DYNAMIC UPDATE PREDICATE LOGIC

DYNAMIC UPDATE PREDICATE LOGIC

László Kálmán and Gábor Rádai

Theoretical Linguistics Programme, Budapest University (ELTE) Research Institute for Linguistics, H A S , Room 120

Budapest L, P.O. Box 19. H-1250 Hungary

E-mail: kalmanCnytud.hu, radaiCnytud.hu

Working Papers in the Theory of Grammar, Vol. 2, No. 6 Supported by the Hungarian National Research Fund ( O T K A )

Theoretical Linguistics Programme, Budapest University ( E L T E ) Research Institute for Linguistics, Hungarian Academy of Sciences

Bu d a p e s t I., P .O . B o x 19. H -1 2 5 0 Hu n g a r y

Te l e p h o n e: (36-1) 175 8285; Fa x: (3 6 -1 ) 2 1 2 2050

D Y N A M IC U P D A T E P R E D IC A T E LO G IC

László Kálmán and

Gábor Rádai

Theoretical Linguistics Programme, Budapest University and

Research Institute for Linguistics, Hungarian Academy of Sciences k a l m a n Q n y t u d . h u , r a d a i @ n y t u d . h u

0. In trod u ction

In current (dynam ic) theories of sem antics, such as G roenendijk a n d S to k h o f’s (1991) Dynamic Predicate Logic (DPL), anap h o ric expressions are tre a te d as b o u n d variables. The m a in draw back of this app ro ach is th a t it relegates th e way in which a n a p h o r/a n te c e d e n t relations are d eterm in ed to som e ex tern al (a n d notoriously unspecified) m odule. For exam ple, consider:

(1) Joe has a cat. It is black.

l3 r ( c a t ( r ) A owns(y, x)) A b la c k ( r)’

T h e D PL form ula in th e above tra n slatio n is in te rp re te d as if th e variable x in b lack (z) was b o u n d by th e ex isten tial quantifier in th e first conjunct. However, w hy x ra th e r th a n any o th e r variable appears in th e tra n sla tio n of th e second sentence is o u tsid e th e scope of th e theory.

No d o u b t, anaphoric reference has certain aspects which, in all probability, can n o t be p u t in sem antic term s. Those aspects m ay be governed by genuinely form al (e.g., syntactic) p ro p e rtie s of th e u tteran ces in which th e an ap h o rs a p p e a r, a n d th ere is n o t much ho p e th a t we can explain th e m w ithin sem an tics.1 B u t th e fact th a t som e properties of an ap h o ric binding fall outside th e scope of sem antics does no t ju stify a tre a tm e n t in which an ap h o r/an te c ed e n t relatio n s are determ ined by m echanism s entirely in d ep en d en t from sem antics, by pure m agic, as it were. In p articu lar, a n an ap h o r always requires th a t th e re b e exactly one salient ind iv id u al

1 For exam ple, the gender agreem ent betw een th e anaphor a n d its antecedent often h as no sem antic c o u n terp art, because g ram m atical gender is a p u rely form al featu re in m any languages. The o th e r well-known case w hen a n a ­ phoric relations d ep en d on form al factors is related to th e d istin ctio n betw een anaphoric vs. reflexive p ro n o u n s in languages like English:

2

0. Introduction

in th e context th a t c a n serve as its an teced en t, a t least am ong th e c an d id a te s no t ex clu d ed by form al fa c to rs (such as gender an d sy n tactic s tru c tu re ). Since th is is a fact a b o u t th e sem antics of a n ap h o rs, it should b e c ap tu red by sem antics. O n th e o th e r hand, a n a p h o rs lexically carry some descriptive content w hich co n strain s th e ra n g e of th eir p o ssib le an tecedents, an d those co n strain ts are also sem antic.

In w h at follows, we will develop a dynam ic th e o ry of an ap h o rs w hich could acco u n t for b o th th e uniqueness requ irem en t an d th e descriptive c o n ten t of a n a ­ p h o rs, as an a lte rn a tiv e to th e b o u n d -v ariab le view. O u r tre a tm e n t is a n a tte m p t to fo rm ally develop, in a dynam ic fram ew ork, ideas on anaphoric b in d in g th a t were p ro p o sed by v ario u s researchers as early as C ooper (1979) a n d E v an s (1980), a n d w hich have b een inform ally p ro p o sed in dynam ic sem antics by K á lm á n (1995) a n d G roenendijk et a1. (1995).

G iven th e fact t h a t th e view of in fo rm atio n sta te s as sets of assig n m en t func­

tio n s in D P L is m o tiv a te d by th e bound -v ariab le view of an ap h o rs th a t we are criticizin g , we will h a v e to develop a different concept of inform ation s ta te s in th e first p la ce (Section 1.1). If we ad o p t th is new concept, we can sto re w h a t entities have b e en in tro d u ced in th e discourse, w ith o u t know ing w h at variables have been u sed for in tro d u cin g th e m . T here is no reason why we should keep tra c k of dis­

course referents by a r b itr a r y nam es (i.e., variables in th e dom ains of assignm ent fu n c tio n s) if we do n o t w ant to use those nam es to refer to th em la te r on. T hen,

(2) Joe likes h im .

*‘Joe likes h im self’

(3) Joe reminded Peter of him .

*‘Joe re m in d e d P e te r of h im self’

In th e se sentences, th e anaphoric p ro n o u n him m ay n o t be co-referential w ith any o th e r noun p h ra s e in th e sentence. T h e fact th a t co-reference h as to be ex p ressed by a reflexive pro n o u n in these sy n tactic configurations pro b ab ly c a n n o t be ex p ressed o r explained in sem antic term s. Finally, th e sy n tactic a n d rh eto rical s tr u c tu r e of an u tte ra n c e som etim es biasses a n ap h o ric relatio n s w ith o u t fully d e term in in g them :

(4) Whenever Joe meets Peter, he greets him .

(5) Joe didn’t recognize Peter, because he had shaved his beard.

T h e preferred re a d in g of (4) is w hen th e g ram m atical role of each a n ap h o r m a tc h e s th a t of its antecedent. In (5), on th e o th e r hand, th e b e st way o f en su rin g th e coherence of th e discourse is to tak e he to be co-referential w ith Peter ra th e r th a n Joe, because th is is th e sim plest way of estab lish in g th e desired rh e to ric a l relatio n betw een th e two clauses. For d e tails on this m ech an ism see P o lan y i (1988) a n d P riist (1992).

0. Introduction 3

in Section 1.2.1, we define a first-order language in terp reted dynam ically in w hich quantification over th e discourse universe can be expressed in an elegant way. T h is is necessary because th e uniqueness of th e antecedents of an ap h o rs m u st b e s a t­

isfied in th e discourse universe ra th e r th a n the entire model. A fter exam ining certain logical p ro p erties of the resu ltin g system (Section 1.2.2), we ‘p a rtia liz e ’ th a t language to account for presuppositions (Section 1.2.3). T h e reaso n for th is is th a t th e requirem ents th a t an ap h o rs impose on th e ir antecedent a re of a p re- suppositional ch aracter. We also exam ine the m ost essential logical p ro p e rtie s of th e system in tro d u ced . Finally, in Section 2, we explain the consequences of o u r tre a tm e n t for various phenom ena related to anaphors, such as donkey sentences an d th e in teractio n of an ap h o rs w ith m odal operators.

1. D yn am ic U p d a te P re d ica te Logic

Possible altern ativ es to th e bound-variable approach to anaphors m u st tre a t a n a ­ phors as quantificational. Consider:

(1) Joe has a cat. It is black.

U nder th e q uantificational view, th e anaphoric pronoun it in th e above piece of discourse m ust be in te rp re te d as th e condensed form of a definite d escrip tio n such as ‘th e non-hum an in d iv id u a l’, which involves a quantifier (‘th e re is exactly one non-hum an in d iv id u a l’) th a t is possibly presuppositional. We will call this ty p e of quantification anaphoric quantification.

T h e in terestin g fact ab o u t expressions of anaphoric quantification is th a t th e y c o n stitu te unstable propositions in th e sense of V eltm an (1981). T h a t is, for in ­ stance, ‘th e re is exactly one n o n -h u m an individual’ m ay be tru e in th e co n tex t in which (1) is u tte re d , b u t false in a subsequent context, in which m ore th a n one non-hum an ind iv id u al has been introduced. T he only dynam ic sem an tic th e o ry in which such u n stab le propositions exist is V eltm an’s (1981, 1990) U pdate Sem an­

tics. U p d a te Sem antics, however, uses a propositional logic, so it can n o t express quantification over individuals. T h e only possible source of in sta b ility in V eltm an (1990) is th e possibility o p e ra to r. Form ulae of th e form ‘<)<p’ (re a d as ‘m ig h t cp’, and in te rp re te d in a n epistem ic m an n er) m ay be tru e in an in fo rm atio n s ta te th a t does no t exclude th e tr u th of </?, b u t can be falsified by subsequent in fo rm atio n . We will in tro d u ce a n o th er ty p e of u n stable propositions, namely, those involving anaphoric quantification. For exam ple, th e anaphoric version of th e q u an tifier

‘th e re is exactly o n e ’ quantifies over th e dom ain of entities alread y in tro d u c e d in a context. T hus, it gives rise to non-upward-entailing quantification, w hich m ay becom e false from tru e as we enlarge th e universe of discourse.

T h e in sta b ility of an ap h o ric quantification will m anifest itself in th e p ro p e rtie s of th e fun ctio n th a t assigns a truth value to every form ula an d in fo rm atio n s ta te .

4

1. Dynamic Update Predicate Logic

F o r exam ple, we w ant to say th a t ‘th e re is exactly one n o n -h u m an in d iv id u a l’ is tr u e in an in fo rm atio n s ta te if an d only if the existence of exactly one n o n -h u m an in d iv id u a l can b e ta k en for g ran ted in th e given in fo rm atio n state. T h u s, a form ula m a y be tru e in an in fo rm atio n sta te w ith o u t being tru e in th e a c tu a l world, even if th e in fo rm atio n s ta te is tru e to th e w orld. O n th e o th e r hand, if ‘th e re is exactly o n e n o n -h u m an in d iv id u a l’ is tru e in a tru th fu l in fo rm atio n sta te an d , m oreover, in a ll its extensions as well (i.e., in th e m ore inform ative in fo rm atio n s ta te s th a t m ay a rise from it), th e n th is p ro p o sitio n m u st indeed be tr u e in th e w orld. Obviously, th e m eaning of an a n a p h o r like it does n o t entail th a t th e world co n tain s ju s t one n o n -h u m a n in d iv id u al.

T h e sy stem th a t we will use in th e following is designed to be th e sim plest o n e th a t can express th e above concepts (including V e ltm a n ’s (1990) concept of

‘u p d a tin g ’). It is called Dynam ic U pdate Predicate Logic (DUPLO for sh o rt), to ex p re ss its close k in sh ip to b o th U p d a te Sem antics a n d D PL. In th e re st of th is sec tio n we first ex p lain th e concept of an inform ation s ta te in D U P L O . T h e n we d escrib e th e sem antics of D U PLO form ulae, which are first-order fo rm u lae w ith a p o ssib ility o p e ra to r ‘<0>’ sim ilar to V eltm an ’s (1990) a n d a p resu p p o sitio n o p e ra to r

‘A ’ sim ilar to B eav er’s (1992).

1 .1 . In form ation S ta te s in D U P L O

As can be ex p ected fro m th e above, in fo rm atio n s ta te s in D U PLO do n o t sto re p o ssib le values of v ariab les (assignm ents are p a ra m e te rs of th e sem antic-value fu n c tio n s as in classical, s ta tic logic), b u t they do keep track of w h a t entities h av e been m en tio n ed a n d w h a t has been le arn t a b o u t th em . An in fo rm atio n s ta te in D U P L O consists of possibilities each of which carries in fo rm atio n a b o u t w h at th e w orld m ay b e like. B u t possibilities are n o t com plete descriptions o f th e world, th e y are p a rtia l in th e sense th a t they express th a t c e rta in facts are know n to be tr u e a b o u t som e e n titie s th e existence of w hich is suggested by previous discourse, b u t n o t every fact is know n ab o u t th em .2

T h e way in w hich we w ill represent possibilities, i.e., p a rtia l in fo rm atio n ab o u t th e w orld, is th e following. E ach possibility contains a n u m b e r of a lte rn ativ es, som e o f th e m m ore co m p lete th a n th e others, possibly w ith com plete possible w orlds a t o n e end of th e scale. T hese altern ativ es will b e called model fragm ents. A m o d e l fragm ent is e x actly like a (first-order) m odel, ex cep t th a t its u n iv erse m ay b e em p ty . It is like a su b se t of the universe of th e m odel, plus an in te rp re ta tio n fu n c tio n th a t is re s tric te d to th a t universe.

2 In w hat follows, we will assum e th a t th e in te rp re ta tio n functions a re first- o rd er, i.e., th e y ju s t assign a set of n -tu p les to each n -a ry p red icate co n sta n t.

E x ten d in g in fo rm atio n sta te s to higher-order in te rp re ta tio n fu n ctio n s w ould ra ise no p ro b lem a t all.

1.1. Information States in DUPLO 5

(6) Definition: Model fragments

A m odel fragm ent / is an ordered pair (A//, J / ) such th a t U/ , th e universe of / , is a set of individuals (tak en from a countable set U con tain in g all individuals), and th e interpretation function of / , X/ , w hich assigns a set of n -tu p les of individuals in V{U J) to every n -ary p re d ic ate co n stan t. We will refer to th e entire set of m odel fragm ents satisfying th e se co n strain ts as J -.

We also need the concept of the informativity of m odel fragm ents:

(7) Definition: Informativity of m odel fragments

A m odel fragm ent = (T/y^Jy,) is at least as inform ative as th e m odel fragm ent f 2 = (W/2,X /a) (w ritten: f x C f 2) iff

(i) Uh C W /n and

(ii) for every predicate con stan t P , X f2(P ) C X fl (P ).

Every possibility p in an inform ation state n is a set of m odel fragm ents. We will talk a b o u t th e core of p, which is th e set of its least in fo rm ativ e elem ents:

(8) Definition: Core of a possibility

If P C T is a possibility, th en th e core of p (w ritten: C (p)) is C ( p ) = { / e p : A ( / E / ' = ► / ' = / ) } •

p e p

T h e universes of th e core elem ents of a possibility represent p o te n tia l discourse universes. W h en u p d a tin g an inform ation sta te in such a w ay th a t a new in ­ dividual is m entioned in th e discourse, the cardinality of every core elem ent of every possibility will increase by one. We will refer to this process inform ally as

‘ex ten d in g th e discourse universe’.

We will also need th e set of m odel fragm ents in a possib ility p th a t are at least as inform ative as a certain m odel fragm ent / £ p:

(9) Definition: The cloak of a model fragment

If p is a possibility in an inform ation state, and / £ p is a m odel fragm ent, th e n the cloak of / in p (w ritten j,pf ) is

I p f = < ! . ( { / 'e p : / ' £ / } .

Finally, we will use th e concept of the set of the least inform ative extensions of a possibility p satisfying d>:

M A X p,c , ( * ) =def {p1 C p: $ & / \ ( W / p " ] k p ' c p") =* p" = p)

p " C p

6 1.1. Information States in DUPLO

w here $ \p ' /p"] is th e sam e as <f>, w ith th e free occurrences of p' s u b s titu te d for by p".

T h e possibilities in a n inform ation s ta te need n o t b e com patible w ith each o th e r; th e m odel frag m e n ts w ith in a possibility need n o t, eith er. T h a t is, th e y need n o t be frag m en ts of th e sam e m odel, so to say. T h e m o st inform ative elem ents of a p o ssib ility can b e co m p lete m odels, nam ely, those th a t could be m odels of th e real w o rld given th e in fo rm atio n represented by th e in fo rm atio n state. T h is need no t o rig in ate fro m th e discourse itself; certain possible w orlds can be excluded alread y a t th e o u ts e t, in th e b e g in n in g of a conversation, so th a t th e single possibility th a t we s ta r t th e conversation w ith need no t co n tain ab so lu tely all possible m odels.

F o r exam ple, it m ay b e th e case th a t ‘all b ird s fly’ is tru e in all m ost in fo rm ativ e m o d el frag m en ts in th e in itia l possibility, alth o u g h no b ird w hatsoever is p resen t in th e universes of th e core elem ents. We will not dwell u p o n th e q u estio n w h at ty p e of in fo rm atio n m ay b e present in this form , nor how it gets there. W e sim ply w ill allow a n y set of po ssib ilities to act as an in fo rm atio n state:

(10) Definition: Inform ation states

T h e set IT of inform ation states is V (V (J-)).

1 .2 . T h e S em a n tics o f D U P L O

T h e lan g u ag e of D U PL O is a first-o rd er language w ith equality, a m o d al o p e ra to r

‘O ’, an d a p re su p p o sitio n o p e ra to r ‘A ’.3 We will proceed in two steps for th e sake o f ex p o sito ry convenience. We will first define a version o f D U PLO w ith to ta l fu n c tio n s (w ith o u t th e A o p e ra to r), th e n we will in tro d u ce th e A o p e ra to r and, a t th e sam e tim e, p a rtia l sem antic-value functions.

1 .2 .1 . T h e T otal V ersion o f D U P L O

W e define tw o im p o rta n t concepts in th e following. T h e first concept is th e truth o f a fo rm u la in an in fo rm atio n sta te (u n d e r a n assignm ent). T h e tr u t h fu n ctio n is a th ree-v alu ed function, b ecau se we w ant to m ake a d istin c tio n betw een ‘know n to b e tr u e ’ ( t r u t h value 1), ‘know n to be false’ ( tr u th value 0) an d ‘n o t know n to b e e ith e r tru e o r false’ ( tr u th value | ) . T h e second cen tral concept is th e resu lt of updating an in fo rm atio n s ta te w ith (th e in fo rm atio n co n ten t of) a fo rm u la (u n d er a n assig n m en t). T h e t r u t h fun ctio n an d th e u p d a te fu n c tio n are defined using sim u lta n eo u s recursion.

3 For th e sake of sim plicity, we will no t in tro d u ce fu n c to rs into th e language.

In p a rtic u la r, we exclude individual co n stan ts. We decided to pro ceed in this way to av o id com plications th a t have no relevance for th e topic of th is p ap er.

W e believe th e system to b e proposed could easily b e enriched w ith fu n cto rs.

1.2.1. The Total Version of DUPLO 7

(11) Definition: DUPLO’s language (total version)

The to ta l version of D U P L O ’s language is an ordered quadruple C' =def (L C ,V ar, Con, Form ).

The set LC of logical constants, the set Var of individual variables a n d th e set Con of non-logical constants are pairw ise disjunct.

LC =dei { 3 ,- \ 0 , - s >,(,)};

Var =def {Xi}i<u>;

Con =def t^J C o n ^ ; l<n<u>

C on(n) —def { P /n ))i< W.

The set Form of formulae is defined as th e sm allest set satisfying th e clauses of th e definitions of atom ic form ulae and of th e tru th -v alu e fu n ctio n below . We will use th e n o ta tio n | • \g for the first-order sem antic-value fu n ctio n for th e atomic formulae of D U PLO . This function yields a classical tr u t h value (1 or 0) for every form ula and m odel fragm ent.

(12) Definition: Atomic formulae and their first-order truth values

If g is an assignm ent function an d / is a m odel fragm ent, th e n th e first-order semantic-value function | • \f is defined in a s ta n d a rd way, as follows:

(i) If . .. , x „ £ Var and P £ Con^n\ th e n ‘P ( x j , . . . , x n ) ’ £ Form is a n atom ic formula of D UPLO.

| P ( x i , . . . , £ „ ) | j dei if ( g ( x i ) , . . . , g ( x n)) £ J / ( P ) ; otherw ise.

( Ü ) If X i,X² € Var, th en ‘(x i = X2)’

PL O .

I ( * l = X 2 ) \ I = d e i

£ Form is an atomic formula of D U - r i i f 2 ( x i ) = g (x 2y,

1 0 otherw ise.

The tr u t h value of a form ula ip in an inform ation s ta te 7r (u n d er an assig n m en t g) will be w ritte n as [<p]^(7r) (th e p a rtia l version of th is function will be [-]s ). T h e resu lt of u p d a tin g an inform ation sta te ir w ith a form ula ip (u n d er an assig n m en t g) is w ritten as [ ^ ^ (tt) (th e p a rtia l version being I ^ J s i 71'))- We will now define b o th th e to ta l tr u t h function and th e to ta l u p d a te function. T h e definition o f th e tr u t h function also contains th e definition of non-atom ic form ulae of D U P L O . W e will rely on th e concept of modified assignments, defined in the u su al way:

8 1.2.1. The Total Version o f DUPLO

(13) Definition: M odified assignment

If g is an assig n m en t function in x £ V ar an d u £ U, th e n g[x:u], th e assignment modified in x for u, is defined as follows:

»[*:«](») = w { “(y) otherwise

for every y £ V ar.

(14) Definition: Total truth function

T h e truth value o f a form ula p> in a n inform ation s ta te 7r u n d e r th e as­

signm ent g in th e to ta l version of D U P L O , w ritten ‘[<^]j(tt) \ is defined as follows:

i. If ip is an a to m ic form ula, th e n

( i if ApgTr A /g p M g = i;

M g(*) =def < 0 if Apg* A /g p = 0;

t \ otherw ise.

T h a t is, fo r a n atom ic fo rm u la to be tru e (false) in an in fo rm atio n s ta te , it has to b e tru e (false) in ev ery m odel fragm ent of every possibility. T h e t r u t h an d falsity of a to m ic form ulae a re stable b ecause, as we will see la te r on, u p d a tin g an in fo rm atio n s ta te can a t m o st eliminate m odel fragm ents (see (20)).

ii. If a: £ Var a n d <p £ Form, th e n ‘Bxip’ £ Form.

( 1 ^ ApgTT A/gC(p) V ug«; =

\ ^ x p \ g( n) =def \ 0 if ApgTT A/gC(p) AugiV/ ({ '!'?/} ) =

^ f otherw ise.

F o r a n e x isten tia l fo rm u la o f th e form 3x<p to b e tru e in an in fo rm atio n s ta te , each core elem ent of each p o ssib ility m ust co n tain a t least one in d iv id u al for w hich th e b o d y ((p) is tru e in th e cloak of the core elem ent. O n th e o th e r h an d , we can only ta k e it for g ra n te d th a t it is false if no core elem ent of any possibility co n tain s such in d iv id u als. T h ere is u n c e rta in ty in th e rem ain in g cases. A n e x isten tia l fo rm u la is T -unstable (it m ay becom e false from tru e ) ju s t in case its b o d y is; it is F-unstable (it m ay becom e tru e fro m false) under n o rm a l circum stances, b ecau se e x ten d in g th e discourse universe m ay in tro d u ce en tities for which <p holds.

iii. If ip £ Form , th e n ‘~ v ’ £ Form.

M K * ) =def 1 - [<p]g(ir).

1.2.1. The Total Version of DUPLO 9

T his is trivial. N ote th a t ->p is T -u n stab le ju st in case ip is F -u n sta b le , and it is F -u n stab le ju s t in case ip is T -unstable.

iv. If ip, ip £ Form, th e n \ p A ip)' £ Form.

As can be seen, the tr u th and th e falsity of conjunction are defined in an asy m ­ m etric way. T h e tr u th of a conjunction is dynamic: If th e first co n ju n ct introduces som e entity, th en th e second conjunct may pick it up. On th e o th e r h an d , if th e first conjunct is T -u n stab le, th en it need not be tru e in 7r provided th e second conjunct falsifies it. Take, e.g., ()p A ->p: its tr u th in n only requires for -up to hold in 7r. So th e in tu itio n behind this definition is th a t a form ula m u st be tru e in an inform ation state th a t could have been produced by u p d a tin g an o th er in fo r­

m atio n sta te w ith it. On th e o th e r hand, our im pression is th a t defining falsity in a dynam ic way is b o th technically im possible and unnecessary.

v. If p,ip £ Form, th en \ p —> t/>)’ £ Form.

using th e o th e r connectives, b u t we m ay need it for tra n sla tin g n a tu ra l-la n g u ag e

T his is alm ost like the usual in te rp re ta tio n of (}p in U p d a te Sem antics, ex cep t [(¥> -» V’) ] i(7r) =def W g(l<p]g(n))-

T h e question arises why we need a separate connective for m a te ria l im plicatio n in stead of using th e stan d a rd definition i~'(p A —'-0)’. As we have seen, A _lV,)]g(7r) = 1’ expresses th a t eith er ip is false or ip is tru e in n. ‘[</? —> xp]g(n) = 1 ’, on th e o th er hand, m eans th a t n eith er -up nor ip need to be tru e in 7r, b u t u p d a tin g 7T w ith ip yields an inform ation sta te in which tp is true. T his can n o t be expressed sentences which express a conditional relation betw een two p ro p o sitio n s th e t r u t h values of which need not be known, as we will see la te r on.

vi. If tp £ Form , th en ‘^ p ' £ Form.

if A

if A , e ,r K « p } ) = 0;

otherw ise.

th a t we stip u la te th a t it m ust be possible to u p d a te every possibility separately (‘d istrib u tiv ely ’) w ith p in ord er for §ip to be true.

T h e u p d a te function is defined in such a way th a t u p d a tin g 7r w ith p yields an inform ation sta te th a t is m inim ally more inform ative th a n n a n d makes p tru e .

10 1.2.1. The Total Version o f DUPLO

(15) T otal update function

If 7r is an in fo rm atio n state, p is a form ula an d g is an assignm ent fu n ctio n , th e n th e to ta l v ersio n of th e re s u lt of updating 7r w ith p u n d e r g, w ritte n

‘VpYgi11)'i *s d efin ed as

M I M = d. t U m a x, . £p( M ; ( {p' } ) = l ) .

p £ n

It is easy to see th a t ip is indeed tru e in (7r)- T h e reaso n is th a t, since every clause in th e tru th fu n c tio n is defined possibility by possibility,

M ^ l ) = M' g M => W]'g{^\ U 7T2) = [<p]'g{n)

for any 7r1,7r2 and <p. O n th e other h a n d , for the sam e reason, (tt) is indeed th e least inform ative e x ten sio n of 7r w hich has this property.

Let u s now review e ac h type of form ulae and e lab o rate on th e ir u p d a te effects.

(16) Facts about the to ta l update function i. If p is an ato m ic form ula, th en

Ivl'gi*) = U {{/ e P: M Í = 1»-

p£n

If p is a n atomic form ula, then th e le ast inform ative ex ten sio n of 7r in w hich p is tru e can b e produced b y leaving ou t every m odel frag m en t from every possibility in which p is false.

= u «/ e p: v (/ e /' & v = 1).

7T / ' € p U^ U j ,

B y th e ab o v e argum ent, th is is indeed th e least inform ative extension of 7r w hich m akes 3x p tru e . W hen u p d a tin g an in fo rm atio n s ta te w ith 3xc^, we g u a ra n te e t h a t every core elem ent o f every possib ility will contain a t least one in d iv id u al t h a t satisfies p , b u t n o t m o re th a n a b so lu tely necessary. If th e in p u t in fo rm atio n s ta te did n o t contain in d iv id u a ls satisfying p , th en every possible value of x will be p resen t in som e fragm ent o f every possib ility in th e o u tp u t sta te . As a re su lt, no n ew p o ssib ilities arise, b u t m an y new core elem ents do w hich only differ in w h at in d iv id u al p lay s the role o f ‘ad, so to say.

1.2.1. The Total Version of DUPLO 11

iii.

= U MAx„.cP(M i({p'})= o).

p £ n

T his is ju s t th e in sta n tiatio n of the definition in (15) for negative form ulae.

iv.

As usual in dynam ic sem antics, conjunction corresponds to function com position.

It is easy to prove th a t this is indeed th e least inform ative extension of 7r in w hich iip A t/d is tru e .

I p -* = U MAXp'Cp([V,]i(M i({p'))) = !)•

J u st like d isju n ctio n (th e negation of a conjunction), conditionals a re also able to m ultiply possibilities w hen we u p d a te an inform ation sta te w ith th e m .

[Ov>rs (ir) = { p e n : M 's ({p}) # 0}.

Unlike in U p d a te Sem antics, m odal formulae do have an u p d atin g effect. As we will see shortly, this is ju s t a technical difference.

The concept of entailment in DUPLO could be defined in th e u su al way:

1= V’ ^ W g ilp i A . . . A ¥>n]J(ír)) = 1

for all 7T a n d g. T his would probably do the job, b u t we can provide a stro n g e r definition, a n d one th a t is closer to th e classical concept of entailm ent:

(17) Definition: Entailment in DUPLO

A sequence of form ulae p>\, . . . , <pnentails a form ula xp ,ipn (=?/>) iff, for all inform ation states n and assignm ents <7,

W g i* ) > W\ A . . . A ¥n)'g{n)-

N ote th a t, in D U PLO , entailm ent is trivially reflexive. In U p d a te S em antics, a form ula like 0 V7 A ->ip does not entail itself. In th a t theory, Ov? A -xp would e n ta il itself if a n d only if, for any 7r, th e result u p d a tin g n w ith O9? A ~'ip su p p o rte d A -<ip (i.e., in our term inology, if A -><p was tru e in it). B u t Ov7 A ~xp h a s th e sam e u p d a te effect as ~'<p, and u p d atin g n w ith ~'p yields an in fo rm atio n s ta t e th a t does n o t su p p o rt In D U PLO , however, we have the following fact:

12 1.2.1. The Total Version o f DUPLO

(18) Fact: Entailment is reflexive ip \= ip for every <p>.

M oreover, e n tailm en t is also triv ially transitive in D U PLO . In first-o rd e r v er­

sions of U p d a te S em an tics, a form ula like 3 x (P (x )) en tails 3y (P (y )), a n d th e la tte r entails P ( y ) , b u t 3x ( P ( x ) ) does not e n ta il P (y ). D U P L O does not have th is ra th e r inconvenient feature.

(19) Fact: Entailment is transitive p | = V ’ &V’ b x = > P l = X -

1.2.2. S o m e P r o p e r tie s o f D U P L O

In w h at follows, we ex am in e certain g en eral p ro p erties of D U PLO . In p a rtic u la r, we are in te re ste d in th o se featu res th a t distinguish it fro m o th e r d y n am ic theories.

However, we defer th e discussion of differences related to an ap h o rs to Section 2.

T h e first p ro p e rty t h a t is usually relevant in th e assessm ent of a dynam ic sem antic th e o ry is elim inativity:

(20) Fact: Elim inativity /V s[v d s(jr) Vp'g?r-P — P •

T h a t is, th e u p d a te fu n c tio n always yields an in fo rm atio n sta te th a t is a t least as in fo rm ativ e as its in p u t. Some version of elim in ativ ity is usually satisfied by dynam ic th eo ries, unless th e y set ou t to account for cases of belief revision.

We h av e m en tio n ed e arlier th a t th e tr u t h and falsity of certain D U P L O for­

m ulae are n o t stable:

(21) Fact: Instability

M s O ) = 1 & = *' b [vYgi*') = l;

M g ( > ) = o & M j O ) = tt' b W O = o.

T h a t is, th e tr u t h a n d falsity of a fo rm u la <p are n o t preserved u n d e r a rb itra ry u p d a te s. F o r exam ple, f)V x P (x ) may b e tru e in n (nam ely, if every p o ssib ility in 7T co ntains su b -p o ssib ilities in which V x P (x ) is tru e). U p d a tin g n w ith -iV xP (x) m ay yield a n o n -em p ty in fo rm atio n s ta te n '. Obviously, <C>VxP(x) is n o t tru e in th e re su ltin g in fo rm atio n s ta te . On the o th e r hand, b ecau se of th e fact m en tio n ed in (14iii), -><0>P is F -u n sta b le .

Since 3xip is F -u n sta b le , universal quantification defined in th e u su al way will com e o u t as T -u n sta b le in D U PLO :

1.2.2. Some Properties of DUPLO 13

(22) Fact: Truth of a universal formula 1. [Vx<^( 7r) = 1

2. [ - 3x- v?]^(tt) = 1 [1, by def.J

3. [3x- < ^ ( 7t) = 0 [2, (14in)]

4. A pen A feC(p) /\ueuf [“'^ [ x r u ^ U p / } ) = 0 [3, (H ii)]

5. Apg7r A/eC(p) A u£Uj [<x]p[x: u] ( ^ [4, (14iii)j

As can be seen, the tr u th of Vx<yS involves universal quantification over ind iv id u als already introduced. So in tro d u cin g new referents m ay alter its tr u t h value.

On the o ther h an d , the F -in sta b ility of \/xip depends on the p ro p e rtie s of (23) Fact: Falsity of a universal formula

1. [Vxp}'g(rr) = 0

2. [ - 3x-v](,(tr) = 0 [1, by def.

3. [3x- ^ ] 's(tt) = 1 [2, (14iii)]

4. Ap€7r A fec(p) Vu&uf 1 13, (14ii)]

5. ApgTT A fec(p) MueUf M s[x :u ](U p /} ) = 0 [4, (14iii)]

If V x P (x ) is false in an inform ation sta te , then it will rem ain false forever, b e­

cause intro d u cin g new individuals m ay no t change th e tr u th of P for th e referents in tro d u ced earlier. However, if VxB\y R (x ,y ) was false in an in fo rm atio n s ta te for 3x->3y R (x ,y ) is tru e in it, th e n introducing a new individual can m ake th e universal form ula true.

In co n trad istin ctio n to D P L a n d sim ilar theories, universal form ulae can also have updating effects in D U PLO .

(24) Fact: Update o f universal formulae 1. [Vx<^(7r)

2. h 3 x - ^ ] '9(7r) [1, by def.

3. U i>€,M A X l,< c p ( [ 3 x ^ ] '( { p '} ) = 0) [2, (16iii)]

4. UpgTT M AXp/Cp(A/eC(p') f\u£U, [_l(d]5[x:u]({J.p'/}) = 0) [3, (14Ü)]

5. Up^TT ^^■■^■p'Cp(A/ec(p') l\ueUj [tr,]s[x:u]({J-p'/}) = ■*■) [4, (14iii)]

T his m eans th a t universal form ulae m ay introduce new individuals into th e dis­

course universe. In p a rtic u la r, th e u p d a tin g effect of a sentence like (25) Every farm er has a donkey

consists in m inim ally trim m in g th e in p u t inform ation state so th a t every fa rm e r in the o u tp u t state has a donkey.

14 1.2.2. Some Properties o f DUPLO

(26) Fact: Update o f V x ( F ( x ) —» 3y D( x , y)) 1 . [ V x ( F ( x ) -*• 3y D ( x , y ) ) f g ( n )

2 . UpgTT M A Xp#Cj> ( A / 6 C(j>') AU&U,

[ F ( x ) - > 3y D ( x , y ) } ' g[ x: u] ( { l p , f } ) = l ) [1, (24)]

3 . U pgjr M A X p 'C p (A /g c (j> ') A u g iq

[3yD (x,y)]'g[x.u]( l F( x ) } g[x:u]( { l p' f})) = 1) [4, (14v)]

T h a t is, th e resulting in fo rm a tio n s ta te will contain as m an y donkeys as th e re are in d iv id u als in th e in fo rm atio n sta te th a t are know n to be farm ers. F u rth erm o re, if a n in d iv id u al already p re s e n t in th e discourse u n iverse tu rn s ou t to b e a farm er la te r on, his donkey will b e in tro d u ced autom atically. (T h e restric tio n to ind iv id ­ u a ls already in tro d u ced s te m s from th e definition of V.) O n th e o th e r h a n d , if we defined m a te ria l im p licatio n in term s of negation a n d co njunction, we w ould get a different resu lt.

(27) Fact: Update of \/x~>(F(x) A ~'3yD(x, y)) 1. [ V x - ( F ( x ) A -i3 y D (x ,y ))]'g(n) 2. UpgTr M A X ,c p ( A /€C ( , ) l\u e u ,

H F ( x ) A - 3 y D (x , y))}'g[x:u]( { l p. f } ) = 1) [1, (24)]

3. U p s* M AXP'C p (A /g c(p ') AueUf [F(x)]lg[tiu]a i P' f } ) = 0 V

[3yD (x ,y )]'g[x:u]( { l p' f } ) = 1) (2, (14iii), (14iv)]

T h is in te rp re ta tio n has n o t m u ch to do w ith a conditional. R ath er, it says ‘every­

o n e (in th e discourse u n iv erse) either is definitely n o t a farm er or positively has a d o n k ey ’. T h is is th e a rg u m e n t th a t we m entioned w h en we defined a sep arate co n n ectiv e for m aterial im p licatio n .

T urn in g back to th e tra n s la tio n in (26), th e m in im al m odification of th e in­

fo rm a tio n s ta te th a t we c a lc u la te d th ere involves in tro d u c in g a donkey for every f a r m e r in th e discourse u n iv e rse if they all own donkeys. (O therw ise, th e em pty in fo rm a tio n s ta te is re tu rn e d .) U p d a tin g w ith (25) c a n n o t in tro d u ce new farm ers, how ever. If a ll farm ers in th e in p u t inform ation s ta te have donkeys, th e n in tro ­ d u c in g new ones would b e a non-minimal m odification; if som e do n o t, th e n th e e m p ty in fo rm atio n would b e re tu rn e d , anyway.

T h e fact th a t m ost d y n a m ic sem antic theories tr e a t sentences like (25) as

‘e x te rn a lly s ta t ic ’ (i.e., n o t g iv in g rise to new discourse referents) stem s from the o b se rv a tio n th a t such a sen te n c e cannot be continued w ith it (as referrin g to a d o n k e y ). We believe this is a non sequitur: a sentence like (25) is p rag m atically in a d e q u a te w hen talking a b o u t ju s t one farm er (w ith ju s t one donkey), so no single d o n k e y is available in its o u tp u t inform ation sta te . Several donkeys m u st b e there,

1.2.2. Some Properties o f DUPLO 15

w hich m akes it in a p p ro p riate, b u t they is fine (except th a t it is am biguous betw een

‘th e farm ers’ an d ‘th e donkeys’).

A n o th er im p o rta n t concept of dynam ic theories of sem antics is distributiv- ity. A form ula is said d istrib u tiv e if and only if u p d a tin g an in fo rm atio n state w ith it consists in u p d a tin g each possibility separately. A language in te rp re te d dynam ically is said d istrib u tiv e if all of its formulae are distributive. As shown by G roenendijk an d Stokhof (1990), a logic th a t is b o th distributive a n d elim inative is n o t really dynam ic. B u t th e ir result does not apply to our theory, b ecau se D U ­ PL O u p d ates do n o t ‘d is trib u te ’ down to th e model fragm ents (except for atom ic form ulae).

(28) Fact: D istributivity in DUPLO

= U p eÄ ( M ) -

T h a t is, th e process of u p d a tin g an inform ation sta te is distrib u tiv e in th e sense th a t we get th e sam e resu lt if we u p d a te th e singleton inform ation s ta te s containing each possibility one by one, th e n take th e union of th e results.

As we m entioned earlier, th e d istrib u tiv ity of th e possibility o p e ra to r is h a rm ­ less in D U PLO . In p a rtic u la r, th e argum ent in G roenendijk et a1. (1994) why possibility should be n o n -d istrib u tiv e does not apply to our tre a tm e n t:

(29) I f someone is hiding in the closet, then he might have done it.

T h e in te rp re ta tio n th a t b o th G roenendijk et ai. (1994) and D U PL O p red ict for th is sentence an d th e D U PL O definitions yield is ‘th e re m ust b e a t least one individual am ong those who m ight be hiding in th e closet who m ig h t have done i t ’. As for th e D U PLO tre a tm e n t of (29), u p d atin g an inform ation s ta te w ith its

«/-clause yields one in w hich every possibility will contain everyone w ho m ay be hiding in th e closet, so looking at possibilities one by one to assess th e tr u t h of th e then-clause yields th e correct result.

O n th e o th e r h an d , D U P L O ’s in terp retatio n of th e necessity o p e ra to r is quite different from th a t of U p d a te Sem antics. In U pdate Sem antics, a fo rm u la like □ </?

(i.e., m ay have no u p d a tin g effect (it yields eith er the in p u t in fo rm atio n sta te , if <p is tru e in ir, or else th e em pty inform ation state). In D U P L O , on th e o th e r hand, u p d a tin g n w ith □ yields an extension of n in which is tru e in a stable way:

(30) Fact: Update effect of □</?

1. [ C M i M

2. [ - ’❖ - y l i M [1, by def.]

3. U , € ,M A X ,.C r( [ « - * > ) i( { p '} ) = 0 ) [2, (16iii)J 4. U p6» M A X p.Cp( M i ( { p ' } ) = 0) [3, (14vi)]

5. U p€,M A X p. c p(A p..Cp. M i « P " » = 1) [4, (15)]

16 1.2.2. Some Properties o f DUPLO

T h e asym m etry b e tw e e n possibility and necessity is n o t u n realistic. C onsider th e follow ing pieces o f discourse:

(31) A man is walking in the park. Maybe he wears a blue T-shirt.

(32) A man is walking in the park. It is known that he wears a blue T-shirt.

A ssu m in g th a t all o u r m o d e l fragm ents are fragm ents of a m odel in w hich two m en exist a lto g e th er, e x a c tly one of w hich wears a b lue T -sh irt, th e piece of discourse in (31) says nothing a b o u t who is w alking in th e p a rk (if we s ta r t from a m inim ally in fo rm ativ e in fo rm atio n s ta te ) .4 It could be e ith e r m an , so b o th p o ssibilities will be p re s e n t in the o u tp u t in fo rm atio n state. T h e piece of discourse in (32), on th e o th e r h a n d , seems to ex clu d e th e possibility th a t th e m a n w alking in th e p a rk is n o t w earing a b lue T -s h irt. U n d er U p d ate S em an tics’ definition of neg atio n , th e piece of discourse in (32) should b e a co n trad ictio n . Since we s ta r t from a m in im ally inform ative in fo rm atio n s ta te , n o th in g is know n ab o u t th e m a n after p ro cessin g a first se n te n c e , and the second sentence w ould claim th a t so m eth in g is know n a b o u t him. U n d e r D U P L O ’s view, however, if co m m u n icatin g ‘ Ov5’ is to co m m u n ic ate ‘it is k n o w n th a t <p is tru e in the w o rld ’, so it m akes sense to consider it synonym ous w ith p (in clu d in g its dynam ic effects) except w hen p is T -u n sta b le .

T h e stabilizing effect of the necessity o p e ra to r ‘ Q ’ can be p u t to use in th e tra n s la tio n of n a tu ra l-la n g u a g e sentences. For exam ple, in m ost cases, overt nega­

tio n is s ta b le in n a tu r a l languages, alth o u g h th e n eg atio n th a t we need in a n a ­ phoric qu an tificatio n a n d universal qu an tificatio n in general is n o t. U n d er th e above tre a tm e n t, we c a n assum e th a t a n a tu ra l lan g u ag e sentence of th e form It is n ot the case that p is always tra n s la te d as which yields th e desired effect. In th is way, we d o n o t have to in tro d u ce a se p a ra te , ‘stab ilizin g ’ neg atio n (to g e th e r w ith a ‘s ta b iliz in g ’ c o u n terp art of universal q u an tificatio n , a n d so on).

Sim ilarly, a n a tu ra l-la n g u a g e sentence of th e form There is exactly one x such that p can b e tra n slated as ‘ 03!;ry>’, b ecau se it is norm ally u n d e rsto o d in a stab le way, unlike a n ap h o ric q u an tificatio n . By th e sam e token, if Every x is such that p is n o t u n d e rs to o d as q u a n tify in g over th e discourse universe, b u t is in te n d e d as an

‘e te rn a l t r u t h ’, its tr a n s la tio n m ust b e prefixed w ith ‘O ’.

N o te th a t double negation can be elim inated in D U P L O . As a consequence, double n e g atio n does n o t m ak e a form ula externally static. For exam ple, we p red ict th e follow ing piece of d isco u rse to be acceptable u n d e r n o rm al circum stances:

(33) I t is not the case th a t Joe does not have a car. I saw it parked next to the entrance.

4 H ere we assume t h a t o u r inform ation ab o u t m en in th e m odel a n d w ho wears a b lu e T -shirt does n o t originate from previous discourse in fo rm atio n .

1.2.2. Some Properties of DUPLO 17

It is not entirely clear w h eth er this is a desirable prediction. P eo p le’s ju d g m en ts diverge on th is piece of discourse, so it probably cannot serve as a key exam ple in th e assessm ent of theories of anaphors.

1.2.3. P a rtia l D U P L O

For the sake of sim plicity, we will not be very precise in the definition of th e p a rtia l version of D U PLO . T h e language C of p artial D U PLO is the sam e as C except th a t th e set of logical co n stan ts contains an additional symbol, th e presupposition operator ‘A ’, and if <p E Form , th e n ‘Ac/?’ £ Form. These form ulae are th e u ltim a te sources of presupposition, i.e., undefinedness of sem antic values. We use th e sym bol V for a n undefined value rath er th a n a special value. T h e p a rtia l tr u th fun ctio n will b e w ritten as [-]5, and the p a rtia l u p d a te fu n ctio n is [- |a . T h e tr u th values of A</? a re as follows:

(14) vii.

[ A y > ]9(7r ) = d e i { * if ') ^ i;

otherw ise.

As for th e o th e r clauses of th e tr u th function, we will ju s t specify w hen its value is undefined. W h en it does assign a tru th value, th e value is alw ays calcu lated in th e sam e way as th a t of th e to ta l version, except th a t we have to use th e p a rtia l versions of th e tr u th an d th e u p d ate functions in th e calculations. T h e definition below a tte m p ts to express stan d ard assum ptions ab o u t p re su p p o sitio n a n d p resu p p o sitio n p rojection, which we will not dwell up o n here, as th ey are ta n g en tial to o u r cen tral concern.

(34) Definition: Partial truth function

i. T h e tr u t h value of an atom ic form ula is always defined. A tom ic form ulae have no presuppositions.

ii.

7r ) = * < ^ d e f V A A M i [ * : « ] ( { p } ) = *•

pe* f€C(P) ueUf

iii.

= * ^ d e i b U ? r ) = *.

[ i f A xj}]g {7r) = * ^ > d e i A = * = > M s O ' ) = * ) • jr'en

iv.

18 1.2.3. Partial DUPLO

v.

W V’lf f « ) = * ^ d e f M 9(7t) = * V ( M U tt) ^ * & [ll>]g(lp]g(7r)) = *).

vi.

[ < M 3(tt) = * <S>def M 3(tt) = * .

T h e p a rtia l u p d a te fu n c tio n is undefined for a form ula ip in a n in fo rm atio n s t a t e 7r if th e le a st in fo rm ativ e extension of 7r in w hich <p is tru e c an n o t b e calcu­

l a t e d because o f p re su p p o sitio n failure. In m o st cases, th is m eans th a t th e tr u t h v a lu e of <p m u st b e defined in 7r in ord er for [ ^ I ^tt) to be defined. T h e re are o n ly two ex cep tio n s from th is rule, namely, co n ju n ctio n an d m a te ria l im p licatio n , b e c a u s e th eir t r u t h in th e ex ten sio n s of 7r depends on n indirectly: an in te rm e d ia te in fo rm a tio n s ta te is involved. B u t, in th e case of m a te ria l im plication, th e u n d e ­ fin ed n ess of th e u p d a te fu n c tio n coincides w ith th a t of th e tr u th fu n ctio n as in th e c a s e o f m ost o th e r types of form ula, because d eterm in in g its tr u t h involves u p d a t­

in g th e in fo rm atio n s ta te w ith th e antecedent. T herefore, we have th e following defin itio n :

(3 5 ) Definition: Partial update function

If 7T is a n in fo rm atio n s ta te , <p is a form ula a n d g is an assignm ent fu n ctio n , then th e re su lt of updating n w ith ip u n d e r g, w ritte n ‘I<r,] s ( 7r) \ is defined as follows.

(i) If ip is of th e form xp A £, th en

M U 70 = * ^ d e f W 3(7r) = * V ( M 9(tt) # * & U ig( W j ( ’r)) = *);

(ii) O therw ise,

M U * -) = * <^def r) = *;

(iii) If I ^ l 3( 7r) 7^ th e n

M M =<w U m ax ,. cp (M s ({ p '}) = l).

pG?r

A ll the facts a b o u t th e to ta l version of D U PL O are valid in th e p a rtia l version, e x c e p t th a t th e proviso ‘w hen defined’ is necessary in m any cases (such as th e d e fin itio n of e n tailm en t). T h is does n o t affect th e im p o rta n t logical p ro p e rtie s of D U P L O , though.

2.

Anaphors in DUPLO 19

2. A n aphors in D U P L O

Obviously, we have in tro d u ced b o th unstable quantification and p resu p p o sitio n s in order to deal w ith an ap h o rs (alth o u g h they m ay be p u t to use for different purposes as well). In term s of w h at we have anticipated in the earlier sections, th e tra n slatio n of a sentence like (1) will be as follows:

(1 ') Joe has a cat. It is black.

‘3 x (c a t(x ) A ow ns(j, x )) A

A (3 !y (n o n h u m an (y ))) A V z(nonhum an(z) —■> b lack (z))’

It is easy to see th a t th e tra n slatio n of the second sentence can b e p ro d u c e d in a com positional m anner. In general, if th e descriptive content of a n a n a p h o r is A , th e n its tra n sla tio n is

(36) Translation o f anaphors

A P (A (3 !x (,4 (x ))) A Vy(.4(y) -» P ( y ))).5

Accordingly, we will use ‘h eM, ‘i t '5 etc. as the translations of a n ap h o rs w ith th e descriptive contents ‘m ale’, ‘n o n -h u m an ’ etc. below.

This tre a tm e n t accounts for one of the m ost obvious ch aracteristics of a n a ­ phors, nam ely th a t, in m ost co n stru cts, th e antecedent can be p resen t in th e initial inform ation sta te even if th e an ap h o r is deeply em bedded:

(37) a. M ary does not know him . b. Mary would know it.

c. I f she was there, she would know it.

In (37a), th e a n ap h o r is w ith in a negative predicate, in (37b), w ith in a m o d al predicate, a n d in (37c), in th e consequent of a (counterfactual) co n d itio n al. Yet in all th ree cases th e antecedent is required to be present in the in itia l in fo rm atio n state. T h e p resu p p o sitio n al tre a tm e n t of anaphors ensures th a t th e an ap h o ric quantifier will have ‘w ide scope’ in th e translations of the sentences in (37) w ith o u t relying on w ildly non-com positional devices.

5 As a m a tte r of course, th is tra n slatio n does no t apply to so-called ‘lazy p ro ­ n o u n s’, which do no t s ta n d for th eir antecedents b u t an analogous entity, as in

The man who gave his paycheck to his wife was wiser than the one who gave it to his mistress.

20 2. Anaphors in DUPLO

O n th e oth er h a n d , th e antecedent does no t have to be p resen t in th e in itial in fo rm atio n state, so we can n o t say th a t the an ap h o ric quantifier alw ays has a

‘w ide sco p e ’:

(38) I f she had, a car, she would lend it to me.

In this sentence, th e a n te c e d e n t of th e an ap h o ric p ro n o u n it is w ith in th e condi­

tio n al; it is difficult to im a g in e a com positional m echanism th a t w ould assign ‘w ide sco p e’ to th e anaphoric q u a n tifie r in (37c), b u t ‘n arro w scope’ to th a t in (38). T h e p re su p p o sitio n a l tre a tm e n t of anap h o ric quan tificatio n , on th e o th e r h an d , deals w ith b o th ‘w ide-scope’ a n d ‘narrow -scope’ cases. In term s of th e definedness con­

d itio n s o f th e u p d ate fu n c tio n , th e presuppositions of form ulae m u st b e satisfied in th e in itia l inform ation s t a t e as a rule. Form ulae of th e form s lip A ip' a n d ‘9? —> i f a re exceptional: the p re su p p o sitio n s of ip m ust b e satisfied by [v7] 3( 7r)- T h a t is, th e p resu p p o sitio n s of th e second m em ber of a co n ju n ctio n or a cond itio n al are to b e satisfied in an in fo rm a tio n sta te th a t th eir first m em b er yields. T h is is essen­

tia lly th e sam e beh av io u r o f ‘presu p p o sitio n p ro je c tio n ’ th a t has been p ro posed by K a r ttu n e n (1973, 1974) a n d K a rttu n e n an d P eters (1979), except for som e m arked cases, w hich are to be d e a lt w ith in special ways (cf. K álm án (1994)).

J u s t th e sam e as th e choice of th e variables corresponding to a n ap h o rs is done b y m agic in D PL, we also need m agic for those co n d itio n s in th e tra n sla tio n s of a n ap h o rs t h a t result fro m fa c to rs ex tern al to sem antics. For exam ple, consider th e following tran slatio n :

(2) Joe likes him.

‘A (3 !x (m ale(;r) A x 7^ j ) ) A V y((m ale(x) A x 7^ j ) —> likes(y, x ) ) ,

H ere it is n o t clear how th e condition ‘x 7í j ' gets in to th e tra n s la tio n of him.

I t is allegedly a condition th a t can be produced by sy n tactic in fo rm atio n (since Joe c-com m ands him, th e y can n o t be co-referential). If we did no t include this co n d itio n o n th e a n tec e d en t, th e sentence could only m ean ‘Joe likes h im se lf’.

O bviously, the p ro p e rtie s of dynam ic logics such as D PL which a re re la ted to th e b o u n d -v ariab le view o f an ap h o rs do not hold for D U PL O . In p a rtic u la r, the so-called donkey equivalences of DPL,

3x(<p) Aip = 3 x (p A ip);

(3x(y>)) —> ip = \/x(ip —> ip)

d o n o t h o ld in D U PLO . ( ‘9? = ip’ m eans th a t b o th [-]3 an d [•]<, yield identical v alu es for ip a n d ip.) W e h a v e argued against th e first of these in th e previous sectio n s. W e will show n ow th a t the second equivalence is also n o t necessary for d e alin g w ith th e relevant fa c ts.

T h e use of anaphors in co n d itio n al sentences is an age-long problem . Consider:

2.

Anaphors in DUPLO 21

(39) I f a farm er owns a donkey, he beats it.

As can be seen, an ap h o rs in th e consequent of a conditional m ay refer to e n titie s in tro d u ced in th e antecedent. M oreover, these anaphors are in th e singular ( a t least in English), which suggests th a t the conditional is a case-by-case s ta te m e n t ab o u t th e entities th a t th e antecedent introduces. According to a pro p o sal in H eim (1990), this phenom enon m ay b e due to the fact th a t conditional sentences involve a (usually im plicit) q uantification over ‘cases’ or ‘situ a tio n s’, which are sele c ted by th e antecedent. For exam ple, (39) should b e in terp reted as ‘in each s itu a tio n in which a farm er owns a donkey, th a t farm er b eats th a t donkey’.

T he m achinery of D U PLO as introduced above offers a n a tu ra l way of c a p ­ tu rin g th e concept of ‘cases’ o r ‘situ a tio n s’. Each ‘case’ or ‘s itu a tio n ’ can b e seen as a possibility in th e in fo rm atio n state. U pdating an inform ation sta te w ith th e antecedent of a conditional sentence yields ju st those ‘m inim al situ a tio n s’ t h a t Heim (1990) m en tio n s.6

As for th e an ap h o rs in th e consequent of (39), th e possibilities th a t u p d a tin g w ith th e antecedent yields are th e ‘biggest’ (i.e., least inform ative) am ong th o s e in which ‘a farm er owns a donkey’. Therefore, th eir discourse universes (i.e., th e universes of th eir core elem ents) will be m inim ally larger th a n those in th e in p u t inform ation state, i.e., they will contain ju st one farm er an d one donkey.7 T h a t is why th e tran slatio n s th a t we proposed for he and it are a p p ro p ria te in th is case:

(39') I f a farm er owns a donkey, he beats it.

3x (F (x ) A 3 y D (x ,y )) —* b e ats(h e ', it') (40) Fact: Truth o f (39')

1. [3x(F(a:) A 3y D (x ,y )) —» beats(he', it, )]s (7r) = 1

2. [beats(he', it')]ff([3 x (E (x ) A 3yD (x, y ) ) ] , ^ ) ) = 1 [1, (14v)]

3. [beats(he',it')]fl(Up€^ MAXp,cP(A/6c(P ') V

u , u ' £ l / f

[F (x ) A D (x, y)]j7[x:u][3/:u,] ( { l p '/ } ) = 1)) = 1 [2, (15)]

6 As a m a tte r of fact, it yields th e m inim al extensions of th e in p u t s ta te s a t ­ isfying th e antecedent. We are aware of th e fact th a t n o t every co n d itio n a l sentence is to be in te rp re te d in this way. For exam ple, th e an teced en t o f a counterfactual selects m inim al situations th a t are no t even co m p atib le w ith th e in p u t inform ation sta te . We believe th a t even norm al con d itio n al se n ­ tences could b e tre a te d in th is way. T h a t is, in th a t case, th e a n te c e d e n t could yield ‘su b -situ a tio n s’ of th e input situations, i.e., m inim al m o d ificatio n s possibly different from its extensions.

7 Unless th e in p u t in fo rm atio n sta te already contains farm ers a n d donkeys; see below.

22 2. Anaphors in DUPLO

Since th e possibilities selected by th e antecedent will co n tain all possible fa rm e r/d o n k e y pairs, we get the in te rp re ta tio n th a t D P L yields if we tra n s la te sentences like (39) as above. T he tra n s la tio n of (39) is tru e if and only if, in every possibility t h a t arises fro m th e in p u t s ta te by in tro d u cin g a farm er a n d a donkey h e owns, th e fa rm e r will b e a t the donkey. W h eth er all conditional sentences in n a tu ra l la n g u ag e are to b e tra n slated in th is way is arguable. In fact, th e uniform

‘d o u b le-u n iv ersal’ readings th a t D P L ’s ‘donkey equivalences’ p red ict have been largely q u e stio n ed in th e lite ra tu re . M ay b e th e ‘dou b le-u n iv ersal’ read in g th a t th e above tra n s la tio n yields is peculiar for such a te m p o ra l sentences, a n d o th e r universal sen ten ces have different tra n slatio n s.

T here is a n in terestin g problem th a t th is analysis raises. If th e in itia l in fo rm a­

tio n state in w hich (39) is u tte re d alread y contains m ore th a n one fa rm e r a n d /o r donkey, th e n th e biggest sub-possibilites th a t satisfy ‘a fa rm e r owns a donkey’ also co n tain sev eral of them . In th a t case, we p red ict he a n d /o r it in th e an teced en t to b e infelicitous. This p red ictio n m ay so u n d odd, b u t — a t least in principle — th e an ap h o ric p ro n o u n s he a n d /o r it c a n n o t in fact be u sed felicitously if th ere a re several p o ssib le an teced en ts in the c o n tex t. T he p ecu liarity of th is p red ictio n in th e case o f (39) m ay b e due to rh e to ric a l reasons. B u t th is p ro b lem would n o t even arise if we ad o p ted th e strateg y described in fo o tn o te 6. T his possibility o p e n s d irectio n s for fu rth e r research.

R efe r e n c e s

B eaver, D. 1992. The K inem atics o f Presupposition. IT L I P rep u b lica tio n Series LP-92-05, U niversity o f A m sterdam .

C o o p er, R. 1979. ‘T he in te rp re ta tio n of p ro n o u n s’. In F. Heny and H. Schnelle, eds., S y n ta x and Sem antics 10. A cadem ic Press, New York.

D ekker, P. 1992. A n Update Semantics for D ynam ic Predicate Logic. IT L I P re ­ p u b lic a tio n Series LP-92-04, U niversity o f A m sterdam .

E v a n s, G. 1980. ‘P ro n o u n s’. Linguistic In q u iry 11.

G ro en en d ijk , J . a n d M. Stokhof, 1990. ‘T w o theories of dynam ic sem a n tic s’. In J . van E ijck, ed., Logics in AI, European Workshop JE L IA ’90. Springer, Berlin, p p . 55-64.

G roenendijk, J . a n d M. Stokhof. 1991. ‘D y nam ic P re d ic a te Logic’. Linguistics and Philosophy 14, 39-100.

G ro en en d ijk , J ., M. S to k h o f an d F. V eltm an. 1994. This M ight Be It. IT L I P re p u b lic a tio n Series LP-94-13, U niversity of A m sterdam .

G roenendijk, J ., M. S tokhof a n d F. V eltm an. 1995. ‘C oreference a n d co n tex tu ally re stric ted q u a n tific a tio n ’. M s., S ep tem b er 11, U niversity of A m sterd am . H eim , I. 1990. ‘E -ty p e p ro n o u n s and donkey a n a p h o ra ’. Linguistics and Philoso­

p h y 13, 137-177.

K á lm á n , L. 1994. Conditionals, Quantifícation and B ipartite Meanings. W orking

References 23

P ap ers in th e T h eo ry of G ram m ar, Vol. 1, No. 3, T heoretical Linguistics P ro ­ gram m e, B u d ap est U niversity (E LTE ) and R esearch In s titu te for Linguistics, H ungarian A cadem y of Sciences.

K álm án, L. 1995. ‘Donkeys w ith o u t donkey equivalences’. P a p e r given at th e Col­

loquium on Designing Logics, CCSO M , U niversity of A m sterdam , Septem ber 8.

K a rttu n e n , L. 1973. ‘P resu p p o sitio n s of com pound sentences’. Linguistic In­

quiry 4, 169-193.

K a rttu n e n , L. 1974. ‘P resuppositions and linguistic co n tex t’. Theoretical Linguis­

tics 1, 181-194.

K a rttu n e n , L. an d S. P eters. 1979. ‘Conventional im p licatu re’. In C. O h an d D. D inneen, eds., Syntax and Semantics. Vol. 11: Presupposition. A cadem ic Press, New York, pp. 1-56.

Polanyi, L. 1988. ‘A form al m odel of the stru ctu re of discourse’. Journal o f Pragmatics 12, 601-638.

P rü st, H. 1992. On D iscourse Structuring, VP Anaphora and Gapping. P h .D . diss., U niversity of A m sterdam .

V eltm an, F. 1981. ‘D a ta sem antics’. In J. G roenendijk, T. Jan ssen a n d M. Stokhof, eds., Formal M ethods in the S tu d y o f Language. Vol. II. M ath em atical C entre, A m sterdam , pp. 541-567.

V eltm an, F. 1990. ‘D efaults in U p d a te Sem antics’. In H. K am p, ed., Conditionals, Defaults and Belief Revision. DYANA rep o rt No. R2.5A, C entre for C ognitive Science, U niversity of E dinburgh, pp. 28-63.

---TTT