Bach, Emmon - The Algebra of Events

8/11/2019 Bach, Emmon - The Algebra of Events

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E M M O N B A C H

T H E A L G E B R A O F E V E N T S

0 . I N T R O D U C T I O N

A n u m b e r o f w r it er s h a v e c o m m e n t e d o n t h e c lo s e p a ra l l e ls b e t w e e n t h e

m a s s - c o u n t d i s t i n c t i o n i n n o m i n a l s y s t e m s a n d t h e a s p e c t u a l c l a s s i f i c a t i o n

o f v e r b a l e x p r e s s i o n s (A l l en , 1 9 6 6 ; T a y l o r , 1 9 7 7 ; M o u r e l a t o s , 1 9 7 8 ; L .

C a r l so n , 1 9 8 1 ; H o e p e l m a n a n d R o h r e r , 1 9 8 0) t h at h as b e e n t h e s u b je c t

o f m u c h a t t e n t i o n i n r e c e n t y e a r s in li n gu i st ic s a n d p h i l o s o p h y . T o t a k e

j u s t o n e c l a ss o f e x a m p l e s f o r n o w , t h e r e i s a p a r a l l e l b e t w e e n t h e t w o

s e ts o f d i s ti n c ti o n s i n th e i r c o o c c u r r e n c e p a t t e r n s w i th e x p r e s s i o n s

d e n o t i n g n u m b e r s o r a m o u n t s , a s in E x a m p l e s ( l a ) - ( 4 b ) :

( 1 )( a) M u c h m u d w a s in e v i d e n c e .

( b ) ( * ) M u c h d o g w a s i n e v i d e n c e .

( 2 ) ( a ) J o h n s l e p t a l o t l a s t n i g h t .

( b ) ( * ) J o h n f o u n d a u n i c o r n a l o t l a s t n i g h t .

( 3 )( a ) M a n y d o g s w e r e i n t h e y a r d .

( b ) ( * ) M a n y m u d s w e r e o n t h e f l o o r .

( 4 ) (a ) J o h n f e ll a s l e e p t h r e e ti m e s d u r i n g t h e n i g h t .

( b ) ( * ) J o h n s l e p t t h r e e t i m e s l a s t n i g h t .

( B y t h e u s e o f ( * ) I i n t e n d t o i n d i c a t e t w o t h in g s : t h a t w e h a v e t o d o a

c e r t a i n a m o u n t o f w o r k t o i m p o s e a s p e c ia l in t e r p r e t a t i o n o n t h e s e n -

t e n c e a n d t h a t t h e i n t e r p r e t a t i o n is s h a p e d b y t h e p r e s e n c e o f t h e n u m b e r

o r q u a n t i t y e x p r e s s i o n . )

T h e b a s ic a i m o f th is p a p e r i s t o t r y t o e l u c i d a t e t h is p r o p o r t i o n :

e v e n t s : p r o c e s s e s : : t h in g s : s tu ff . T h e a c c o u n t d r a w s h e a v i l y o n a r e c e n t

p a p e r b y G o d e h a r d L i n k o n th e c o u n t - m a s s - p l u ra l d o m a i n ( L in k , 1 9 8 3)

a s w e l l a s o n t h e w o r k o f a n u m b e r o f w r it e rs w h o h a v e c o n t r i b u t e d a

g r e a t d e a l t o o u r u n d e r s t a n d i n g o f v e r b - c l a s s i f i c a t i o n . 1 I n S e c t i o n 1 , I

r e v i e w b r i e f l y t h e c l a s s if i c a ti o n a n d i n S e c t i o n 2 L i n k ' s a n a l ys i s f o r t h e

n o m i n a l d o m a i n . I n S e c t i o n 3 , I se t f o r t h o u r p r o p o s a l s a b o u t e v e n t s a n d

p r o c e s s e s a n d in S e c t i o n 4 t a k e u p a n u m b e r o f p r o b l e m s , s o m e w i th ,

s o m e w i t h o u t , s o l u t i o n s .

Linguistics and Philosophy

9 1 9 8 6 ) 5 - 1 6 .

@ 1 9 8 6 by D Reidel Publ ishing Comp any


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6 E M M ON B AC H

1 . E VE NT S PR OC E SS E S ST AT E S

Here s a scheme of the kinds of distinctions we want to deal with (based

on L. Carlson, 1981, but using our terminology in part):

s t a t e s

d y n a m i c o ] s t a t i c b )

e v e n t u a l i t i e s

n o n - s t a t e s

processes

c )

events

p r o t r a c t e d d ) momen taneous

happen ings e ) cu trn ina t ions

f )

Typical examples are:

(a) sit, stand, lie + LO C

(b) be drunk, be in New York, own x, love x, resemble x

(c) walk, push a cart, be mean (Agentive)

(d) build x, walk to Boston

(e) recognize, notice, flash once

(f) die, reach the top

I will take it as given that it is necessary to have at least this much of a

classification if we are to deal adequately with the syntax and semantics

of English. A great deal of evidence for this point has been given in the

last several years, for example in connection with attempts to understand

the English progressive and similar constructions in other languages. 2

Most recently, Hans Kamp (1981) and E. Hinrichs (1981) have shown

the necessity for these distinctions for interpreting narrative structures.

2 M ASS C OUNT AND PL UR AL I N T HE NOM I NAL SY ST E M

In the work alluded to above, G. Link (1983) argues for the adoption of a

somewhat more richly structured model than those made available, for

example, in Montague s work. 3 In this section, I will briefly sketch the

outlines of Link s system.

The main idea in Link s semantics is to give more structure to the

domain of individuals. Along with ordinary individuals like John and

Mary as in standard interpretations of the predicate calculus or in


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T HE AL GE B R A OF E VE NT S

M o n t a g u e ' s w o r k w e a r e t o h a v e p l u r a l i n d iv i d u a ls li k e t h o s e d e n o t e d b y

the children o r John and Mary a s w e l l a s q u a n t i t i e s o f s t u f f o r m a t t e r

t h a t c o r r e s p o n d s t o i n d i v i d u a l s o f b o t h k i n d s , s u c h a s t h e g o l d i n T e r r y ' s

r i n g o r t h e s t u f f t h a t m a k e s u p t h e p l u r a l i n d i v i d u a l J o h n a n d M a r y . 4

M o r e o v e r , c e r t a i n r e l a t i o n s a m o n g t h e s e v a r i o u s s u b d o m a i n s a n d t h e

e l e m e n t s m a k i n g t h e m u p a r e p r o p o s e d . I p r e s e n t t h e e s s e n t i a l s i n a n

i n f o r m a l w a y ( f o r p r e c i s e d e t a i l s th e r e a d e r i s r e f e r r e d t o L i n k , 1 9 8 3 ).

S t a r t w i t h a s e t A ~ o f i n d i v i d u a l s o f t h e m o r e f a m i l i a r s o r t , f o r e x a m p l e ,

J o h n , M a r y , t h is t ab l e , T e r r y ' s r in g . W e e x t e n d t h i s d o m a i n b y m e a n s o f a

j o i n o p e r a t i o n t o d e f i n e a s u p e r s e t E a s f o ll o w s :

(i)

(ii)

~_Ei

I f a , f l ~ Ei t h e n t h e / - j o i n ( i n d i v i d u a l j o i n : aUifl o f o l an d

/ 3~E

S o t h e / - j o i n o f J o h n a n d M a r y i s i n E~ i f e a c h o f J o h n a n d M a r y i s. W e

e s t a b li s h a p a rt i a l o r d e r i n g o n t h e m e m b e r s o f E~ ( - ~ ) b y s a y i n g t h a t a is

l e s s t h a n o r e q u a l t o ( o r i s a n i n d i v i d u a l p a r t ( / - p a r t ) o f ) 13 j u s t i n

c a s e t h e / - j o i n o f o t a n d 13 i s j u s t / 3 i ts e l f. T h u s t h e i n d i v i d u a l J o h n i s a n

/ - p a r t o f t h e p l u r a l i n d i v i d u a l s J o h n a n d M a r y o r T e r r y ' s r i n g a n d J o h n .

T h e i n d i v i d u a ls f r o m w h i c h w e s t a r t e d a r e a t o m s in th e b i g s t r u c t u r e t h a t

w e a r e b u i l d i n g .

A m o n g t h e e l e m e n t s o f A~ ( a n d h e n c e E i ) t h e r e i s a su b s e t w h i c h

f o r m s a sp e c i a l s u b s y s t e m o f i ts o w n . T h e s e a r e t h e p o r t i o n s o f m a t t e r o r

s t u f f , f o r e x a m p l e , t h e g o l d o f w h i c h T e r r y ' s r i n g i s c o m p o s e d . T h i s

s u b s y s t e m h a s it s o w n jo i n a n d p a r t i a l o r d e r i n g ( m - j o i n : U r n; m - p a r t :

--< ,1). C a l l t h i s s e t D~ . F i n a l l y , we n e ed t o s p ec i fy t h e r e l a t i o n s h i p

b e t w e e n t h e s y s t e m o f D ~ a n d t h e r e s t o f t h e d o m a i n . W e d o t h i s b y

a s s u m i n g a m a p p i n g hi- f r o m i n d i v i d u a l s ( a t o m i c a n d p l u r a l ) t o t h e s t u f f

o u t o f w h i c h t h e y a r e c o m p o s e d . T h i s m a p p i n g s h o u l d sa t is f y t h e

r e q u i r e m e n t t h a t t h e o r d e r i n g ---~ a m o n g t h e i n d i v i d u a l s b e p r e s e r v e d i n

t h e o r d e r i n g -


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E M M O N B A C H

t h e e x t e n s i o n o f t h e s i n g u l a r c o u n t n o u n m a n , s ay J o h n , b u t t h e r e i s a l s o

a p l u r a l i n d i v i d u a l f a l l i n g u n d e r t h e ex t en s i o n o f t h e p l u r a l n o u n

cells

s u ch t h a t t h e v a l u es fo r h i . g i v en t h e t w o a rg u men t s a r e i d en t i ca l . T h e

t w o i n d i v i d u a l s a r e m e m b e r s o f t h e e q u i v a l e n c e c l a s s i n d u c e d b y t h e

re l a t i o n o f m a t e r i a l i d en t i ty . L i n k ca l ls a s y s t em o f t h is s o r t a B o o l ea n

m o d e l s t r u c t u r e w i t h h o m o g e n e o u s k e r n e l

boosk) .

S o m e c o n s e q u e n c e s o f L i n k ' s c o n s t r u c t i o n t h a t I f in d i n t e re s t in g a n d

a p p o s i t e f o r t h e p r e s e n t c o n t e x t a r e t h e s e ( I h a v e n ' t g i v e n e n o u g h d e t a i l s

t o s h o w t h a t t h e s e c o n s e q u e n c e s f o l l o w ) :

( 5) S u p p o s e H e n g s t a is a h o r s e a n d H e n g i s t is a h o r s e . T h e n t h e

p l u ra l i n d i v i d u a l H en g s t a an d H en g i s t i s n o t a h o r s e , b u t i s i n

t h e e x t e n s i o n o f

horses

( co n t r a s t mas s t e rms ) .

(6) Su p p o s e t h e p l u r a l i n d i v i d u a l A an d B i s i n t h e ex t en s i o n o f

horses a n d l i k e w i s e C a n d D . T h e n t h e p l u r a l i n d i v i d u a l A , B ,

C , an d D i s a l s o i n t h e ex t en s i o n o f

horses

(c f . mass t e rms) .

( 7) E v e n i f t h e i n d i v id u a l t h a t is t h e q u a n t i t y o f g o l d c o m p o s i n g

T e r r y ' s r i n g i s o l d , T e r r y ' s r i n g n e e d n o t b e .

( 8) T h e t w o m e a n i n g s o f s e n t e n c e s l ik e

John and Mary l i f ted the

box ( e a c h v s . to g e t h e r ) c a n b e n i c e l y r e p r e s e n t e d i n L i n k ' s

s e m a n t i c s b y a d d i n g t h e i n t e r p r e t a t i o n p r o v i d e d f o r t h e p l u r a l

i n d i v i d u a l t o t h e i n t e r p r e t a t i o n p r o v i d e d , s a y , i n M o n t a g u e ' s

P T Q .

3 T H A L G E B R A O F E V E N T S A N D P R O C E S S E S

W e n o w w a n t t o t r y o u t L i n k ' s i d e a s i n t h e d o m a i n o f e v e n t u a l i t i e s , t h a t

i s , t o c h a r a c t e r i z e t h e s t r u c t u r e o f t h e m o d e l w h e n w e e x t e n d i t t o t h e

d o m a i n o f e v e n t s a n d p r o c e s s e s , w h i c h f o r t h e m o m e n t I w i l l c o n s i d e r

j u s t a s n ew k i n d s o f e l emen t s i n t h e ( s o r t ed ) d o ma i n . I w i l l s t a r t b y

c o n s i d e r i n g e v e n t s t o b e a n a l o g o u s t o t h e s i n g u l a r a n d p l u r a l i n d i v i d u a l s

a n d b o u n d e d p r o c e s s e s ( ' b i t s o f p r o c e s s ' ) a n a l o g o u s t o t h e p o r t i o n s o f

m a t t e r t h a t m a k e u p t h e ' m a t e r i a l e x t e n s i o n s ' o f th o s e i n d i vi d u a ls .

O u r n e w s y s t e m w i l l t h e n i n c l u d e t h e f o l l o w i n g :

(1) Ee: t h e s e t o f e v e n t s w i t h j o i n o p e r a t i o n s U e a n d p a r t i a l

o r d e r i n g - < e ( a c o m p l e t e a t o m i c B o o l e a n a l g e b r a ) ;

(2)

Ae ~ Ae :

a t o m i c e v e n t s ;

(3) D e _ A e : b i t s o f p ro ces s w i t h j o i n I t e an d p a r t i a l o rd e r i n g -< p

( a c o m p l e t e j o i n s e m i l a t t i c e ) ;


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T H E L G E B R O F E V E N T S

(4 ) I n a d d i t i o n , w e w i ll n e e d t w o t e m p o r a l r e l a t i o n s o n E e X E e :

~ : s t r i c t l y p r e c e d e s ( tr ., i rr ., a s y m m . ) ,

: o v e r l a p s ( n o n t r . , r e fl ., s y m m . ) ( cf . B a c h , 1 9 8 1 ; K a m p ,

1 9 8 0 ) ;

(5) a h o m o m o r p h i s m he f r o m (E e , U e , < e , ~ , ) t o

(De , U p , --< v, ~ , ) s u c h t h a t

(i) he (a ) = a i f f a c De ,

(ii) h e ( a Ue/3) = he(a) Uphe([3) , a n d

(iii) a R l 3 ~ h e ( a ) R h e ( [ 3 ) fo r R = -< e , oc , o an d R ' = _< p , o c,

r e s p e c t i v e l y .

F o r p u r p o s e s o f i l l u s t r a t i o n , I w i l l a s s u m e t h a t t e n s e l e s s c l a u s e s o f

E n g l i s h a r e t o b e i n t e r p r e t e d a s d e n o t i n g s e t s o f e v e n t u a l i t i e s , i . e .

m e m b e r s o f t h e d o m a i n E e ( f o r s o m e d i s c u s s i o n o f t h e g e n e r a l k i n d o f

m o d e l s t r u c t u r e I a s s u m e , s e e B a c h , f o r t h c o m i n g ) . S o h e r e a r e s o m e

e x a m p l e s o f t h e k i n d s o f e v e n t u a l i t i e s t h a t c o r r e s p o n d t o t h e a b o v e

d i s t i n c t i o n s :

(9) J o h n k is s M a r y : a t o m i c e v e n t

( 10 ) M a r y s t u m b l e a n d M a r y t w is t h e r a n k l e : p l u ra l e v e n t

( 11 ) M a r y s t u m b l e : a t o m i c e v e n t

( 12 ) P e o p l e d i s c o v e r t h e h i d d e n c o v e : p l u ra l e v e n t

( 1 3) S a l l y b u i l d a c a b i n : a t o m i c e v e n t

( 14 ) S a l ly p o u n d i n a n a i l: a t o m i c e v e n t

( 15 ) J o n e s p o i s o n t h e p o p u l a c e : a t o m i c e v e n t

( 16 ) J o n e s p o u r p o i s o n i n t o t h e w a t e r m a i n : a t o m i c e v e n t

O u r h o m o m o r p h i s m h ( h e n c e f o r t h I w i ll d r o p s u b sc r ip t s o n a ll s y m b o l s

w h e r e i t i s c l e a r f r o m c o n t e x t w h i c h d o m a i n w e a r e c o n s i d e r i n g ) w i l l

d e l i v e r u p f o r u s t h e b o u n d e d b i t s o f p r o c e s s c o r r e s p o n d i n g t o i n s t a n c e s

o f e a c h o f t h e s e e v e n t t y p e s . J u s t a s i n t h e c a s e o f t h e n o m i n a l d o m a i n i t

is e x c e e d i n g l y d i ff ic u l t t o f i n d E n g l i s h e x p r e s s i o n s w h i c h c o r r e s p o n d t o

t h e s e 'p u r e p r o c e s s e s ' ( c f. o u r r e m a r k s o n ( * ) a f t e r o u r f ir st e x a m p l e s ) .

S o m e i n tu i ti o n s I w a n t t o c a p t u r e w i th r e g a r d t o th e a b o v e e x a m p l e s a r e

t h e s e :

A d

( 1 0) a n d ( 1 1 ) : a p l u r a l e v e n t o f t y p e ( 1 0 ) h a s ( n e c e s s a r i l y ) a

s i n g u l a r e v e n t o f t y p e ( 1 1 ) a s a n / - p a r t , a n d t h e p r o c e s s e s a s s o c i a t e d b y h

w i t h t h e l a t t e r is a p - p a r t o f t h e p r o c e s s a s s o c i a t e d w i t h t h e f o r m e r .

A d

( 13 ) a n d ( 14 ): a n e v e n t o f t y p e ( 1 4 ) m i g h t v e r y w e l l b e s u c h t h a t i ts

p r o c e s s i s a p - p a r t o f t h e p r o c e s s a s s o c i a t e d w i t h a n e v e n t o f t y p e ( 1 3 ) .


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1 E M M O N B A C H

A d (15) and (16): Events of these two types might be materially

(processually) equivalent while the events themselves are different. Thus,

Jones might very well intentionally pour poison into the water main (in

order to rid waterbeds of bedfish) and not intentionally poison the

populace (cf. Davidson, 1980,

passim .

Just as in the nominal domain (Link, 1983), I will assume that our

interpretation assigns various predicates to different classes according as

they fall under the sort of classification outlined above. (HOW we decide

or do this will not be my concern in this paper.) So Dying names an

atomic kind of event, Running doesn t, and so on. Familiar properties of

these various kinds of eventualities will follow, such as indivisibility and

additivity (cf. L. Carlson, 1981; Bach, 1981): no proper p-part of a dying

is a dying; the fusion of two runnings is a running, but no two dyings are

a dying. I will return below to some interesting problems connected with

such facts.

4 S O M E P A R A L L E L S A N D P U Z Z L E S

We have found it quite instructive to think about parallels and differen-

ces obtaining between the two domains. In a number of places questions

and observations about one of the domains has led us to consider prob-

lems in the other domain in a new light.

4 .1 . P a cka g in g a n d Gr in d in g

It has frequently been observed that practically any count noun or name 5

can be used as a mass term: There was dog splat tered al l over the road;

M u ch miss io n a ry wa s ea ten a t th e f e s t i va l

(David Lewis s Universal

Grinder, cf. Pelletier, 1979). Moreover, the opposite switch occurs as

well:

m u d s

= kinds of mud ,

i c e - c rea m

= portions of ice-cream (Uni-

versal Packager). In each case, we have a change of meaning with no

overt marking in the form of the word.

In the verbal domain, we find the same sort of phenomenon. Dowty

(1972) observed that practically any process verb can be used event-

ually , given the right context.

One of his examples was the process verb look for. One of the

characteristics of process verbs is that they don t occur comfortably in

the context

N P f in ished Vin g: ? I f in ishes look ing for a un icorn .

Yet in the

context of a library with a well-defined search procedure, a sentence like

I f in ished look ing for a book seems perfectly ordinary.


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T H E L G E B R O F E V E N T S 11

I n E n g l i sh , t h e w a y o f s w i tc h i n g b a c k a n d f o r t h b e t w e e n c o u n t a n d

m a s s , e v e n t a n d p r o c e s s t y p i c a l l y i n v o l v e s n o c h a n g e i n t h e f o r m s

i n v o l v e d . T h e d i f f e r e n c e is r a t h e r i n d u c e d b y t h e c o n t e x t . I n o t h e r

l a n g u a g e s , o v e r t m o r p h o l o g i c a l p r o c e s s e s o r r e la t io n s h i p s a r e a v a i la b l e

o r o b l i g a t o r y , f o r e x a m p l e , i n t h e p e r f e c t i v e - i m p e r f e c t i v e c o n t r a s ts in

S l a v i c l a n g u a g e s . T h i s ra i s e s i m p o r t a n t q u e s t i o n s o f p r i n c i p l e f o r t h e

a n a ly s is o f E n g li s h . D o w e w a n t t o i n v o k e f o r m a t i o n r u le s w i t h z e r o -

m o r p h o l o g y ( i d e n t it y o p e r a t i o n s in t h e s y n t a x o f w o r d s ) , a s in L i n k s r u l e

f o r f o r m i n g t h e m a s s - t e r m c o u n t e r p a r t t o a c o u n t n o u n l ik e

apple

O r d o

w e w a n t t o s o m e h o w g i v e m e a n i n g s f o r w o r d s t h a t a r e u n s p e c i f ie d a l o n g

t h i s d i m e n s i o n ?

I t s e e m s t o m e t h a t t h e r e is a n a s y m m e t r y in t h e s e r e la t i o n s b e t w e e n

c o u n t a n d n o n - c o u n t m e a n i n g s t h a t ru n s in t he s a m e d i r e c t i o n i n t h e t w o

d o m a i n s . T h a t i s, i f w e s t a rt w i th a c o u n t m e a n i n g a n d d e r i v e t h e

n o n - c o u n t m e a n i n g (a s in L i n k s r u l e) t h e r e s e e m s t o b e a r e g u l a r a n d

p r e d i c ta b l e m e a n i n g . T h e m a s s - t e rm

apple

s e e m s t o m e a n t h e s t uf f s u c h

t h a t t h e r e i s a t l e a s t o n e a p p l e s u c h t h a t t h a t s t u f f s t a n d s i n t h e

c o n s t i t u t i o n - r e la t i o n t o t h e a p p l e ( b u t s e e b e l o w , S e c t i o n 4 . 3 f o r s o m e

r e m a i n i n g p r o b l e m s w i t h t h i s a c c o u n t ) . O n t h e o t h e r h a n d , g o i n g i n t h e

o t h e r d i r e c t i o n , t h e c o n n e c t i o n s e e m s m u c h l e s s s y s t e m a t i c , a s a l r e a d y

n o t e d . A b e e r m a y b e a s e r v i n g o f b e e r o r a k i n d o f b e e r . S i m i la r ly , i n t h e

v e r b a l d o m a i n , w h e n w e p u t a p r o c e s s e x p r e s s i o n i n t o a c o u n t c o n t e x t ,

w e m u s t c o m e u p w i th s o m e k i n d o f c o r r e s p o n d i n g e v e n t , b u t ju s t w .h at i t

is is r e l a t i v e l y f r e e , p e r h a p s t h e b e g i n n i n g o f th e p r o c e s s in q u e s t i o n , o r

s o m e b o u n d e d p o r t i o n o f it . T h i s a s y m m e t r y is p re d i c t e d b y o u r f o r m a l

s e t - u p : t h e r e i s a f u n c t i o n ( h o m o m o r p h i s m ) f r o m t h e c o u n t e l e m e n t s t o

t h e n o n - c o u n t o n e s , b u t i t is a m a n y - t o - o n e m a p p i n g s o th a t w e c a n t in

g e n e r a l e x p e c t a u n i q u e a n s w e r w h e n w e a s k w h a t c o u n t e l e m e n t t hi s

p o r t i o n o f n o n - c o u n t s t u f f m i g h t c o r r e s p o n d t o .

C o u n t e l e m e n t s c o m e a s a l re a d y b o u n d e d a n d d i sc r e te it em s . T h e r e -

f o r e w e c a n c o u n t t h e m . N o n - c o u n t e l e m e n t s d o n t a n d t h e r e f o r e n e e d

s o m e a d d i t io n a l s p e c i f i c a ti o n i n o r d e r t o b e u s e d a s c o u n t a b l e e x p r e s s i o n s

w i t h p l u ra l s o r n u m b e r s . F u r t h e r , e x p r e s s io n s w h i c h c a r v e o u t m e a s u r e s

o r q u a n t i t i e s o f s t u f f , - p o u n d s o f , p o r t i o n s o f , e t c . - c a n n o t g o w i t h p u r e

c o u n t - i t e m s i n t h e s in g u l a r, b u t d e m a n d i n t e r p r e t a t i o n o f t h e c o u n t - i t e m

a s m a s s - t e r m o r p r o c e s s c o u n t e r p a r t . M o r e o v e r , f o r p l u ra l s siz e a n d

m e a s u r e a r e r e l e v a n t t o d e t e r m i n i n g n a t u r a l n e s s a n d u s e f u l n e ss o f t h e

p a r t i c u l a r e x p r e s s i o n s ;

two tons of horses

i s o d d f o r p r a c t i c a l r e a s o n s i n a

w a y t h a t

two tons o f beans

o r

f i f ty tons of horses

a r e n o t ( c f . L . C a r l s o n ,

1 9 8 1 , o n t h e s e a n d m a n y o t h e r d e ta i l s) . T h e r e a r e i n t e r e st i n g p u z z le s

a b o u t c o u n t i n g t h a t w e w i ll r e t u r n t o b e l o w ( S e c t i o n 4 .4 ) .


8/12

2 E M M O N B A C H

4 . 2 . The Part i t ive Puzzle

D o w t y ( 1 9 7 8 ) a n d o t h e r s h a v e d i s c u s s e d t h e s o c a l l e d ' i m p e r f e c t i v e

p a r a d o x ' ( I p r e f e r t o c a l l it a p u z z l e ) . B r i e f l y , t h e p u z z l e i s t h is : h o w c a n

w e c h a r a c t e r i z e t h e m e a n i n g o f a p r o g r e s s i v e s e n t e n c e s l i k e ( 1 7 ) o n t h e

b a si s o f t h e m e a n i n g o f a s i m p l e s e n t e n c e l i k e (1 8 ) w h e n ( 17 ) c a n b e t r u e

o f a h is t o r y w i t h o u t ( 1 8 ) e v e r b e i n g t r u e ?

( 1 7 ) J o h n w a s c r o s s i n g t h e s t r e e t .

( 18 ) J o h n c r o s s e d t h e s t r e e t.

( S e e V l a c h , 1 9 8 1 ; D o w t y , 1 9 7 9 .)

N a t u r a l l y , w e w a n t t o u s e t h e a p p a r a t u s w e h a v e s e t u p t o p r o v i d e a n

a c c o u n t o f t h e E n g l i s h p ro g r e s s iv e , p e r h a p s a l o n g t h e l in e s o f V l a c h

( 1 9 8 1 ) . T h i n k i n g a b o u t h o w t o d o t h i s h a s l e d u s t o s e e t h a t t h e r e i s a

p e r f e c t l y pa r a ll e l p r o b l e m in t h e n o m i n a l d o m a i n , w h i c h w e c a l l t h e

' p a r t i t i v e p u z z l e ' .

C o n s i d e r L i n k ' s a c c o u n t o f t h e f o l lo w i n g s e n te n c e :

( 1 9 ) T h e r e is a p p l e i n t h e s a l a d .

L i n k ' s i n t e r p r e t a t i o n a m o u n t s t o t h i s : t h e r e a r e s o m e a p p l e s , s u c h t h a t

s o m e o f t h e s t u f f m a k i n g t h e m u p i s p r e s e n t i n th e s a l id . N o t e t h e

e x i s t e n ti a l q u a n t i f i c a t i o n o v e r a p p l e s; th e s e n t e n c e c o u l d n o t b e t r u e o f a

h i s to r y w h i c h n e v e r h a d a n y a p p l e s i n i t. T h i s s e em s r e a s o n a b l e e n o u g h

f o r t h i s s e n t e n c e , b u t c o n s i d e r t h e f o l l o w i n g :

( 20 ) T h i s is p a r t o f a p a p e r o n n a t u r a l l a n g u a g e m e t a p h y s i c s .

( 21 ) W e f o u n d p a r t o f a R o m a n a q u e d u c t .

I t s e e m s a s if ( 20 ) c o u l d b e t r u e e v e n t h o u g h ( al as ) t h e p a p e r i n q u e s t i o n

n e v e r r e a c h e d f u l f i l m e n t , a n d ( 21 ) t r u e w h e n t h e r e n o l o n g e r i s a n

a q u e d u c t o r e v e n i f p r o g r e s s o n t h e c o n s t r u c t i o n w a s i n t e r ru p t e d f o r e v e r

b y h o r d e s o f b a r b a r i a n s f r o m t h e n o r t h .

L e t u s lo o k m o r e c l o se l y a t L i n k ' s a c c o u n t . T h e d e n o t a t i o n o f t h e m a s s

t e r m c o r r e s p o n d e n t m p o f a p r e d i c a t e P is g i v e n a s th i s (p . 3 0 9 ):

~rnp~ : = {x e D I x -- su p [h~e~]}.

( H e r e , sup s t a n d s f o r supremum. T h a t i s , t h e d e n o t a t i o n o f apple ( u s e d

a s a p r e d i c a t i v e m a s s t e r m ) i s t h e s e t o f q u a n t i t i e s o f m a t t e r t h a t a r e

m - p a r t s o f t h e v a l u e o f h a p p l i e d t o t h e s e t o f a p p l e s in t h e w o r l d . T h u s ,

n o a p p l e s , n o a p p l e . B u t t h e r e c o u l d s u r e l y b e a w o r l d i n w h i c h i t w a s

p o s s ib l e t o a r t if i c ia l ly m a n u f a c t u r e a p p l e w i t h o u t t h e r e b e i n g a n y a p p l e s ,

o r f o r le s s f a r f e t c h e d e x a m p l e s , c o n s i d e r a g a i n E x a m p l e s (4 ) a n d ( 5 ).


9/12

T H E L G E B R O F E V E N T S

13

S u c h e x a m p l e s s h o w t h a t w e n e e d t o a ll ow f o r a m o r e i n d i r e c t r e l a t io n

b e t w e e n t h e d e n o t a t i o n o f a m a s s p r e d i c a t i v e m a s s t e r m a n d t h e c o r -

r e s p o n d i n g c o u n t p r e d i c a t e . B a s i ca l ly , w e n e e d t o b e a b le t o s ay w h e n

c e r t a i n s t u f f i s o f t h e r i g h t k i n d t o q u a l i f y a s f a l l i n g u n d e r t h e e x t e n s i o n

o f t h e m a s s t e r m , o r b e t t e r w e n e e d t o a s s u m e t h a t w e c a n s a y w h e n t hi s

i s t h e c a s e . T o a c t u a l l y g i v e c r i t e r i a i s n o p a r t o f l i n g u i s t i c s ( c f . P u t n a m ,

p a s s i m .

F u r t h e r , a l t h o u g h w e h a v e a s s u m e d t h a t t h e tw o d o m a i n s o f t h in g s a n d

s tu f f a r e s e p a r a t e , i t s e e m s t o m e t o b e r e a s o n a b l e t o a s s u m e t h a t o u r

k n o w l e d g e o f w h a t q u a l if ie s as a q u a n t i t y o f a p p l e o r m u d o r g o l d is

b a s e d o n o u r u n d e r s t a n d i n g o f w h a t is m e a n t b y t h e t e r m p h r a s e s

apples ,

m u d ,

o r

go ld ,

u n d e r s t o o d a s n a m e s f o r ki n d s ( G . C a r l s o n , 1 9 7 7 ) o r

p r o p e r t i e s ( C h i e r c h i a , 1 9 8 2 ) . B o t h C a r l s o n a n d C h i e r c h i a a r g u e t h a t

s u c h t e r m s a r e m o r e i n t e n s io n a l th a n t h e p r o p e r t i e s o f M o n t a g u e , w h i c h

a r e f u n c t i o n s f r o m w o r l d - t i m e p a i r s t o i n d i v i d u a l s . W e m a y s a y t h a t a

p r o p e r t y o r k i n d d e t e r m i n e s s u c h a fu n c t i o n , w h i c h t h e n m a y b e u s e d t o

g e t t h e d e n o t a t i o n s o f t h e c o r r e s p o n d i n g p r e d i c a t i v e s app le , go ld a n d

m u d . S o t o s a y t h a t t h e r e is a p p l e i n t h e s a l a d i s t o s a y t h a t t h e r e is s o m e

s t u f f i n t h e s a l a d o f t h e r i g h t s o r t a s t o q u a l i f y a s a p p l e a n d t h e l a t t e r

i n v o l v e s a p p e a l i n g t o o u r k n o w l e d g e o f t h e k i n d o f a p p l e s o r th e

p r o p e r t y o f b e i n g a n a p p l e . I t s h o u l d t h e n f a l l o u t o f o u r t h e o r y t h a t

p a r t i c u l a r a p p l e s a r e m a d e o f a p p l e a n d s o o n .

4 .3 . H o w O l d I s t h e G o l d ?

L i n k p r o v i d e s a n i c e a n a ly s is o f t h e p u z z l e p r e s e n t e d b y a s e n t e n c e l ik e

th is :

( 2 1 ) T h e g o l d m a k i n g u p T e r r y ' s r i n g i s o l d b u t t h e r i n g i t s e l f i s

n e w ( n o t o l d ) .

P u z z l e s l i k e t h i s o n e a r e a m o n g t h e b e s t e v i d e n c e f o r n o t i d e n t i f y i n g

t h i n g s w i t h t h e i r m a t e r i a l c o u n t e r p a r t s . B u t t h e r e i s s t i l l a p r o b l e m .

T h e i n t e r p r e t a t i o n o f (2 2 ) is t hi s: T h e x s u c h th a t x m a k e s u p T e r r y ' s

r i n g a n d i s g o l d i s o l d b u t T e r r y ' s r i n g i s n o t o l d . N o c o n t r a d i c t i o n , s i n c e

x a n d T e r r y ' s r i n g a r e n o t th e s a m e t h i n g , x is j u s t th e v a l u e o f h w i t h

T e r r y ' s r in g a s a r g u m e n t . B u t n o w c o n s i d e r a s e n t e n c e l ik e (2 3 ):

( 2 3 )

T h e s n o w m a k i n g u p t h is s n o w m a n i s q u i t e n e w b u t t h e H2

m a k i n g i t u p i s v e r y o ld ( a n d t h e H a n d O e v e n o l d e r ) .

T h e i n t e r p r e t a t i o n o f th is s e n t e n c e c o m e s o u t li ke th is : T h e x s u c h t h a t x


10/12

4 E M M O N B A C H

c o n s t i t u t e s t h e s n o w m a n a n d x is s n o w i s n e w b u t t h e y s u c h t h a t y

c o n s t i t u te s t h e s n o w m a n a n d y is w a t e r is v e r y o l d ( n o t n ew ) . T h i s

is

c o n t r a d i c t o r y a c c o r d i n g t o L i n k ' s a c c o u n t s in c e x a n d y m u s t b e i d e n -

t i c a l ( s in c e h is a f u n c t i o n ) . I f w e f o l l o w L i n k ' s s a g e a d v i c e - o u r g u i d e

i n o n t o l o g i c a l m a t t e r s h a s t o b e l a n g u a g e i t s e l f ( L i n k , 1 9 8 3 , p p . 3 0 3 f .) -

t h e n i t s e e m s t o m e t h a t w e h a v e t o s e t th i n g s u p i n s u c h a w a y t h a t w e

c a n r e f e r t o in d i v id u a l s u n d e r a d e s c r i p t i o n , s o m e h o w . T h i s p u z z l e is

c l o s e ly c o n n e c t e d t o t h e n e x t o n e .

I f w e f o l l o w L i n k ' s a d v i c e , t h e n w e m u s t a c k n o w l e d g e t h a t tw o t h i n gs

w i t h c o n t r a d i c t o r y p r o p e r t i e s c a n n o t b e id e n t ic a l . T h u s t h e s n o w m a k i n g

u p t h e s n o w m a n a n d t h e H 2 0 m a k i n g i t u p m u s t b e d i ff e re n t , a n d n e i t h e r

c a n b e e q u a t e d w i th t h e u n d i f f e r e n t i a t e d q u a n t i ty o f m a t t e r g i v e n b y

L i n k 's h o m o m o r p h i s m . W h a t t o d o?

T h e f ir st p o s s i b il it y is t o a c k n o w l e d g e t h a t o u r l a n g u a g e a l lo w s u s t o

t a l k a b o u t c h a i n s o f c o m p o s i t i o n , s o t o s p e a k . T h e s n o w i n t h e s n o w m a n

i s i t s e l f m a d e u p o f t h e w a t e r i n t h e s n o w m a n ( p l u s a i r ) a n d s o o n . W h a t

t h e e x a m p l e s h o w s , t h e n , i s t h a t w e c a n n o t u s e t h e c o n s t i t u t i o n r e l a t i o n

d i r e c t l y in a n i n t e r p r e t a t i o n o f a p h r a s e l i k e

t h e sn o w ma k in g u p th e

s n o w m a n .

O u r i n t e r p r e t a t i o n o f su c h p h r a s e s m u s t b e s u c h t h a t i t d o e s

n o t h o l d t h a t i f x m a k e s u p a a n d y m a k e s u p a t h e n x = y . W e c a n

e s s e n ti a ll y k e e p a l l o f L i n k ' s a p p a r a t u s i n c l u d i n g t h e h o m o m o r p h i s m a n d

t h e e q u i v a l e n c e c la ss es g e n e r a t e d b y i t b u t m e r e l y a m e n d t h e w a y i n

w h i c h E n g l i s h w o r d s l i k e

make up, const i tu te

o r p h r a s e s l i k e

the gold in

the ring

a r e i n t e r p r e t e d .

A s e c o n d w a y w o u l d b e t o r e m o v e a l t o g e t h e r t h e e n t i t ie s in D f r o m

t h e d o m a i n o f in d i vi d u a ls . I e x p l o r e o n e w a y o f d o i n g t hi s b a s e d o n

C r e s s w e l l 's ( 1 9 7 3 ) m e t a p h y s i c s o f p o s s ib l e w o r l d s a n d i n d i v i d u a l s in

B a c h , f o r t h c o m i n g . T h i s w o u l d a m o u n t t o s a y in g s o m e t h i n g l ik e th is :

Sto f f an s ich

( j u s t l i k e t h e

Din g a n s i ch

c a n h a v e n o p r o p e r t i e s , a t l e a s t

a s f a r a s o u r l a n g u a g e i s c o n c e r n e d .

4 .4 .

H o w M a n y T h i n g s A r e T h e r e i n t h e R o o m ?

I n b o t h d o m a i n s m a n y w r i t e r s h a v e p o i n t e d t o c h a r a c t e r is t i c p r o p e r t i e s

l i k e a d d i t i v i t y , s u b d i v i s ib i l it y , a n t i a d d i t i v i t y a n d a n t i s u b d i v i s i b i l i ty w h i c h

p l a y c l e a r ro l e s in g i v i n g a n a c c o u n t o f e n t a i l m e n t r e l a t io n s a m o n g

s e n t e n c e s : H e n g i s t ' s e a r c a n ' t b e a h o r s e , m u d p l u s m u d is m u d , a h o r s e

p l u s a h o r s e i s n 't a h o r s e , a n d s o o n . B u t i n b o t h d o m a i n s t h e r e a r e c l e a r

a n d o r d i n a r y e x a m p l e s o f c o u n t i t e m s t h a t d o n ' t f o l lo w t h e s e r e s tr i c ti o n s .

T h e s e a r e w o r d s l i k e

th ing , even t , happen ,

a n d s o o n . S u p p o s e i t i s t r u e

t h a t s o m e t h i n g h a p p e n e d , t h e n in th e n o r m a l c a s e th e r e a r e s m a l l e r


11/12

T H E A L G E B R A O F E V E N T S ] 5

s u b e v e n t s t h a t m a k e u p t h e b ig th i n g t h a t h a p p e n e d t h a t a r e al so

h a p p e n i n g s . S i m i la r ly f o r t h i n g s . I n b o t h d o m a i n s w e a r e at s o m e t h i n g o f

a lo s s t o t r y t o a n s w e r q u e s t i o n s l ik e t h e s e :

( 24 ) H o w m a n y t h in g s a r e t h e r e i n t h e r o o m ?

( 25 ) H o w m a n y e v e n t s t o o k p l a c e i n t h e l a st h o u r ?

G u p t a ( 1 9 8 0 ) h a s s t r e s s e d t h e i m p o r t a n c e o f c r i te r i a o f r e i d e n t i f i c a t i o n

a n d i n d i v i d u a t io n i n t h e lo g i c o f c o m m o n n o u n s . O u r d i s c u s s i o n h e r e

s h o w s t h a t p r i n c i p le s o f i n d i v i d u a t i o n a r e c r u c i a l fo r e x p r e s s i o n s a n d

c o n c e p t s i n t h e v e r b a l d o m a i n a s w e l l . W e f o l l o w L i n k i n not r e q u i r i n g

t h a t t h e s u b d o m a i n s D i a n d D e b e a t o m i c B o o l e a n a l g e b r a s . T h i s is a s i t

s h o u l d b e . I t is n o t p a r t o f li n g u i s t i c s t o d e c i d e w h e t h e r a ll m a t t e r is

a t o m i c o r a l l h a p p e n i n g s a r e r e d u c i b l e t o li tt le g r a n u l e s o f p r o c e s s .

I n d e e d , i f c o n t e m p o r a r y p h y s i c a l th e o r i e s a r e t o b e b e l i e v e d , s u c h

u l t im a t e q u e s t i o n s a r e b a s i c a l ly i n c o h e r e n t . E v e n t s a n d p r o c e s s e s a r e

d i s jo i n t, a n d t h i s s e e m s t o b e m o r e a n a r t i f a c t o f o u r l a n g u a g e o r

c o n c e p t u a l i z a t i o n s o f t h e w o r l d t h a n s o m e t h i n g a b o u t t h e w o r l d i ts el f, so

t h a t p r o b a b l y h e r e t o o o u r s t r i c t l y s e m a n t i c t h e o r i e s s h o u l d r e m a i n s il en t .

A C K N O W L E D G E M E N T S

A l a r g e p a r t o f t h e s u b s t a n c e o f t h i s p a p e r d e r i v e s f r o m j o i n t w o r k w i t h

B a r b a r a P a r t e e , f ir st p r e s e n t e d b y u s at a c o l l o q u i u m a t S t a n f o r d U n i -

v e r s i t y , h e n c e t h e f r e q u e n t w e . I t a k e fu l l r e s p o n s i b i l i t y f o r e r r o r s o r

i n fe l i c i ti es . G r a t e f u l a c k n o w l e d g m e n t is t e n d e r e d t o t h e M a x P l a n c k

I n s t i t u t fi ir P s y c h o l i n g u i s t i k i n N i j m e g e n , w h i c h s u p p o r t e d p a r t o f t h e

w o r k r e p o r t e d o n h e re .

N O T E S

Besides the writers mentioned in the first paragraph, see for example, the excellent survey

in Do w ry (1979) and the references cited there. T he classic m od ern works dealing with

verb-classif icat ion are Kenny (1963), and Vendler (1957). To my knowledge, Verkuyl

(1972) was the first extensive work which recognized the importance of these distinctions

for linguistic theory.

2 D ow ty (1977), Vlach (1981), for example.

4 It is important to notice that Link's proposals differ crucially from previous attempts to

deal with plurals which cons tructe d interpretations for plurals in purely set-theoretic ways,

as for example in Bennett 's (1974) classic treatment.

4 Strictly speaking, w e w ill hav e the corresponding N P denotations, that is, pro per ty sets

for the individuals mentioned. I ignore this complication throughout the paper.

s Link does not deal with the use of names in mass contexts: They put five pounds of Porky

into the stew


12/12

6 EMMON BACH

R E F E R E NC E S

Allen, R. L.: 1966, The Verb System o f Present-Day English Mouton, The Hague.

Bach, Emmon: 1981, On Time, Tense, and Aspect: An Essay in English Metaphysics , in

Peter Cole (ed.), Rad ical Pragmatics Academic Press, New York, pp. 62-81.

Bach, Emmon: forthcoming, The Metaphysics of Natural Language , to appear in Pro-

ceedings of the Seventh International Congress on Logic M ethodology and Philosophy of

Science.

Carlson, Gregory: 1977, Reference to Kinds in English Ph.D. dissertation, University of

Massachusetts, Amherst.

Carlson, Lauri: 1981, Aspect and Quantification , Syntax and Semantics 14.

Chierchia, Gennaro: 1982, Nominalizations and Montague Grammar: A Semantics

Without Types for Natural Language , Linguistics and Philosophy 5, 303-354.

Cresswell, Max J.: 1973, Logics and Languages , Methuen, London.

Davidson, Donald: 1980, Essays o n Actions and E vents Clarendon Press, Oxford.

Dowty, David R.: 1972, Studies in the Logic o f V erb Asp ect and Tim e Reference in English

unpublished Ph.D. dissertation, The University of Texas, Austin.

Dowty, David R.: 1977, Toward a Semantic Analysis of Verb Aspect and the English

Imperfective Progressive ,

Linguistics and Philosophy

1, 45-79.

Dowty, David R.: 1979,

Word Meaning and Montague Gram mar

Reidel, Dordrecht.

Gupta, Anil: 1980, The Logic o f Co mm on Nouns Yale University, New Haven.

Hinrichs, Erhard: 1981, Temporale Anaphora im Englischen University of Tuebingen,

Magisterarbiet.

Hoepelman, J. and C. Rohrer: 1980, On the Mass Count Distinction and the French

Imparfait and Passe Simple , in C. Rohrer (ed.), Tim e Tense and Aspect Tuebingen,

Niemeyer, pp. 629-645.

Kamp, Hans: 1980, Some Remarks on the Logic of Change , Part I, in Christian Rohrer

(ed.), Tim e Tense and Quanti fiers Niymeyer, Tuebingen.

Karnp, Hans: 1981, l~v6nements, repr6sentations diseursives et r6f6rence temporelle ,

Language 64, 39-64.

Kenny, Anthony: 1963, Action Em otion and Will Humanities Press, New York.

Link, Godehard: 1983, The Logical Analysis of Plurals and Mass Terms , in R. Bauerle,

Ch. Schwarze, and A. von Stechow (eds.), Meaning U se and Interpretation of Lan gua ge

pp. 302-323.

Mourelatos, Alexander, P. D.: 1978, Events, Processes, and States , Linguistics and

Philosophy 2, 415-434.

Pelletier, F. J.: 1975, Non-Singular Reference , in F. J. Pelletie~ (ed.), Mass Terms: Some

Philosophical Problem s

Reidel, Dordrecht, pp. 1-14.

Putnam, Hilary: 1975, Mind Langu age and Reali ty Philosophical Papers, Vol. 2,

Cambridge University Press, Cambridge.

Vendler, Zeno: 1957, Verbs and Times , Philosophical Re view 56, 143-160.

Verkuyl, Henk: 1972, On the Compositional Nature of the Aspects Reidel, Dordrecht.

Vlach, Frank: 1981, The Semantics of the Progressive , Syntax and Semantics 14, 271-292.

D e p a r t m e n t o f L i n g u i st ic s

U n i v e rs i t y o f M a s s a c h u s e t t s A m h e r s t

A m h e r s t M a s s a c h u s e t t s 0 1 0 0 3 U . S . A .

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Bach, Emmon - The Algebra of Events