barcan marcus modalities and intensional languages

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R U TH B R C N M R C U S

M O D A L I T I E S A N D I N T E N S IO N A L L A N G U A G E S *

The subject o f th is pape r i s the found at ions of moda l log ic . By founda-

t ions, we genera l ly mean the under ly ing assumpt ions , the underp innings .

Ther e is a norma t ive sense in which it has been c la imed that moda l log ic

i s wi thout fou ndat ion . P rofessor Quine , in W o r d a n d O b j e c t suggests tha t

i t was conceived in s in: the s in of confus ing use and ment ion . T he or ig inal

t ransgressors were Russel l and Whi tehead . Lewis fo l lowed su i t and

cons t ructed a log ic in which an opera tor corresponding to 'necessar i ly '

opera tes on sen tences whereas ' i s necessary ' ought to be v iewed as a

pred icate of sen tences . As Professor Quine recons t ruct s the h i s tory of the

enterpr i se , I the op era t iona l use of modal i t ies p rom ised only one advan-

tage: the poss ib i l i ty of quant i fy ing in to modal contex t s . This severa l o f

us z were e nt iced into doing. But the evi ls of the sentent ial calculus were

found out in the funct ional ca lcu lus , and wi th i t - to quote again f rom

W o r d a n d O b j e c t -

' t h e v a ri ed s o r ro ws o f m o d a l i ty t r an s p o s e ' .

I do no t in tend to c la im that m odal log ic i s whol ly wi thout sorrows , bu t

only that they are no t those w hich Professor Quine descr ibes . I do c la im

that moda l log ic is wor th y of defense , fo r i t is usefu l in con nect ion w i th

man y in teres ting and im por ta n t ques t ions such as the analys i s o f causa-

t ion , en ta i lment , ob l igat ion and bel ief s t a tements , to nam e o nly a few.

I f we ins is t on equat ing fo rmal log ic wi th s t rongly ex tens ional funct ional

calcu li then S t rawson a i s correc t in say ing that ' t he analy t i ca l equipment

(of the form al log ic ian) is inadequate for the d i ssection of mos t o rd in ary

types of empi r ica l s t a tement . '

I N T E N S I O N L L N G U G E S

I wil l begin with the not ion of an intensional language. I wil l make a

fur ther d i s t inc t ion between those which are expl ic i t ly and impl ic i t ly

* Presented at themeeting of theBoston Colloquium fo r the Philosophy of Science,

February 7, 1962 in conjunction with a commentary by Prof. W. V. Quine. It was

announced under the more general title Foun dation o f Modal Logic.

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in tens iona l . Our not ion of in tens iona l i ty does not divide languages into

mutu a l ly exc lus ive c lasses bu t r a ther orders them loose ly as s t rongly or

weakly intens iona l . A language i s expl ic i t ly in tens iona l to the degree

to which i t does not equa te the ident i ty re la t ion wi th some weaker form

of equiva lence . We wi l l assume tha t every language must have some

constant objec ts of r e fe rence ( things) , ways of c lass i fying and order ing

them, w ays of mak ing s ta tements , and w ays of separa t ing t rue s ta tements

f rom fa lse ones . We wi l l not go into the ques t ion as to how we come to

regard some e lements of exper ience as th ings , but one c r i te r ion for sor t ing

out th e e lements of exper ience which we regard as th ings i s tha t they m ay

ente r in to the ident i ty re la t ion. In a formal ized language , those symbols

which na me things wi l l be those for which i t i s meaningful to asse r t tha t

I holds be tween them, where I names the ident i ty re lat ion.

Ordinar i ly , and in the fami l ia r const ruc t ions o f formal sys tems, the iden-

t i ty re la t ion mu st be he ld appropr ia te for individua ls. I f x and y a re

ind iv idua l na me s t he n

(1) x y

is a sentence, and if they are ind ividu al var iables, then (1) is a sentential

func t ion. Whether a language confer s th inghood on a t t r ibutes , c lasses ,

proposi t ion s i s no t so muc h a mat te r of whether var iables appropr ia te

to them can be quant i f ied upon (and we wi l l r e turn to th is la te r ) , but

ra ther whether (1) i s mean ingful where x and y ma y take as va lues

names of a t t r ibutes , c lasses, proposi t ions . W e note in pass ing tha t the

meaningfulness of (1) wi th respec t to a t t r ibutes and c lasses i s more

f requent ly a l lowed than the meaningfulness of (1) in connec t ion wi th

proposi t ions .

Returning now to the not ion of expl ic i t in tens iona l i ty , i f ident i ty i s

appropriate to proposit ions, a t tr ibutes, c lasses, as well as individuals,

then any weakening of the ident i ty re la t ion wi th respec t to any of these

ent i ties may be t hou ght o f as an extens iona l izing of the language . B y a

weakening o f the ident i ty re la tion i s mea nt equa t ing i t wi th some weaker

equivalence relat ion.

On the leve l of individua ls , one or p erhaps two equiva lence re lat ions a re

custom ar i ly present : ident i ty and indiscernibi li ty . This does n ot prec lude

the int roduc t ion of others such as s imi la r i ty or congruence , but the

s t rongest of these i s identi ty . Where ident i ty is def ined ra ther than taken

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MOD LITIES ND INTENS ION L L NGU GES

as primit ive, i t i s cust oma ry to define it in terms o findiscern ibi l i ty , one for m

of which i s

(2) X Ind y =a ¢ (fp)(¢ x eq ~0y)

In a sys tem of mater ia l impl ica t ion

Sm) ,

eq is t aken as ~ . In modal

systems, eq ma y be take n as :::- -:. In mo re s t rongly intensional systems eq

may be t aken as the s t ronges t equ ivalence re la t ion appropr ia te to such

expressions as (px . In separat ing (1) and (2) I should l ike to suggest the

possibi l i ty that to eq uate (1) and (2) may already be an expl ici t weak ening

of the ident i ty relat ion, and co nsequ ent ly an extensional izing principle.

This was f i rs t sugges ted to me by a paper o f Ram sey . 4 Tho ugh I now

regard h i s par t i cu lar argument in suppor t o f the d i s t inc tion as unconvinc-

ing, I am reluctant to reject the possibi l i ty . I suppose that at bot tom my

appeal i s to o rd inary l anguage, s ince a l though i t i s obvious ly absurd to

talk o f two things being the same thing, i t seems not qui te so absur d to

ta lk o f two th ings being ind i scern ible f rom one another . In equa t ing (1)

and (2) we are saying th at to be dis t inct is to be discern ibly dis t inct in the

sense of there being one proper ty no t c omm on to bo th . P erhaps i t is

unnecessary to m ent ion that i f we conf ine th ings to ob ject s wi th spat io -

temporal his tories , then i t makes no sense to dis t inguish (1) and (2).

And indeed , in my ex tens ions o f modal log ic , I have chosen to def ine

ident i ty in terms o f (2). Ho we ver the possibi l i ty of such a dis t inct ion

ought to be ment ioned before i t i s ob l i t era ted . Exc ept fo r the weakening

of (1) by equat ing i t with (2), extensional i ty principles are absent on the

level of individuals .

Proceed ing now to funct ional ca lcu l i wi th th eory o f types , an ex ten-

s ional ity p r incip le i s o f the form

(3) x eq y -+ xl y .

The a r row may be one o f the impl ica t ion re la t ions presen t in the sys tem

or som e metal inguist ic condi t iona l , eq is one of the equivalence relat ions

approp r ia te to x and y , bu t n o t iden t i ty . Wi th in the sys tem of mater ia l

implicat ion, x and y may be take n as symbols for classes , eq as class

equal i ty ( in the sense of hav ing the same members ) ; o r x and y may be

taken as symbols fo r p ropos i t ions and eq as the t r ip le bar . In ex tended

moda l systems eq may be t aken as the quadruple bar where x and y

are symbols fo r p ropos i t ions . I f the ex tended modal sys tem has symbols

for c lasses , eq may be t aken as hav ing the same memb ers o r a l t erna-

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t ive ly , necessar ily hav ing the same m embers , which ca n be expressed

wi th in such a l anguage. I f we wish to d i s tinguish c lasses f rom at t r ibu tes

in such a sys tem (a l though I regard th i s as the perpetua t ion of a confus ion) ,

eq m ay be t ake n as necessar ily app l ies to the same th ing , which i s

direct ly expressible within the system. In a languag e, which pe rmits

ep i s temic contex t s such as bel i ef con tex t s , an even s t ronger equ ivalence

re la t ion would have to be presen t , than e i ther mater ia l o r s t r i c t equ iv-

a lence . Takin g that s t ronger re la t ion as eq , (3 ) would s ti ll be an ex ten-

sional izing principle in such a s t rongly intensional language.

I should now l ike to tu rn to the no t ion of impl ic i t ex tens ional i ty , which i s

boun d up wi th the k inds of subs t i tu t ion theorem s avai l ab le in a l anguage.

Conf in ing ourse lves fo r the sake o f s implic i ty o f expos i t ion to a sen ten t i a l

ca lcu lus , one form of the subs t i tu t ion theorem i s

(4) x eq l y -+ z eq2 w

where x , y , z , w are wel l - fo rmed, w i s the resu l t o f rep lac ing one or

more occurrence s of x by y in z , a nd -+ symbol izes impl ica t ion or a

meta l ingui s t i c condi t ional . In the sys tem of mater ia l impl ica t ion Sm o r

QSm), (4 ) i s p rovab le where eq l and eq2 are bo th t aken as mater ia l equ iv-

a lence for app ropr ia te values o f x , y , z , w. Th at i s

5) x -_- y) =

No w (5) i s dea r ly fa l se i f we are go ing to a l low contex t s involv ing beli ef ,

log ica l necess ity , phys ica l necess ity and so on . W e are fami l i ar wi th the

examples . I f x ~ s t aken as John i s a feather less b iped , and y as John i s

a rat io nal a nima l , then (5) fai ls . Our choic e is to re ject (5) as i t s tands, or

to rejec t al l contexts in which i t fai ls. If the lat ter c hoice is made, the

language i s impl ic i t ly ex tens ional s ince i t cannot counten ance pred icates

or con tex t s which might be permiss ib le in a more s t rongly in tens ional

l anguage. P rofessor Quine s so lu t ion i s the l a tt er . Al l such contex t s are

dum ped ind i scr iminate ly on to a shel f l abell ed referen t i a l opaci ty o r

more preci se ly contex t s which co nfer referen t i a l opaci ty , and are d i s -

posed of . But the conten t s o f tha t shel f are o f enorm ous in teres t to some

of us and we would l ike to exam ine them in a sys temat ic and form al

manner . Fo r th i s we need a language which i s appro pr ia te ly in tens ional .

In the m odal ca lcu lus , s ince there are two k inds of equ ivalence which m ay

hold between x and y , (4 ) represen t s four poss ib le subs t i tu t ion theorems ,

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M O D L I T I E S N D I N T E N S I O N L L N G U G E S

some o f which a re provable . W e wi l l r e turn to th is shor t ly . S imi la r ly , i f

we are going to permit epistemic contexts, the modal analogue of (4) wil l

f a i l in those contexts and a more appropr ia te one wi ll have to supplement i t.

IDENTITY ND S UBS TITUTIO N IN QU NTIF IED MOD L LOGIC

In the l ight of the previous remark s I wo uld l ike to turn specifically to the

cr i t ic i sms ra ised aga ins t extended modal sys tems in connec t ion wi th

identi ty and substi tut ion . In p ar t icular , I wil l i'efer to m y 5 extension o f

Lewis 6 4 which consis ted of in t roduc ing quant i f ica t ion in the usua l

ma nne r a nd t he a dd i t ion o f the a x iom 7

(6 ) O O x) A - - 3 0x ) ~ ,4 .

I wil l cal l this system QS4. QS4 does n ot have an explic it axiom of ex-

tens iona l i ty , a l thou gh i t does have an impl ic i t extens iona l iz ing pr in-

c iple in the fo rm of the subs t i tu t ion theorem. I t would appear tha t

for ma ny uses to which mod al calculi may be put , 5 is to be preferred 8.

In an extended 4 , Pr ior 9 has show n tha t (6) i s a theorem. M y sub-

sequent r emarks , unless otherwise indica ted, apply equa l ly to QS5.

In

Qs4

(1) is def ined in terms of (2) . (2), and cons equen tly (1) , adm it

of a l terna t ives where ' eq ' may be taken as mate r ia l or s t r ict equiva lence :

' I ra ' and ' I ' respectively. The following are theorems of

QS4:

(7) (xlmy) ~ (xly) a n d

(8) (xly) ~ [] (xly)

where 'D ' i s the modal symbol for logica l necess i ty . In (7) 'Ira a nd ' I '

a re s t r ic tly equiva lent. W i thin such a m odal language , they a re there fore

indis t inguishable by vi r tue of the subs t i tu t ion theorem. Cont ingent

identi t ies are d isal lowed b y (8).

(9) (xly). 0 ~ (xly)

is a contradict ion.

Profess or Qu inO ° f inds these results offensive, for he sees (8) as 'pur ifyin g

the universe. ' Concrete enti t ies are said to be banished and replaced

by pa l l id concepts . The a rgu ment i s f ami l ia r :

(10) The evening star eq the mo rnin g star

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i s sa id to express a t rue iden t i ty , ye t they are no t va l id ly in ter subs t i tu t -

ab le in

(11 ) I t i s necessary tha t the even ing s tar i s the even ing s tar .

The r ebu t ta l s a r e a l so f ami l iar . Rather than ted ious r epet i t ion , I w i l l t ry

to r es ta te them more per suas ively . Th is i s d i f f icu l t , f o r I have never

appre cia te d the fo rce o f the o r ig inal a rgu men t , In r es ta t ing the case ,

I would l ike to cons ider the fo l lowing in fo rmal argument :

(12) I f p i s a t au to lo gy , and p eq q , then q i s a t au to log y

w h er e eq n am es s o m e eq ui v a l en ce r e l a t io n ap p r o p r i a t e t o p an d q .

In

Sm

i f eq is tak en as ~ then a restr icted (12) is ava ilable where p ~ q

i s p rovab le .

One migh t say in fo rmal ly tha t w i th r espect to an y language, i f (12) is sa id

to f ai l , then we mu s t be us ing tau to lo gy in a very pecu l iar way , o r what i s

taken as eq i s no t su f f icien t equ ivalence r e la t ion ap prop r ia te to p and q .

C o n s i d e r t h e c l a i m t h a t

(13) alb

i s a t rue iden t i ty . No w i f (13) is such a t rue iden t i ty , then a and b are the

s am e t h i n g . I t d o es n t s ay t h a t a an d b a r e t w o t h i n g s w h i ch h ap p en ,

th rough some acciden t , to be one . True , we are us ing two d i f f eren t names

f o r t h a t s am e t h i n g , b u t w e m u s t b e ca r e f u l ab o u t u s e an d m en t i o n . I f ,

then , (13) i s t rue , i t mus t say the same th ing as

(14)

aIa

But (14) i s su re ly a tau to logy , and so (13) mus t su re ly be a tau t o logy a s

wel l . Th is i s p rec i se ly the impor t o f nay theorem (8) . We would therefo re

expect , indeed it would be a consequence o f the t ru th o f (13) , tha t a i s

r ep l aceabl e b y b i n an y co n t ex t ex cep t t h o s e w h ich a r e ab o u t t h e n am es

a a n d b .

N o w s u p p o s e w e co m e u p o n a s t a t em en t l i k e

( I 5 ) Sco t t i s the au t hor o f

Waverley

an d w e h av e a d ec i s io n t o m ak e . T h i s d ec is i o n can n o t b e m ad e i n a f o r m a l

v acu u m , b u t m u s t d ep en d t o a co n s i d e r ab l e ex t en t o n s o m e i n f o r m a l

cons id era t ion as to w hat i t is we are t ry ing to sa y in (10) and (15) . I f we

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d ec i de t h a t t h e even i n g s t a r an d t h e m o r n i n g s t a r a r e n am es f o r t h e

s am e t h in g , an d t h a t Sco t t an d t h e au t h o r o f

Waverley

a r e n am es f o r

the sam e th ing , then they mus t be in ter subs t i tu tab le in every con tex t . In

fac t i t o f ten happen s , in a g rowing , changing language, tha t a de scr ip t ive

phrase comes to be used as a p roper name - an iden t i fy ing tag - and the

descr ip t ive meaning i s los t o r ignored . Somet imes we use cer ta in dev ices

such as cap i ta l iza t ion and dropping the def in i te ar t ic le , to ind ica te the

ch an g e in u s e. T h e ev en in g st a r b eco m es E v en i n g S t a r , t h e m o r n i n g

s t a r b eco m es M o r n i n g S t a r , an d t h ey m ay co m e t o b e u sed a s n am es

for the same th ing . Singu lar descr ip t ions such as the l it t le corpora l , the

Pr i n ce o f D en m ar k , t h e s ag e o f C o n co r d , o r t h e g r ea t d i s s e n t e r ,

a r e a s w e k n o w o f t en u s ed a s a l te r n a t iv e p r o p e r n am es o f N ap o l eo n ,

H am l e t , T h o r eau an d O l i v e r Wen d e l l H o l m es . O n e m i g h t ev en d ev i s e a

cr i te r ion as to when a descr ip t ive phrase i s be ing used as a p roper name.

Su p p o s e t h r o u g h s o m e a s t r o n o m i ca l ca t ac l y s m , V enu s w as n o l o n g e r t h e

f i rs t s tar o f the even ing . I f we con t inued to ca l l i t a l te rnat ive ly Even ing

S t a r o r t h e ev en i n g s t a r t h en t hi s w o u l d b e a m eas u r e o f t h e co n v e r s i o n

of the descr ip t ive phras e in to a p ro per n ame. I f , however , we would

then r egard (10) as fa l se , th i s would ind ica te tha t the even ing s tar was no t

u s ed a s an a l t e rn a t i v e p r o p e r n am e o f V enu s . We m i g h t m en t i o n i n

p as s i n g t h a t a l t h o u g h t h e co n v e r s i o n o f d e s c r i pt i o n s i n t o p r o p e r n am es

ap p ea r s t o b e a s y m m et r i c , w e d o f i nd p r o p e r n am e s u s ed i n s i n g u l ar

descr ip t ions o f some th ing o ther th an the th ing name d, as in the s ta te-

me nt M ao Tse- tung i s the Sta l in o f Ch ina , where one in tends to asser t a

s imi lar i ty be tween the en t i t i es named .

T h a t an y l an g u ag e m u s t co u n t en an ce s o m e en t it ie s a s t h in g s w o u l d ap p ea r

to be a p recondi t ion fo r language. But th i s i s no t to say tha t exper ience

is g iven to us as a co l lec tion o f th ings , fo r i t would ap pea r tha t the re are

cu l tu ra l var ia t ions and accompanying l ingu is t ic var ia t ions as to what

sor t s o f en t i ti es ar e so s ingled ou t . I t would a l so ap pea r to be a p recondi -

t ion o f language th a t the s ing l ing ou t o f an en t i ty as a th ing i s acco m-

pan ie d by man y - and perh aps an indef ini te o r in f in i te num ber - o f

un ique descr ip t ions , fo r o therwise how would i t be s ing led ou t? But to

g ive a th ing a p roper name i s d i f f eren t f rom g iv ing a un ique descr ip t ion .

Fo r s u p p o s e w e t o o k an i n v en t o r y o f a ll t h e en t it i es c o u n t en an ced a s

th ings by some par t icu lar cu l tu re th rough i t s own language, wi th i t s

o w n s e t o f n am es an d eq u a t ab l e s i n g ul a r d e s c r i p ti o n s , an d s u p p o s e t h a t

309


num ber were f ini te ( th is assu mpt ion i s for the sake o f s impf i fying the

exposi t ion) . And suppose we randomized as many whole numbers as we

needed for a one- to-one cor respondence , and thereby tagged each thing.

This ident i fying tag i s a proper n ame of the thing. In taking ou r invento ry

we discovered tha t many of the ent i t ies countenanced as th ings by tha t

language-cul ture comp lex a l ready had proper names, a l thoug h in ma ny

cases a singu lar descr iption ma y hav e been used. This tag, a pro per n ame ,

has no meaning. I t s imply tags . I t i s not s t rongly equa table wi th a ny

of the s ingula r descr ipt ions o f the thing, a l thoug h s ingula r descr iptions

ma y be equa table ( in a weaker sense) wi th each other where

(16) Des cl eq Desc2

means tha t Desc l and Desc2 descr ibe the same thing. But here too,

wha t we a re asse r t ing would dep end on ou r choice of eq . The

pr inc iple of indiscernibi l i ty m ay be thou ght o f as equa t ing a proper nam e

of a th ing wi th the tota l i ty of i t s descript ions.

Perhaps I shou ld ment io n tha t I am no t unaw are of the awful s impl ic i ty

of the tagging procedure I descr ibed above . The assump t ion of f in i tude ;

and even i f th is were not assumed, then the assum pt ion o f denum erabi l i ty

~of the c lass o f th ings . Also, the assum pt ion tha t a l l th ings co untena nced

by the language-cul ture complex a re nam ed or descr ibed. But m y point i s

only to dis t inguish tagging f rom descr ibing, proper names f rom descrip-

t ions . You may descr ibe Venus as the evening s ta r and I may descr ibe

Venus as the morning s ta r , and we may both be surpr ised tha t as an

empir ical fact , the same thing is being descr ibed. But i t is no t an em pir ical

fac t tha t

(17) Venus I Venus

and i f a i s anoth er proper nam e for Venus

(18) Venus I a .

No r i s i t ext raordinary, tha t we o f ten conver t one o f the descript ions of a

thing into a proper name. Perhap s we ough t to be more consis tent in our

use of upper -case le t te r s , but th is i s a ques t ion of r e forming ordinary

language . I t ought not to be an insurmountable problem for logic ians .

Wh at I have been a rguing in the pas t severa l minutes i s, tha t to say of an

ident i ty (in the s t rongest sense of the word) tha t i t i s true , i t must be

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MOD LITIES ND INTENSION L L NGU GES

t au to log ical ly t rue or analy t i ca l ly t rue . The cont rovers ia l (8 ) o f QS4 no

more ban ishes concre te en t it i es f rom the un iverse than (12) ban ishes f rom

the un iverse red-b looded prop os i t ions .

Let us re turn now to (10) and (15). I f they express a t rue iden t i ty , then

'Scot t ' ought to be anywhere in tersubs t i tu tab le for ' the au thor o f Waver-

ley

and s imi lar ly for ' the m orn ing s tar ' and ' the evening s tar ' . I f they are

not so universal ly intersub st i tutable - that is, i f our decision is that they

are no t s imply proper names for the same th ing ; tha t they express an

equivalence which is possibly false, e.g. , someone else might have wri t ten

Waverley, the s tar f i r s t seen in the evening might hav e been d i f feren t f rom

the s tar f i r s t seen in the morn ing - then they are no t iden t i t i es . One

solut ion is Russel l 's , whose analysis provides a t ransla t ion of (10) and

(15) such that the t r u th of (10) and (15) does no t com mit us to the log ical

t ru th o f ( I0) and (15) , and cer ta in ly no t to t ak ing the 'eq ' o f (10) as

iden t i ty , except on the expl ic i t assumpt ion of an ex tens ionaliz ing ax iom.

Other and re la ted so lu t ions are in t erms of membersh ip in a non-e mpty

uni t c lass , o r appl icab i l i ty o f a un i t a t t r ibu te . But w hatever the choice

of a solut ion, i t wil l have to be one which permits intersubst i tutabi l i ty,

o r some analogue o f in tersubs t i tu tab i l i ty for the mem bers o f the pai rs :

'S co t t ' an d ' t h e au t h o r o f

Waverley',

and ' the evening s tar ' and ' the

morn ing s tar ' , w hich i s shor t o f being un iversa l. In a l anguage which i s

impl ic it ly s t rongly ex tens ional ; th a t i s where a l l con tex t s in which such

subs t i tu t ions fa i l a re s imply eschewed, then of course there i s no har m in

equat ing iden t i ty wi th weaker forms of equivalence . But why res t r i c t

ourselves in this way when, in a more intensional language, we can st i l l

mak e a l l the subs t i tu t ions permiss ib le to th is weake r form o f equivalence ,

yet admi t contex t s in which such a subs t i tu t iv i ty i s no t permi t t ed . To

show th i s , I would l ike to tu rn to the ins tances of (4) which are prov able 11

in QS4. I wi l l again conf ine my remarks , fo r the purpose of expos i t ion ,

to 4 , a l though i t i s the genera l i za t ions for QS4 which are ac tual ly

proved . An unres t r i c ted

(19) x ~ y---~z---- w

is clearly not pr ova ble w heth er '---~' is tak en as mater ial imp licat ion,

s trict implicat ion or a metal inguist ic condit ional . I t would involve us in a

cont rad ic t ion , i f our in terp re ted sys tem al lowed s ta tements such as

311


(20) (x ~ y) ,- , [] (x ~ y)

as i t must i f i t is not to reduc e i tself to the system of mate rial implicat ion.

Indeed , the unde r ly ing assumpt ion abou t equivalence which i s impl ici t in

the whole 'even ing s tar morn ing s tar ' con t roversy is tha t there are

equivalences (misleadingly cal led ' t rue ident i t ies ') which are cont ingent ly

true. Let x a nd y o f (19) be tak en as so me p and q which sat isfies (20).

Let z be Q p=__p) and w be D( P ~ q) . Then (19) is

(21) p-=- q) --- ~ • p =- p) ~ D p == q)) .

Fro m (20) , s impli f ica t ion , mod us ponens and D (P ~ P) , which is a

theore m of $4 , we can deduce [ ] (p ~ q) . The l a t t er and s impl i f ica t ion of

(20) and conjunct ion l eads to the cont rad ic t ion

(22) [ ] (p ~ q) . ~ [ ] (p ~ q) .

A res t ri c ted form of ( t9 ) i s p rovable . I t i s p rovable i f z does no t conta in

any modal opera tors . And th i s i s exact ly every contex t a l lowed in Sin,

without a t the same t ime banish ing modal contex t s . Indeed a s l igh t ly

strong er (19) is prova ble. I t is prova ble i f x do es not fall within the scope

of a modal oper a tor in z .

Where in (4), eql and eq~ are both taken as s t r ict equivalence, the

subs t i tu t ion theorem

23) x ~ y) -+ z ~= w)

is provable without restrict ion, and also where eql is taken as s t r ict

equivalence and eq~ is taken as material equivalence as in

(24) (x _--5 y) -+ (z :~r w ).

But (23) is also an extensional izing principle, for i t fai ls in epistemic

contex t s such as contex t s involv ing 'knows that ' o r 'be l i eves that ' . For

cons ider the s ta tement

25)

and

(26)

When Professor Quine rev iewed the paper on iden t i ty

in QS4, he knew that I-

aIr~b ~ aIrab .

When Professor Quine rev iewed the paper on iden t i ty

in QS4 he knew that k aIb ~ aln~b.

312




Although (25) is true, (26) is false for (7) holds in QS4. But rather than

repeat the old mistakes by abandoning epistemic contexts to the shelf

labelled referential opacity after having rescued modal contexts as the

most intensional permissible contexts to which such a language is

appropriate, we need only conclude that (23) confines us to limits of

applicability of such modal systems. If it should turn out that statements

involving knows that and believes that permit of formal analysis, then

such an analysis would have to be embedded in a language with a st ronger

equivalence rela tion than strict equivalence. Carnap s intensional isomor-

phism, Lewis analytical comparability, and perhaps Anderson and Belnap s

mutual entailment are attempts in that direction. But they too would be

short ofidenti ty, for there are surely contexts in which substitutions allow-

ed by such stronger equivalences, would convert a t ruth into a falsehood.

It is my opinion 12 that the identity relation need not be introduced for

anything other than the entities we countenance as things such as individ-

uals. Increasingly strong substitution theorems give the force of universal

substitutivity without explicit axioms of extensionality. We can talk of

equivalence between propositions, classes, attributes, without thereby

conferring on them thinghood by equating such equivalences with the

identity relation. QS4 has no explicit extensionality axiom. Instead we

have (23), the restricted (19), and their analogues for att ributes (classes).

The discussion of identity and substitution in QS4 would be incomplete

without touching on the other familiar example:

(27) 9 eq the number of planets

is said to be a t rue identity for whcih substitution fails in

(28) [] (9 > 7)

for it leads to the falsehood

(29) [] (the number of planets > 7).

Since the argument holds (27) to be contingent (~ [] (9 eq the number of

planets)), eq of (27) is the appropriate analogue of material equivalence

and consequently the step from (28) to (29) is not valid for the reason that

the substitut ion would have to be made in the scope of the square. It was

shown above that (19) is not an unrestricted theorem in QS4.

On the other hand, since in QS4

313

R U T H B R C N M R C U S

(30) (5 -k 4) = s 9

where =~ i s the appropr ia te ana logue for a t t r ibutes (classes) of s t r ic t

equivalence, 5 -k 4 m ay replace 9 in (28) in accord ance with (23). I f ,

however , the square were dropped f rom (28) as i t va lid ly can for

(30a) [] p -3 p

is provab le, the n by the restr icted (19), the very same sub sti tut ion available

to m is available here.

THE INTERP RET TION OF QU N TIF IC TION

The second prom inent a rea o f c ri t ici sm of quant i f ied modal logic involves

inte rpre ta t ion o f the opera t ions o f quant i f ica t ion when combined w i th

modal i t ies. I t appears to m e tha t a t l eas t some of the problems s tem f rom

an absence of an ad equa te , un equivoca l , col loquia l t r ans la t ion o f the

opera t ions of quant i f ica t ion. I t is of ten not quant i f ica t ion but our choice

of reading and impl ic i t in te rpre t ive consequences of such a reading

which leads to diff icult ies. Such diff icult ies are not confined to modal

sys tems. The most common reading of exis tent ia l quant i f ica t ion i s

(31) There i s ( exist s ) a t l eas t one ( some) th ing (person) which (w ho ) . . .

Strawso n, 13 for e xample, does no t even a dm it o f signif icant al ternatives,

f o r he sa ys o f (31 ): . . .w e migh t t h ink i t s t ra nge t ha t t he w ho le o f

modern formal logic a f te r i t l eaves the proposi t iona l logic and before i t

c rosses the boundary in to the ana lys is of mathemat ica l concepts , should

be con f ined to the e labora t ion of se t s of ru les g iving the logica l in te r re la -

t ions of formulae which, however complex, begin wi th these few ra ther

s t ra ined and aw kward phrases . Indeed, taking (31) a t f ace va lue , S t raw-

son ge ts in to a m udd le abou t tense ((31) i s in the present tense) , and the

ambigui t ies of the word exis t . Wh at we would l ike to have and d o no t

have , i s a d i rec t, unequivoca l co l loquia l r eading of

(32) (q x) ~0x

whieh gives us the force o f e i ther of the fol lowing:

(33) Som e subs ti tut ion instance of ~0x is t rue

o r

There is at least one value of x for which ~0x is t rue.

314




I am not suggesting that (33) provides translat ions of (32) , but only

that what is wanted is a translat ion with the force of (32) .

As seen from (33) , quantif icat ion has pr imari ly to do with truth and

falsi ty, and op en sentences. R eading in accordance with (31) ma y entangle

us unnecessar i ly in ontological perplexit ies. F or if quantif icat ion has to

do w ith things an d if var iables for at tr ibutes or classes can be quantif ied

upon , then in acco rdance with (31) they are things. I f we st il l want to

dist inguish the identifying from the classifying function of language, then

we are involved in a classification of different kinds of things and the

accompanying platonic involvements. The solution is not to banish

quantif icat ion on var iables other than individual var iables, but only not

to be taken in by (31) . We do in fact have some colloquial counterpar ts

of (33) . The non-te mpo ral sometimes or in some cases o r in at least

one case , which have greater ontological neutral i ty th an (31) .

Some of the arguments involving modali t ies and quantificat ion are closely

connected with quest ions of substi tut ion and identi ty. At the r isk of

bor edo m I wil l go through one again. In QS4 the fol lowing defini t ions are

introduced: 14

(34 ) (~0 = m ~ ) ---=aS x) ~ox =~ g/x)

35) ~0 = , ~) = a s D ~ = m ~)

Since the equality in (10) is contingent, (10) may be written as

(36) ( the evening star =m the morning star) .

I t is a lso the case that

(37) O ~ ( the evening star =m the morning star) .

One way o f writ ing (11) is as

(38) [ ] ( the evening star =m the evening star) .

By existential generalization on (38), it follows that

(39) (3 ~0) [] (cp -----m he eve nin g st ar ).

In the words of (31) , (39) becomes

(40) There is a thing such that it is necess ary that it is equal to

the evening star .

315


The s tubborn unla id ghost r i ses aga in . Which th ing, the evening s ta r

which by (36) is equa l to the m orning s ta r? But such a su bs t i tu t ion would

lead to the fa l sehood

(41) [ ] ( the evening s ta r = m the morn ing s ta r ) .

The a rgum ent m ay be repea ted for (27) throu gh (29).

In QS4 the solution is clear . For since (37) holds, and since in (39) (0

occurs within the scope of a square, then we cannot go from (39) to (41) .

On the other hand the step from (38) to (39) (existential instantiat ion) is

entirely valid. Fo r surely there is a value of ~o for w hich

[] (~0 = the ev ening star)

is t rue. In par t icular , the case where ~0 is replaced by the evening

star .

There is also the specif ic problem of interpret ing quantif icat ion in (6) ,

which i s a pos tula te of QS4. Read in accordance wi th (31) as

(42) I f i t is logically possible that there is someth ing wh ich ~0 s,

then there i s somethin g such th a t i t i s logica l ly poss ible

tha t i t ~0 s,

i t is admi t tedly odd. Th e antecede nt seems to be ab out w hat i s logica l ly

poss ible and the consequent about what there i s . How can one go f rom

poss ibi l ity to exis tence? Read in accordance w i th (33) we have the c lumsy

bu t no t so pa ra dox i c a l

(43) I f i t is logically possible that q~x for som e value of x, then

there is some v alue of x su ch tha t i t is logically possible

tha t ~0x.

Al though the emphasis has now been shi f ted f rom things to s ta tements ,

and the onto logica l consequences o f (42) are absent , i t is st il l indirect and

awkw ard. I t would ap pear th a t ques t ions such as the acceptabi l i ty or non-

acceptabi l ity of (6) a re bes t solved in te rms of som e seman t ica l cons t ruc-

t ion. This wi l l be re turned to in conc lus ion, but f i rs t some min or mat te r s .

A defense of modal logic would be incomple te wi thout touching on

cr it ic isms of modal i t ies which s tem f rom confus ion about w hat is or i sn t

provable in such sys tems. One example i s tha t of Ro senb loom 15 who

se ized on the fac t tha t a s t rong deduc t ion theorem is not ava i lable in

316




QS4, as a r eason for d iscarding s tr ict impl ica t ion as in any w ay re levant

to the d educ ibi l ity re la t ion. He fa i led to note z6 tha t a w eaker an d

perhaps more appropr ia te deduc t ion theorem is ava i lable . Indeed,

And erson and Belnap, 17 in the i r a t temp t to formal ize enta i lment wi th out

modal i t ies , r e jec t the s t rong form o f the deduc t ion theorem as counte r -

in tui t ive for enta i lment .

Ano ther example occurs in

Word and Object

18 which can be sum marize d

as fol lows:

(44) Mod ali t ies yield talk of a difference between necessary and

c on t i nge n t a t t r i bu t e s .

(45) Mathem at ic ians ma y be sa id to be necessar ily ra t iona l and

no t necessar i ly two-legged.

(46) Cyclists are necessar i ly two-legged and no t necessar i ly rat ion al .

(47) a is a ma them aticia n and a cyclis t .

(48) Is this concrete individ ual necessar i ly rat ion al or contin-

gently two-legged or vice versa?

(49) Talk ing referential ly oT the object with no special bias

tow a r d a ba c kgr ound g r oup ing o f ma the ma t ic i a ns as

aga ins t cyc l i s t s . . , the re i s no semblance of sense in ra t ing

some o f h is a t t r ibutes as necessary and others as cont ingent .

Professo r Quine says tha t (44) thro ugh (47) are supposed to evok e the

appropr ia te sense of bewi lderment and they sure ly do. For I kno w of no

inte rpre ted mod al sys tem which countenances necessary a tt r ibutes in the

manner suggested. Translat ing (45) through (47) we have

( 50) ( x ) ( gx

-3 Rx ) ~ (x) [] (M x = Rx ) ~ (x) ,,~ 0 (M x ,,~ Rx )

which is conjoine d in (45) with

(51) (x) ,~ [] (M x = Tx) ~- (x) 0 ~ (M x = Tx) =- (x) 0 (Mx N Tx) .

Also

(52) (x )( Cx -3 Tx) ~ (x) [] (C x = Tx ) ~ (x) ,~ ~ ( Cx ,,~ Tx)

which is conjoine d in (46) with

317

RUTH B RC N M RCUS

53) x) ~ [] Cx = Rx)_~ x) 0 ~ Cx = Rx) ~ x ) 0 Cx. ~ Rx) .

An d in (48)

(54) M a . C a

Am on g the conc lus ions we can draw f ro m (49) throu gh (53) a re

[] (M a = Ra) , ~ ~ (M a ,,~ Ra) , 0 (M a. ~ Ta) , ~ [] (M a = Ta) ,

[] (Ca = ra , ~ 0 (Ca ~

ra), 0 (Ca ~ Ra), ~ []

(Ca ~ Ra) ,

Ta , Ra , Ta • Ra

But nothing to answer ques t ion (48) , or to make any sense of (49) . I t

would appear tha t Professor Quine i s assuming

(55) (p --3 q) -3 (p -3 [] q)

i s provable in QS4, bu t i t is not , except where p ~ [ ] r for som e r .

Keeping in min d tha t we a re dea l ing wi th logica l modal i ties , non e of the

a t t r ibutes (M, R, T , C) in (50) throu gh (54) taken separa tely , or con-

joined, a re necessary. I t i s not tha t sor t of a t tr ibute which mod al logic,

even der iva t ive ly , countenances as be ing necessary. A word i s appro-

pr ia te here abo ut the der iva tive sense in w hich we can speak of logica lly

necessary and c ont ingen t a t t r ibutes .

In QS4 abs t rac ts a re in t roduced such tha t to every func t ion there

cor responds an abs t rac t , e .g .

(56)

x @ A

= as B, where B i s the resul t of subs t i tu ting every f ree

oc c ure nc e o f y i n A by x .

I f r is some abs t rac t then we can def ine

(57) xe [] r =a S []

(xer ), I- [] r

=aS (x) (xe [ ] r )

a n d

(58)

xe ~) r

----=aS

O( xe r) , t- ~ r - --as (x) (xe <3, r)

I t is c lear tha t am on g the abstracts to whi ch ~ [] ma y v alidly be attLxed,

wi l l be those cor responding to tautologica l func t ions , e .g . ,

p ( y I y ) ,

p(rVxv ~ ~0x) , e tc . I t wou ld be app ropria te to cal l these necessary

a t t r ibutes , and the sym bol @ is a der iva tive way of applying modal i t ies

to at tr ibutes.

318




Simi lar ly , a ll o f the a t t r ibu tes o f (50) th rou gh (54) cou ld in the sense of

(58) be cal l ed cont ingent , where ~ i s the der ivat ive modal i ty fo r con t in -

gency of a t t r ibu tes . How ever , i f (50) is t rue , then the a t t r ibu te o f being

ei ther a mathemat ic ian or no t ra t ional cou ld appropr ia te ly be ca l l ed

necessary , fo r

(59) (x) [] (xe29(MxV ~ Rx )) .

SEM NTIC CONSTRUCTION S

I wo uld l ike in conclus ion to suggest tha t the po lemics o f mod al log ic are

perhaps bes t car r i ed ou t in t erms of some expl ic i t senaan t ica l cons t ruct ion .

As we have seen in connect ion wi th (6) i t i s awkw ard a t bes t and a t wo rs t

has the charac ter o f a qu ibb le , no t to do so .

Let us reapprai se (6 ) in t erms of such a cons t ruct ion . 19 Cons ider fo r

example a l anguage (L) , wi th t ru th funct ional connect ives , a modal

ope rat or (~>), a f ini te num be r o f individu al constan ts , an infini te

num ber of ind iv idual var iab les , one two-p lace pred icate (R) , quan-

t i f ica t ion and the u sual cr i t er ia fo r being wel l - fo rmed. A d oma in

(D) of ind iv idual s i s then cons idered which are nam ed by the con-

s tan ts o f L . A mode l o f L i s def ined as a c lass o f o rdere d couples

(poss ib ly empty) o f D, The m embers o f a model are exact ly those

pai rs be tween which R ho lds . To say therefore tha t the a tomic

sen tence

R(ala2)

of L ho lds o r i s t rue in M , i s to say that the ordere d

couple (b l , b2) i s a m emb er of M, where a l and a2 are the names

in L of b l and b~. I f a sen tence A of L i s o f the form ~

B, A

is t rue

in M i f and o n ly i f B i s no t t rue in M . I f A i s o f the fo rm B1 • Bz then A

is t rue in Mi f and on ly if bo th B1 and B2 are t rue in M. I r A i s o f the for m

(3x) B, then A i s t rue in M i f and on ly i f a t l eas t one subs t i tu t ion ins tance

of ~ is t rue (holds) in M. If A is ~ B then A is t rue in M i f and only i f B

i s t rue in some m odel M1.

We see tha t a t rue sen tence of L i s def ined re la t ive to a model an d a

doma in of ind iv iduals . A log ical ly t rue sen tence i s one which would be

t rue in every model . W e are now in a pos i t ion to g ive a rough pro of o f (6 ) .

Suppose (6) i s fa l se in some M. Then

~ 0 3x) ~ox- ~ 3x) 0 ~ x)

319

R U T H B R C N M R C U S

i s fa lse in M. The refor e

<> <> Ox ) vx . ~ 3x) <> cx )

is t rue in M. So

<>Ox)~x • ~ Ox) O ~x

i s t rue in some M1. There fore

60)

an d

61)

O O x ~ox

~ 3x) <) ~x

are t rue in M1. Conse quent ly , f rom 60)

62) 3x ) ~0x

is t rue in some m odel Mz There fore there i s a member of D b) such that

63)

is t rue in M2. But fro m 61)

~b

Ox) ~px

i s no t t rue in M1. Consequen t ly there is no m embe r of D such that

64) 0 ~b

is t rue in M1. So the re is no mod el M~ such that pb is t rue in M~. But t iffs

resu l t cont rad ic t s 63) . Consequent ly , in such a cons t ruct ion , 6 ) mus t be

t rue in e very model .

I f th i s i s the sor t o f cons t ruct ion one has in mind then we are persuaded

of the p laus ib i li ty of 6 ) . Indeed , go ing back to 43), it can be seen that

t if fs was the sor t o f cons t ruct ion which was being assumed. I f 6 ) is to be

regarded as of fens ive in a way o ther and here I am b orrowin g an image

from Professor Whi te) than the manner in which we regard ea t ing peas

wi th a fo rk as of fens ive , i t mus t be in t erms of some sem ant ic cons t ruct ion

which oug ht to b e m ade expl ici t. ~0

We see , tha t though the roug h out l ine above correspon ds to the

Leibniz ian d i s t inc t ion between t rue in a poss ib le wor ld and t rue in a l l

possible worlds, i t is also to b e not ed that there a re n o specifical ly inten-

320



M O D A L I T IE S A N D I N T E N S I O N A L L A N G U A G E S

siona l ob j ec t s . No new en t i t y i s spaw ned in a poss ib l e wo r ld t ha t i sn ' t

a l r eady in t he dom a in in t e rms o f which the c l a ss o f m ode l s i s de f ined .

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

no t wi th t h ings . O n th i s i n t e rp re t a t i on , 21 Profe s sor Qu ine ' s ' f l i gh t f rom

in tens ion ' ma y have b een exh i l a ra t ing , bu t unnecessa ry . 22'

Departme nt o f Philosophy, Roosev elt U niversity, Chicago, Illinois, US A

N O T E S

1. W. V. Quine, Word and Object, 1960, pp. 195-196.

2. a . F .B. Fi tch, Symbolic Logic, New York, 1952.

b. R. Carnap, 'Modal i t ies and

quantification, Journal of Symbolic Logic,

Vol. XI (1946), pp. 3 3-64.

c . R. C. Barcan (Marcus) , ' A funct ional ca lculus of f irst order based on

strict implication, ' Journal of Symbolic Logic, Vol. X I (1946), pp. 1-16.

d. R. C. Barca n (Marcus), 'T he identity of individuals in a strict function al

calculus of first orde r, ' Journal of Symbolic Logic, Vol. XII (1947), pp. 12-15.

3. P. F. Strawson, Introductionto Logical Theory, Lon don , 1952, p. 216.

4. F. PI Ram sey, The Foundations of Mathematics, L o n d o n a n d N e w Y o r k ,

i931, pp. 30-32.

5. Op. cit,

notes 2c , 2d .

6. C. I . Lewis and C. H. Langford, Symbolic Logic, New York, 1932.

7. See A. N. Prior, Time and Modality, Oxfo rd, 1932, for an extended discus-

sion o f this axiom.

8. 5 results fro m addin g to 4.

p---3 E3 (>p.

9. A. N. Pr ior , 'Mo dal i ty and Quant i f ica t ion in 5, ' Journal of Symbolic Logic

March, 1956.

10. W. V. Quine, From a LogicalPoint of View, Cambridge, 1953,pp. 152-154.

11. Op. cir.,

note 2c. Theorem XIX* corresponds to (23). The restricted (19),

given the conditions of the restriction, although not actually proved, is

c learly provable in the same manner as XIX*.

12. See R. Barcan Marcus, 'Extensionality, '

Mind,

Vol. LXIX, n.s. , pp. 55-62

which overlaps to som e extent the present paper.

13. Op. cit., note 3.

14. Op. cit., note 2c. Abstracts are introduced and attributes (classes) may be

equated wi th abst rac ts . Among the obvious fea tures of such a ca lculus of

attributes (classes), is the presence of equivalent, non-identical, em pty

attrib utes (classes). If the null attrib ute (class) is defined in terms of identity,

then i t wi l be intersubst i tut ible wi th any abst rac t o n a con tradic tory funct ion.

321

RUTH BARCAN MARCUS

15.

P. Rosenbloom The Elements of Mathematical Logic New York

1950,p. 60.

16. R. Barcan Marcus, 'Str ict implication, deducibility, and the deduct ion

theorem,'

JournalofSymbolic Logic

Vol. XV III (1953), pp. 234-236.

17. A. R. Anderson and N. D. Belnap,

The Pure Calculus of Entailment

(pre-

print).

18.

Op. cit.

note 1, pp, 199-200.

19. The construction here outlined corresponds to that of R. Carnap,

Meaning

and Necessity

Chicago, 1947. The statement of the construction is in

accordance with the method of J. C. C. McKinsey. See also, J. C. C. Mc-

Kinsey, 'On the syntactical construction o f systems of modal logic,'

Journal

of Symbolic Logic

Vol. X (1946), pp. 88-94; 'A new definition of truth,'

Synthese

Vol. VII (1948/49), pp. 428-433.

20. A criticism of the construction here outlined is the assumption of the

countability of members of D. McKinsey points this out in the one chapter

I have seen (Chapter I, Vol. II), of a projected (unpuNished) two volume

study of modal logic, and indicates that his construction will not assume the

countability of members of D. Whereas Carnap's construction leads to a

system at least as strong as $5, McKinsey's (he claims) will be at least as

strong as $4 (without (6) I would assume). I've not' seen, nor been able to

locate any other parts of this study in which the details are worked out

out along with completeness proofs for some of the Lewis systems. See also,

J. Myhi l,

Logique et analyse

1958, pp. 74-83; and S. Kripke,

Journal of

Symbolic Logic

Vol. XXIV (1959), pp. 323-324 (abstract).

22. If one wishes to talk about possible things then of course such a construc-

tion is inadequate.

22. This paper was written while the author was under N.S.F. Grant 24335.

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barcan marcus modalities and intensional languages

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