tio n o f a p re d ic a te c o n s ta n t is o fte n c a lle d its ex te n sitm . T h e s o -c a lle d o r d e r o f a lo g ic is n o t d e te rm in e d b y th e o rd e rs o f th e c o n s ta n ts p re se n t, b u t b y th e o rd e rs o f th e v a ria b le s o f q u a n tific a tio n .
W e a s su m e th a t th e fo rm u la /* is c lo se d (n o o c c u rre n c e s o f fre e v a ria b le s).* T o a c c o u n t fo r free v a ria b le s in s u b -fo rm u la s th e m e a n in g m u st b e g iv e n w ith re s p e c t to a n a ssig n m e n t a (a ftm ction fro m v a ria b le s to m a th e m a tic a l s tru c tu re s o f o b je c ts) b e sid e s th e m o d e l M , b u t th is c o m p lic a tio n c a n b e ig n o re d h e re . A s fo r c o n sta n ts th e re ca n b e m a n y k in d s o f v a ria b le s. In first o rd e r lo g ic , q u a n d fic a tio n is a llo w e d o v e r in d iv id u a l v a ria b le s o n ly , w h e re a s in h ig h e ro r d e r lo g ic q u a n d fic a d o n is a llo w e d o v e r p re d ic a te v a ria b le s to o .
E n ta ilm e n t is d e fin e d as:
<1
t n
Dynamic Semantics
D y n a m ic s im p ly th e (o m tin u o u s o r d isc re te ) c h a n g e o f s o m e th in g - in c a se o f n a tu ra l la n g u a g e se m a n tic s w e s u g g e st to c o n s id e r th e c h a n g e fro m o n e in fo n n a tio n s ta te to a n o th e r in fo n n a tio n sta te o f a n a g e n t (tu u n a n o r p ro g ra m ) p ro c e s s in g th e te x ts W e firs t p re s e n t th e g e n e ra l id e a le a v in g th e d e ta ils o f in fo rm a tio n sta te s u n s p e c ifie d . W e th e n c tm s id e r d iffe re n t s p e c ific s tru c tu re s a s in fo rm a tio n sta te s.
W e in tu itiv e ly h a v e th e fo llo w in g d istin g u ish e d in fo rm a tio n sta te s:
V a c u o u s in f o r m a t io n s ta t e
T h e a g e n t h a s n o in fo rm a tio n a t all.
A b s u r d in f o r m a t io n s ta t e
T h e a g e m h a s c o n tra d ic to ry in fo rm a tio n .
D e fin ite i n f o r m a tio n s ta t e
T h e a g e n t h a s m a x im a l in fo rm a tio n .
O fte n th e re is o n ly o n e v a c u o u s an d o n e a b su rd in fo rm a tio n s ta te , w ritte n /q a n d loo, w h e re a s th e re a re n u m e ro u s d e fin ite in fo rm a tio n s ta te s.f^ W e o fte n h a v e a p a rtia l o rd e rin g o f in fo rm a tio n sta te s c o rre sp o n d in g to g e ttin g *‘b etter* ’ in fo rm e d .
In d y n a m ic se m a n tic s w e le t [r] b e a fu n c tio n fro m in fo rm a tio n s ta te s to in fo rm a tio n sta te s. O b se rv e th a t th e re is n o n e e d to s e e th e m e a n in g re la tiv e to a m o d e l a s in th e s ta tic se m a n tic s (alth o u g h it is p o ssib le to d o so ). I f th e in itia l in fo rm a tio n s ta te is /q w e h a v e th e fo llo w in g in fo rm a tio n sta te c h a n g es:
r [ ^ l ] , J 1^*1 T [^ > 1 [ ^ « 1 T [ * • ) T
Jo ---* Jl ---* J2 ---* • • • ---* Jfi ---* J*
W e d e fin e th e re le v a n t re s u ltin g in fo rm a tio n sta te s as follow s:^ ^
/ „ = [ < „ ] ( ... [ < i] (/o )) /* = [ M ( / „ )
T h e re a re se v era l d iffe re n t d e fin itio n s o f e n ta ilm e n t p o s s ib le . W e h e re fo llo w th e s lo g a n : e n ta ilm e n t m a k e s e x p lic it im p lic it in fo rm a tio n . M o re p rec isely : T h e c o n c lu s io n u is im p lic it in th e p re m iss e s
t i...i f no (im p o rta n t) c h a n g e in in fo rm a tio n sta te o c c u rs w h e n r , is p ro c e s s e d a fte r t], h a v e b e e n p ro ce sse d :
<n
H e re -> is a su ita b le e q u iv a le n c e re la tio n th a t c o m p a re s in fo rm a tio n sta te s f o r (im p o rta n t) c h a n g e s; i f a ll c h a n g e s a re im p o rta n t th e n w e c a n u s e = f o r « .
Information States as Sets o f Alternatives
L e t i4 s ( a ] , <22» <<3> —) b e a se t o f n o t fu rth e r s p e c ifie d a lte rn a tiv e s an d le t a n in fo rm a tio n state I
b e a s e t o f a lte rn a tiv e s ( / c ^4). T h e d e fin itio n s o f th e v a c u o u s a n d a b su rd in fo rm a tio n sta te s are:
/q kA I o o m 0
T h e d e fin ite in fo rm a tio n sta te s a re I = [a], a e A.
T h e in fo rm a tio n s ta te s h a v e a p a rtia l o rd e rin g v ia th e s u b s e t re la tio n c - F o r e x a m p le:
{ ® i} Q ^
{01,02} 2 {02,03}
F o r a p a rtic u la r in fo rm a tio n s ta te c h a n g e [r] th e re a re so m e in te re s tin g p ro p e rtie s it m ay h av e:
D i s tr i b u ti v e i n f o r m a t io n s ta t e c h a n g e s =
U W ({® })
^« € / E l im in a t iv e i n f o r m a t io n s ta t e c h a n g e s [ f] (7 ) C / f o r a l l /
D is trib u tiv e in fo rm a tio n sta te c h a n g e s m e a n th a t th e a g e n t w o rk s “p o in t-w is e ” , i.e. b a se d o n the d e fin ite in fo rm a tio n sta te s. T h is p ro p e rty d o e s n o t h o ld fo r a g e n ts d e a lin g w ith d e fa u lts etc. A n illu s tra tiv e e x a m p le :
t E J o h n m i g h t lie .
W ( / )
\ loo Ol
t h e r e is a n o € / s u c h t h a t “ J o h n l ie s ” , o t h e r w i s e .
N o te th a t i f o n e h a s th e in fo rm a tio n th a t “Jo h n d o e s n o t lie ” it is a b su rd to p ro c e s s r. It is e a sily seen fro m th e a b o v e d e fm itio n th a t th e in fo rm a tio n sta te c h a n g e is n o t d istrib u tiv e , sin c e th e te st a c / d o e s n o t w o rk p o in t-w is e . H o w e v e r, th e in fo rm a tio n sta te c h a n g e is e lim in a tiv e , sin c e / » c / Tor a l l / .
E lim in a tiv e in fo rm a tio n s ta te c h a n g e s m e a n s th a t th e a g e n t g e ts “ b e tte r in fo rm e d ” to w a rd s th e d e fin ite in fo rm a tio n sta te s (w ith th e e x c e p tio n th a t th e o n ly c h a n g e fro m a d e fin ite in fo rm a tio n sta te is to th e ab s u rd in fo rm a tio n state). T h is p ro p e rty d o e s n o t h o ld fo r a g e n ts d e a lin g w ith re v is io n etc.
W h e n th e s e tw o p ro p e rtie s a re “u n iv e rs a l” - th a t is fulfiU ed f o r all te x ts t c o n sid e re d - o p tim is a tio n s a re p o ssib le :
• D is trib u tiv e in fo rm a tio n sta te c h a n g es:
In ste a d o f m e a n in g [t] a s a fu n c tio n fro m in fo rm a tio n sta te s to in fo rm a tio n sta te s, th a t is a fu n c tio n fro m se ts o f a lte rn a tiv e s to se ts o f a lte rn a tiv e s, w e j u s t s p e c ify a re la tio n —* t b e tw e en a lte rn a tiv e s :
>t a
e [f]({a})
T h e m e a n in g [r] is th e n o b ta in e d as:
T h is g iv e s d istrib u tiv e in fo rm a tio n sta te c h a n g e s, sin c e th e d iffe re n t e n v iro n m e n ts ca n b e seen p o in t-w is e . I f w e a p p ly th e a b o v e m e n tio n e d o p tim isa tio n fo r d istrib u tiv e in fo rm a tio n sta te ch a n g es th e n w e g e t n o n -e lim in a tiv e in fo rm a tio n sta te c h a n g e s, sin c e th e d e fin ite in fo rm a tio n sta te s m u st a lso b e u s e d an d n o t o n ly th e in fo rm a tio n sta te s re su ltin g fro m p ro c e s s in g th e tex ts.
N o te : T h e c h a in o f in fo rm a tio n sta te s /q, ^ e lim in a tiv e in fo rm a tio n sta te c h a n g e s w ith re s p e c t to r j,.../« ; h e n c e f o r 1 ^ i < n w e h a v e :
Representation in Logical Type Theory
B e s id e s th e b a s ic ty p e t o f tru th v a lu e s w e in tro d u c e th e fo llo w in g b a s ic ty p es:
e E n titie s
i In d ic e s
T h e v a ria b le s a n d v.w a re o f ty p e s e a n d i resp e c tiv e ly .
T h e w o rld s a re re p re s e n te d v ia m o d e ls f o r lo g ic a l ty p e th e o ry a n d tiie ty p e o f en tities.^^ T h e e n v iro n m e n ts a te re p re s e n te d v ia th e ty p e o f in d ic e s u sin g a sp e cia l c o n sta n ts f o r ea ch sto re , th a t is fo r e a c h (a n a p h o ric ) d isc o u rs e m a r k e ri'^
mi,...,m4: r —» e
T h e fo llo w in g d e fin itio n ( fo r e x p re sse s th e e q u a lity o f tw o e n v iro n m e n ts o n th e v a lu e s o f all sto re s e x c e p t s to re j:
~ i = X v w { / \ TUj V = TTlj w )
i< i< *
T h e f o l l o w i n g a x io m s ( f o r l ^ i ^ k ) e x p r e s s e s t h a t w e c a n c h a n g e e a c h v a lu e o f e a c h s to r e i n a ll e n v i r o n m e n t s i n d e p e n d e n tl y :
V vV x 3 w (v w A rriiW = x )
Example using a Categorial Grammar
W e u se th e fo llo w in g c a te g o ria l g ra m m a r (h e re w ritte n a s a p h ra s e s tru c tu re g ra m m a r, b u t s e e [1 ] f o r d e ta ils a n d a m o re su b sta n tia l fra g m e n t):
s
NP VP
Sentence
VP
—» walk 1 whistle | . . .
Verb Phrase
NP
DET CN 1 hex 1 . . . 1 he*
Noun Phrase
DET -» ai 1 . . . | a*
Determiner
C o n s id e r th e fo llo w in g tin y e x a m p le (m o ip h o lo g y ig n o re d ):
I ■ A| man walks.
T ra n sla tio n o f le x ic a l e n trie s (v a ria b le s X .Y o f ty p e e —» i —> i —» r):
( a ,) * = X X Y X v w 3 v ' v " { v v ' A X ( m i v ') v ' v ” A Y ( m i v ') v " w )
( h c i) * = AA^Av u7 (X ( m i v ) r « ; ) (/>)• = AxAvu»(t> = w A p ^ x )
w h e r e p* = w a l k | w h i s t l e | m a n | . . . a r e c o n s t a n t s o f t y p e e t c o r r e s p o n d i n g t o
p = w a lk I w h is tle | m a n | . . .
B a se d o n d ie g ra m m a r, w h e re strin g c o n c a te n a tio n c o rre s p o n d s to fim c tio n a p p lic a tio n (sh o w n a s j u x ta p o a tio n a s u s u a l), w e o b ta in th e fo llo w in g m e a n in g a fte r X -co n v ersio n s:
—»j = (a i)* ( m a n ) * (w a lk )* = A v u '( v tn A m a n ( m i w ) A w a l k ( m i w ) )
A n a lo g o u sly :
I’ ■ Hei whistles.
= X
vm<
v=
wawhistle(m] .
w))
S e n te n c e c o n ju n c tio n u s e s th e o p e ra to r 0 k)^qX v^^B v'(pv^^A qV 'w ) (v a ria b le s p ,q o f ty p e a n d u sin g th e a b o v e -m e n tio n e d a x io m s w e h a v e th e d e s ire d resu lt:
— = © t* t'* = A v iu (v ~ i tn A m a n ( m i t n ) A w a l k ( m i w ) A w h i s t l e ( m i t n ) )
A ll th e re re m a in s is to sp e ll o u t th e d e ta ils o f a su ita b le re la tio n > a n d w e h a v e o u r e n ta ilm e n t ^ u s in g a sy ste m atic a l a n d c o m p o sitio n a l a n a ly sis in a c la ssic a l lo g ic .
Intensionality
A c o m m o n ^ jp r o a c h to in te n sio n a lity re s ts o n th e n o tio n o f “ p o s s ib le w o rld s ’* (s e e 1), u s u a lly m a n ip u la te d b y n e w o p e ra to rs lik e □ an d 0 D ia m o n d . S u c h a “p o s s ib le w o r ld " c o rre s p o n d s to th e w o rld c o m p o n e n t o f o u r a lte rn a tiv e s in an in fo rm a tio n state. H o w e v e r, in o u r re p re s e n ta tio n in lo g ic a l ty p e th e o ry w e h a v e k e p t a fix e d w o rld v ia m o d e ls f o r th e lo g ic a l ty p e th e o ry an d th e ty p e o f e n titie s (th e ty p e o f in d ic e s is u se d fo r th e e n v iro n m e n ts to g e th e r w ith th e d isc o u rs e m a rk e rs ), b u t th is s e tu p w ill n o t w o rk a n y lo n g e r. O n e w a y o u t is to in tro d u c e a n e w b a s ic ty p e w o f w o rld s a n d le t p^ fro m a b o v e b e c o n sta n ts o f ty p e w - > e - » r in ste a d o f ty p e e - * t j u s t A sp e cia l m a rk e r mworid o f ty p e i - » w w o u ld th e n s e le c t th e c u rre n t w o rld . W e in te n d to ^ 1 1 o u t th e d e ta ils in a fu tu re p a p e r.
Notes
1) N o tic e th a t “in te n tio n " m e a n s p u rp o se , a im , e tc . w h e re a s “ in te n s io n " h e re (a s w e ll a s in lo g ic an d th e p h ilo s o p h y o f la n g u a g e ) is th e d ire c t o p p o s ite o f “e x te n s io n " (cf. th e d is tin c tio n b e tw e e n se n se an d referen ce)!
Jø rg e n V illa d se n
D e p a rtm e n t o f C o m p u te r S c ien c e T e c h n ic a l U n iv e rs ity o f D e n m a rk ID /D T H . B u ild in g 344
D K -2 8 0 0 L y n g b y , D e n m a rk
