Steps Toward a Constructive Nominalism - Nelson Goodman and W. V. Quine

8/10/2019 Steps Toward a Constructive Nominalism - Nelson Goodman and W. V. Quine

http://slidepdf.com/reader/full/steps-toward-a-constructive-nominalism-nelson-goodman-and-w-v-quine 1/18

The J oí'un a l or Stmbouc Loaic V o lu m e 1 2 , N u m b o r 4 , D e c e m b e r 1W 7

STEPS TOWARD A CONSTRUCTIVE NOMINALISM

N E L S O N G O OD MA N a n d W . V . Q U IN E

1. Renunciation of abstract entities. W e d o n o t be lie v e i n a b s t r a c t e n t it ie s .

N o o n e s u p po s e s t h a t a bs t r a c t . e n t i t ie s — c la s s e s , r e l a t io n s , p r o p e r t i e s , e t c . —

e x i s t i n s pa c e- t im e ; b u t w e m e a n m o r e t h a n t hi s . W e r e no un c e t h e m a l to g e t he r .

W e s ha l l n o t f o r e g o a l l us e o f pr e d ic a t e s a n d o t h e r w o r ds t h a t a r e o f t e n ta k e n

t o ña m e a b s t r a c t o b je c t s . W e m a y s t i ll w r i te 'x i s a d o g , ’ o r ‘x i s b e t w e e n y

a n d z ' \ f o r h e r e ‘is a d o g ’ a n d ‘is b e t w e e n . . . a n d ’ c a n b e c o n s t r u e d a s s y n c a te -

g o r e m a t ic : s i g n if ic a n t i n c o n t e x t b u t n a m i n g n o t h in g . B u t w e c a n n o t us e v a r i

a ble s t h a t c a li f o r a b s t r a ct ob je c ts a s v a lúe s . 1 I n ‘x is a d o g , ’ o n l y c o nc r e t e o b

je c ts a r e a p p r o p r ia t e v a lúe s o f t h e v a r ia b le . I n c o n ír a s t , t h e v a r ia b l e in ‘ x is

a z o o l o g i c a l s pe c ie s ’ c a lis f o r a b s t r a c t o b j e c t s a s v a l ú e s ( un le s s , o f c o ur s e , w e c a n

s o m e h o w i d e n t i f y t h e v a r i o u s z o o lo g i c a l s pe c ie s w i t h c e r t a in c o nc r e t e o bj e c t s ).

A n y s y s te m t h a t c o unt e n a n c e s a b s t r a c t e n t it ie s w e d e e m u n s a t is f a c t o r y a s a f in a l

p h i l o s o p h y .

R e n u n c i a t i o n o f a b s t r a c t o b je c t s m a y l e av e u s w i t h a w o r l d c o m po s e d o f phy s -

i ca l o b je c t s o r e v e n t s , o r o f u n i t s o f s e ns e e x p e r ie n ce , d e p e n d i n g u p o n d e c is i on s

t h a t n e e d n o t b e m a d e he r e. M o r e o v e r , e v e n w h e n a b r a n d o f e m p i r ic i s m is

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

e v e n t s , 2 t h e p h i l o s o p h y o f m a t h e m a t i c s s t i l l f a c e s e s s e n t i a l l y t h e s a m e p r o b l e m

t h a t i t do e s w h e n a l l un iv e r s a l s a r e r e p u d ia t e d . M e r e s e n s o r y q u a l it ie s a f fo r d

n o a d e q u a t e ba s is f o r t h e u n l i m i t e d u niv e r s e o f n u m b e r s , f u n c t io n s , a n d o t he r

c l a s s e s c l a i m e d a s v a l ú e s o f t h e v a r i a b l e s o f c l a s s i c a l m a t h e m a t i c s .

W h y d o w e r e f us e t o a d m i t t h e a b s t r a c t o b je c ts t h a t m a t h e m a t ic s ne e ds ?

F u n d a m e n t a l l y t h i s r e fu s a l is b a s e d o n a p h il o s o p hic a l i n t u i t i o n t h a t c a n n o t b e

jus t i f ie d b y a p p e a l t o a n y t h in g m o r e u l t ím a t e . I t is f o r t i f ie d , ho w e v e r , b y c e r

t a i n a postcriori c o n s i de r a t i o ns . W h a t s e e m s t o b e t h e m o s t n a t u r a l p r i n c ip ie

f o r a b s t r a c t in g c la s se s o r p r o p e r t ie s le a d s t o p a r a d o x e s . E s c a p e f r o m t he s e

pa r a d o x e s c a n a p p a r e n t l y b e e ff e c te d o n l y b y r e c our s e t o a l t e r n a t i v e r u le s w h os ea r t if i c ia J i t y a n d a r b i tr a r i ne s s a r o us e s u s p ic i o n t h a t w e a r e lo s t in a w o r l d o f m ak e -

b e l i e ve . 3

Received Sept. 2, 1947.1 T hat i t is in the valúes of the variables, and not in the supposed designata of constant

terms , tha t the ontolog y of a theory is to be soug ht, has been urged by W . V . Quine in A 'otes

on ezistence and necessity, The journal of philosophy, vol. 4 0 ( 1 9 4 3 ) , pp. 1 1 3 - 1 2 7 ; also in Designation and existence, ibid., vol. 3 6 ( 1 9 3 9 ) , pp. 7 0 1 - 7 0 9 .

1 A s for ex ample in Ne ls on G oodm an's A study o f q ualities ( 1 9 4 1 , typescript, HarvardUnivers ity L ibrar y ). Q ualitative (“ abstract” ) particles of experience and spatio-teraporally bounded ("concrete ” ) particles are there regarded as e qually acceptable basicelement.8 for a sy stem. Dev ices described in the present paper w ill probably niake it pos*sible so to revise tha t st udy th a t no constr uction will depend upon the existence of classes.

aThe simple principie of class abstraction, which leads to Itussell's paradox and others,is this : G iv en an y for mula containing the v ariable ‘z 1, there is a class whose memhers areall and only the objects x for w hich tha t formula holds. See W. V . Quine , Mathem atical

logic, p p . 128-130. Fo r a brief survey of sys tama designed to exelude the paradox es, seep p . 163-166, o p . ci t . ; a lso Elemenl and number, this J o u r n a l , vol. 6 ( 1 9 4 1 ) , pp. 1 3 5 - 1 4 9 .

105



2. Renunciation of infinity. W e decline to assume tha t there are infmitelymany objects. N o t only is our own experience finite, but there is no general

agreement among physicists that there are more than finitely many physical

objects in a ll space- time.4 I f in fa ct the concrete vvorld is finite, acceptance ofany theory tha t presupposes infinity w ould require us to assume tha t in addition

to the concrete objects, finite in number, there are also abstract entities.Classical ar ithme tic presupposes an infinite r ealm of numbers. Henee if, in

an effort to reconcile arithmetic with our renunciation of abstract entities, we

were to unde rta ke to id e ntify numbe rs ar bitr ar ily w it h certa in thing s in theconcrete world, we should thereby drastically curtail classical arithmetic; for,

we cann ot as sume there are infm ite ly m any such thing s.Classical syntax, like classical arithmetic, presupposes an infinite realm of

objects; for it assumes that the expressions it treats of admit concatenaron toform longer expressions w ithout end. B ut if ex pressions mus t, l ike ev ery thing

else, be found w ithin the concrete w or ld, then a limitles s r ealm of ex pressionscannot be assumed. Indee d, ex pressions construed in the customar y w ay as

abstract typographical shapes do not exist at all in the concrete world; the lan-

guage elementó in the concrete w orld are ra ther inscriptions or mar ks , the shapedobjects r ather th an the shapes.5

The stock of available inscriptions can be vastly increased if we include, not

only those w hich have colors or sounds contrasting w ith the surr oundings , but

all appro priately shaped spatio- temporal regions even though they be indis-ting uishable from the ir surrounding s in color, sound, tex ture, etc. B ut the number and leng th of inscriptions w ill s till be limited insofar as the spatio- temporal

w or ld itself is lim ite d. Cons equently we cann ot say tha t in g eneral, g iv en any

two inscriptions, there is an inscription long enough to be the concatenation of

the two.Furthermore, there can be at most only as many inscriptions as concrete

objects. Henee, if concrete objects are finite in number , there are bound to be

some for w hich there are no ñame s or descr iptions whatever. Otherw ise every

concrete object w ould have to be the ñame or description of a unique a nd distinctconcrete objec t; and we should thus be deprive d of a ll predicates and connectives,

to say nothing of sy nonyms, duplícate inscr iptions, and non- inscriptions.3. T he nom inalist’s problems . B y renouncing abstract entities , we of course

exelude all predicates which are not predicates of concrete individuáis or ex-

plained in terms of predicates of concrete individuáis . Moreove r, we reject anystatement or definition—even one that explains some predicates of concrete

indiv iduáis in terms of others— if it com mits us to abstrac t entities. For ex-ample, until we find some way of construing ‘is an ancestor of’ in terms of ‘is a

4 A c c o r d in g t o q u a n t u m p hy s i c s , e a c h p h y s i c a l o b j e c t c o n s is t s o f a f i n it e n u m b e r o f

s p a t io - t e m p or a l ly s c a t t e r e d q u a n t a o f a c t i o n . F o r t h e r e to be i n f i ni te l y m a n y p hy s i c a l

o b j e c t s , t h e n , t h e w o r l d w o u l d h a v e t o h a v e i n f i n i t e e x t e n t a l o n g a t l eas t. o n e o f i t s s p a ti o-

t e m p o r a l d i m e n s i o n s . W h e t h e r i t ha s i s a q u e s t i o n u p o n w h ic h t he c u r r e n t s p e c u l a t io n

o f p h y s i c i s t s s e em s t o be d i v i d e d .6 A n o m i n a l is t i c s y n t a x la n g u a g e m a y , o f c o ur s e , s t i l l c o n t a i n s h a pe - p r ed ic a te s , e n a b l i n g

u s t o s a y t h a t a g i v e n i n s c r i p t i o n i s , f o r e x a m p le , d o t- s h a pe d , do tt e d- l in e - s h ap e d, O dy s s e y -

sha ped . See §5 an d §10.



parent of’ other than the way the ancestral of a relation is usually defined insy steins of logic,6 the re lationship between these predica tes remains for us un-explained.

W e shall, then, face problems of reducing pre dicates of abs tr ac t entities topredicates of concrete individuáis, and also problems of constructing certainpredicates of concrete indiv iduáis in terms e ither o f cer tain others or of any otherstha t sa tisfy some more or less well- defined criter ia. A part from those predicatesof concrete objects vvhich are permitted by the terms of the given problem toappear in the definiens, nothing m ay be used but indiv idual- variables, quanti-fication w ith respect to such variables , and truth- functions. Dev ices like re-cursive definition and the notion of ancestral must be excluded until they them-selves have been satisfactorily explained.

W e are not as nominalists concerned w ith the motive s behind the demand thata g ive n predícate o f concrete indiv iduáis be defined in terms of ce rtain other suchpredicates. N atur ally the demand may of ten arise from a feeling that the latterpredicates are in some sense the clearer, and we may as persons often share thisfeeling; b ut pure ly as nom inalists we k now no differences of clar ity among predi-cates of concrete indiv iduáis.7 Our problem is solely to prov ide, where definítions are called for, definí tions tha t are free of any terms or devices t ha t aretainted by belief in the abstract. We shall natur ally first try to find definitions

where, for var ied reasons, we feel they are mos t urg ently needed; a nd we shall not

waste time look ing for definitions in terms of pre dicates tha t we suppose to beambiguous or self- contradictory. B ut, as has perhaps been illustrate d by thecase of ‘ancestor’ and ‘pare nt,’ it cannot be said th a t the ex planation of one predícate in terms of another is of interest only if the latter is regarded as clearer.Indeed, if we have o nly a pseudo- ex planation (invo lving abstr act entities) relat-ing predicates of indiv iduáis, the problem of replacing it by a g enuine construc-tion has as immediate interest as the problem of defining a given predícate interms of others w hich come up to a certain standar d of clar ity, or the problem ofexplaining a predícate of abstract entities.

4. Some nominalistic reductions. Some statements tha t seem to be aboutabstr act entities can be rephrased in w ell- known w ays as stateme nts about concrete objects. T hus, where ‘A 9and ‘B 9are thoug ht of as fixed terms and not asbindable variables, the statement:

Class A is included in Class B

may be rephrased as:

Ev ery thing that is an A is a B .

6 T he u s u a l de f in i t io n , w hich wa 8 f i rs t s et fo r t h b y Freg e in 187 9 ( B eg rif fs ch rif t , p. 60),h a s b e co m e w e l l - k n ow n t hr o u g h W h i te h e a d a n d R u s s e l l a n d o t h e r w r i t e r s . I t i s p r e s e nt e d

o n ce m o re in t he n ext s ect io n .

7 I t m i g h t b e s u pp os e d t h a t t h e n o m i n a l is t m u s t r e g a r d a s u n c le a r a n y p r e d ic a t e o f

i n d i v i d u ái s f o r w h i c h t he r e i s no e x p l a n a ti o n t h a t d o e s n o t i n v o l v e c o m m i t m e n t t o a b s t r a c t

e n t i ti e s . B u t un le s s “ e x p l a n a t io n ” a s he r e i n te n d e d de p e nd s u p o n s t a n d a r d s o f c l a r i t y ,

w hic h do n o t conc e r n the n o m in a lis t as n o m in a l is t , a s uita ble e x pla n a t io n c a n a lw a y s

be s u pp l ie d t r i v i a l l y b y e q u a t i n g t he p r e d ic a t e i n q u e s t i o n w i t h a n y a r b i t r a r i l y c o nc o c te d

s in g le wo rd.



The phrases ‘is an A ’ and ‘is a B ’ here are predicates of concrete objects, and

are regarded as naming nothing in themselves; tha t is to say , the positions which

they occupy are treated as inaccessible to bound variables.

Ce rtain statements w hich ev en invo lve ex plicit quantification over classes arereplaceable by equivalent statements which conform to the tenets of nominalism.

T o take a simple example, the statement:

Class A is included in some class other tha n A

is equiva lent to:

Something is not an A .

Statements purporting to specify sizes of finite classes of concrete objects are

also easily accommodated. T hus the stateme nt:Class A has three members

may be rendered:

There are distinct objects x , y , and z such tha t any thing is an A i f and only if

it is x or y or z\

i.e.:

( 3 z ) ( 3 y ) ( 3 2 ) ( x j ¿ y . y 7 ¿ z . x j ¿ z . ( w ){ A w = : w = x .v . w = y . v . w = z))'

Obviously an y s tatement aff irming or denying tha t there are just, or a t least, ora t most, a ce rtain numbe r of concrete individuáis satisfy ing a g iven predícate canbe readily translated in similar fashion, provided the translation is short enough

to fit into the universe.8The definition of ancestorhood in terms of parenthood according to Frege’s

method seems to inv olve a class- variable ev en more ess entially. T he definiens

of '6 is an anccstor of c’ would run thus:

b is distinct from c; and, for every classx , i f c is a member of x and all parents

of members of x are members of x then b is a member of x\

i.e.:

M c . ( i ) ( c a . ( y )(z)(z €x . Parent yz . 3 . y €x ) b t x ).

B ut we can transíate this sentencc also w ith help of the no tation ‘P ar t st, ’ mean-ing that the individual s is part (or all) of the individual t .* W e need on ly re-

•T h e n o m i n a l i s t n e e d n o t n e c e s s a r i ly r e g a r d s u c h a s e n t e nc e a s ‘ T h e r e a r e 1 01000 o b j e c t s

i n t he u ni v e r s e ’ a s m e a n i n g l e s s , e v e n t h o u g h t h e r e b e n o t r a n s l a t i o n a l o n g th e s e li ne s . F o r ,

t h i s s e n t e n c e c a n b e t r a n s l a t e d a s ‘T h e u n i v e r s e ( a s a n i n d i v i d u a l ) h a s 1 0 1000 o b j e c t s a s

p a r t a ’ w h e r e ‘ h a s 10 1 0 0 0 o b j e c t s a s p a r t s ’ i s t a k e n a s a p r i m i t i v e p r e d í c a t e o f i n d i v i d u á i s .

B u t w h il e t h i s t r a n s l a t i o n s a tis f ie s p u r e l y n o m i n a l is t i c d e m a n d s , t he r e m a y b e ex t ra -

n o m i na l Í8 t i c re a s o ns o f e c o n o m y o r c l a r i t y f o r w a n t i n g a t r a n s l a t io n t h a t c o n t a i n s n o s u ch

p r e d íc a t e . A n d w h e r ev e r a n d f o r w h a te v e r r e a s on s a t r a n s l a t io n o f a n e x p r e s s io n i s w a n t e d

i n t e r m s o f c e r t a i n pr e d ic a t e s o r a c e r t a i n k i n d o f p r e d ic a t e s , th e s e a rc h f o r s u c h a t r a n s l a

t i o n i s a p r o b l e m f o r t h e n o m i n a l is t — t h o u g h o f c o ur s e n e i t h e r h e ñ o r a n y o n e e ls e c la im s

t h a t e v e r y p r e d íc a t e c a n b e d e f in e d i n t e r m s o f e v e r y p o s s ib le s e t o f o t he r s .

•A e y s t e m a t i c t r e a t m e n t o f ‘p a r t ’ a n d k i n d r e d t e r m s w i ll b e f o u n d i n The cálc idus of

indiv iduáis and i t s uses by H e n r y S . L e o n a r d a n d N e ls o n G o o d m a n i n th is J o u r n a l , v o l . 5



place ‘class’ by ‘individual/ and ‘member* by ‘part,’ provided we also stipulatethafc 6 be a parent and c hav e a pare nt. T his added s tipulation insures tha t b and c be single whole organisms, ra ther tha n frag menta or sums of organisms. In

symbols, ‘6 is an ancestor of c*becomes:

b c . (3 u) Parent b u . (3w) Parent w c . (x ) j Part e x .(y )(z)(Parto.Parent yz . 3 P ar t yx ) . O P ar t bx}.

Clearly the above method of tr anslation presupposes tha t an indiv idual maybe spatio- temporally scattered, or discontinuous. I t presupposes tha t continu-

ity is not necessary fo r concreteness. A broke n dish is no less concrete tha n a w hole one , but merely has mor e com plicated boundar ies; and any to ta lity ofindiv iduáis, howev er disperse in space and time, counts as an indiv idual in tum .

Individuáis, thus liberally construed, serve some of the purposes of classes, asis ev ident from the above trea tment of ‘ancestor.* B ut it is by no means truetha t we can in general simply identify any class of indiv iduáis w ith a scattered

single indiv idual, and reconstrue ‘member* as ‘part.* T he indiv idua l composedof all persons, e.g., has many parts which are not persons; some of these partsare parts of persons, and some consist of many persons or of parts of many persons. In the above analy sis of ‘ancestor,* we were able to overcome this diffi-

culty by inserting the clause ‘(3u) Parent bu . (3w) Parent wc.* Commonly,however, this kind of difficulty admits of no such simple solution.

T he two- place predícate ‘is ances tor of * is, t o borrow ter minolog y fr om theplatonistic logic of relations, the (proper) ancestral of the two- place predícate‘is pare nt of.* W e have seen, above, how it can be defined. B ut the schemeused there does not w ork f or the a ncestral of ev ery two- place predicate of individuáis. I t w orks so long as every individual has at least some pa rt w hich itshares w ith none of the indiv iduáis w hich are its “ ancestors** (w ith respect to the

predicate in question). A t the present w riting we know no w ay of defining theancestral of every two- place predicate of indiv iduáis nominalistically .

A rather different problem is rais ed by such s tate ments as:

There are more cats than dogs.

A s pointéd o ut ear lier, we are alr eady able to deal w ith s uch stateme nts as ‘T here

is at least one cat and not at least one dog* ands‘There are at least two cats andno t at J eas t tw o dogs.* A n alter nation of enough such successive stateme nts

w ill be true if and only if ther e ar e mor e cats tha n dog s, because i t w ill contain atleast one component stateme nt that is true in view of the actual number of catsand of dogs. Use of this method requires, firs t, knowledge tha t in a ll space-time there are not more than so many (say fifty trillion) dogs, and second, a

prodigious amount of w riting or talk ing. E v en though the requisite knowledgebe av ailable, the practical difficulties of ac tually w riting or speaking the translation of the statement about cats and dogs would be prohibitive.

(1 9 40 ), p p . 4 5- 5 5. E a r l i e r v e r s i o ns w e r e p u b li s h e d b y T a r s k i a n d L e é n i e w s k i . A l t h o u g h

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

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

w hic h the te r m s in que s t io n a r e p u t he re.



A be tter method of tr ans la tion makes use o f the predic ate ‘is pa r t of ’ and

another simple aux iliary pre dicate: ‘is big ge r tha n.’ T he predicate ‘is a b it ’ isthen so defined tha t it applies to every object tha t is jus t as big as the smallest

animal among all cats and dogs. In other words, lx is a b it ’ is defined to meantha t for every y , if y is a c at or a dog and is bigg er than no other cat or dog, thenneither is x bigger than y ñor is y bigger then x . Fo r brev ity we shall cali x

a bit o f z w henx is a b it and is part of z. Now if and only if there are more catsthan dogs will it be the case that every individual that contains at least one bit

of each cat is big ge r than some indiv idual th at contains a t least one b it of eachdog. (Such an indiv idual w ill of course be spatio- temporally scatter ed.) Ac-

cordingly we may transíate the sentence ‘There are more cats than dogs’ as fol-

lows:

Every individual that contains a bit of each cat is bigger than some individual

that contains a bit of each dog.

(Sy mbolic transcr iptions are omitted here, as they w ill be g iven later fo r paraJ-lel cases: §6, D9- 10.)

This method of translation has the great advantage, over the first methodsuggested, that there is no practical difficulty about writing down an actual

trans lation, regardless of the m ultiplic ity of indiv iduáis concemed. B ut, like

our method of defining the ancestral, it is not completely ge neral. I t w ill stil l

w ork if , in place of ‘is a c a t’ a nd ‘is a d o g / we choose any oth e r tw o predicate seach of which is such that the individuáis fulfilling it are discrete from oneanother. T hus it holds good for such a case as:

There are more human cells than humans,

and indeed for most cases where such numerical comparisons are made in or-dinary discourse. It has an im por tant use in nominalistic sy ntax , as we shall

see later. More ov er, by a re lative ly simple change it can be made general enough

to w ork wherev er each individua l fulfil ling either of the tw o predicates has a part

tha t it shares w ith no other indiv idual fulfill ing tha t predicate. A nd in addi-tion there are w ays of modify ing the m ethod to take care of ce rtain cases where

even this latter condition is not satisfied. B ut we have not found any generalfor mulation th a t w ill cover a ll cases regardless of how the indiv iduáis concerned

overlap one another.T he method w ill, how ever , help us in finding a nominalistic r eduction for even

so platon istic 10- sounding a s tate m ent as:

T here are mor e age- classes tha n grade- classes in the W hite S chool.

W e jus t replace this by :

T here are more age- wholes than grade- wholes in the W hite S chool,

where an ag e- whole is the individual composed of all pupils in the school who were bo m dur in g a sing le cale nd ar y ear, a nd a grade - w hole is an in div id ua l com-

1# W e ua e ‘p l a t o n i s t i c ’ a s t h e a n t i t h e s i s o f ‘n o m i n a l is t i c . ’ T h u s a n y l a n g u a g e o r t h e o r y

t h a t i n v o lv e s c o m m it m e n t t o a n y a b s t r a c t e n t i t y i s p l a to n is t ic .



posed of all pupils who receive equally advanced instr uction. T he new sentenceis then readily translated in the same way as the one about cats and dogs.

A combination of devices already described enables us to tr ansíate a statement

like:T here are ex actly one- third as many Canadians as Mex icans.

Letting ‘the Mexican whole’ stand for the individual that is comprised of allMexicans, the translation runs:

T here are some mutually discrete wholes x t y , andz such tha t each is comprisedof Mexicans and such that x + y + z = the Mexican whole; and there are exac tly as many Canadians (in all) as there are Mex icans in x and as in y and as inz. T he last clause may then be further trans lated by a s light v ar iation of the method

used in the example of cats and dogs.The foregoing samples will illustrate some of the means that remain in our

hands for interpreting statements that prima facie have to do with abstractentities. Cer tainly we have not as ye t reached our goal of knowing how to deal

w ith ev ery stateme nt we are not ready to dispense w ith altog ether. B ut thereis as y et no convincing reason for supposing the goal unattainable. Some of thedevices used above are r ather pow erful, and by no means all the possible methodshave been explored.

Since, however, we have not as yet discovered how to transíate all statements

that we are unwilling to discard as meaningless, we describe in following sectionsa course that enables us—strictly within the limitations of our language and

w ithout any retreat from our position— to talk about certain statements withoutbeing able to transíate them.

6. Elements of nominalistic syntax. It may na turally be asked how , if weregard the sentences of mathematics merely as strings of mark s w ithout meaning,

we can account for the fact tha t mathematicians can proceed w ith such remark-able agreement as to methods and res ults. Our answer is tha t such intellig ibilityas mathematics possesses derives from the syntactical or metamathematical

rules governing those marks. A ccordingly we shall tr y to develop a sy ntaxlanguage tha t w ill tre at mathematical expressions as concrete objects—as actualstrings of phys ical mark s.11 Since one mar k is as concrete as another, we candeal w ith such marks and strings as V and *(!>)(» « v | v « v )’ quite as well as

w ith ones lik e *(* or ‘Eiífel T ower. ’ B ut our s y ntax lang uag e mus t itself be purelynominalistic; it must make no use of terms or devices which involv e commitmentto abstract entities. I t m ig ht seem tha t this prog ram could be carried out w ithout any difliculty once we have specified tha t we are dealing w ith concrete marks;but actually classical syntax has depended so heavily upon platonistic devices

in constructing its definitions that the nominalist is faced with the necessity offinding new means of definition at almost every step. N ot only subsidiary terms,

11 W e m ig ht , equal l y c onsis tent l y w i th nominal ism, construe mark a p henomenal l y , as

ev ents in the v isual (or in the aud itory or . tac tual ) f ie ld . Moreove r, a l though we shal l

reg ard a n ap p rop riate ob jec t d uring i ts ent ire ex is tence as a s ingl e m árk , we c ould equal ly w ell —a n d ev en adv a ntag e ous ly i f we w a nt t o incr eas e the s upply o f m a r k s — cons tr ue a

ma rk as c omp ris ing the ob jec t in quest ion d uring onl y a s ingl e mome nt of time.



bu t such k ey terms as ‘for m ula,’ ‘s ubs titution,’ and ‘theore m’ have t o be defined12

by quite new routes.T he platonistic object language tha t our nominalistic sy ntax is to tr ea t of must

contain notations for truth- functions, quantification, and members hip. A ll we nee d for these purpos es ar e par entheses, v ar iable s , the str ok e ‘| ’ of a lte m ativ edenial, and the sign V of membership. Parentheses w ill serve both fo r enclos-

ing a lter nativ e denials to indícate g rouping s a nd for enclosing v ariables to formuniversal quantifiers. T o simplify our sy ntactical trea tme nt, let us require tha t

each altemative denial be enclosed in parentheses—even when it stands apart

from any broader contex t . A s var iables we may use V , V ’, lvn\ etc., so that

the simple ty pographical shapes of the object languag e reduce to six : V ,‘)\ ‘I’, and V .

A s already mentio ne d, th e characters of our lang uag e ar e no t these abs tr actshapes—which we, as nominalists, cannot countenance—but rather concretemark s or.insc riptions . W e can, howe ver, apply shape- predicates to such in

dividuáis; thus ‘Vee x ’ will mean that the object x is a vee (i.e., a V - shapedinscr iption), and ‘A c x ’ w ill mea n tha t s is an accent (i.e., a " ’- shaped inscr iption),and ‘LPar x ’ w ill mean t ha t a; is a lef t parenthesis, and ‘R P a r x ’ w ill mean tha t

s is a right parenthesis, and ‘S tr z ’ w ill mean tha t x is a s trok e (a ‘|’- shapedinscription), and ‘Ep x 1 w ill me an th a t x is an epsilon.

B ut it happens actually tha t left parentheses and r ight parentheses are alike

in shape, and distinguishable only by their orientation in broader contexts.It would appear therefore that instead of writing ‘LPar x\ to mean that x is

intrinsically a left parenthesis, we should write ‘LPar xy\ meaning that x is aleft parenthesis from the po int of view of its or ientation w ithin the longer inscription y\ and corre spondingly for ‘R P a r .’ Since how ever this ex ceptional

trea tme nt is m ade necessary solely b y a typog raphical idiosy ncrasy, we ma y dis-

regard it. T he reader may , if he likes, restore an intrinsic distinction betweenleft and r ig ht parentheses by think ing of each lef t parenthesis as comprisíng

w ithin its e lf the s tr aig ht un ink e d lin e jo in in g its tips .

Our nominalistic s y ntax m ust conta in, besides the six shape- predicates, somemeans of expressing the concatenation of ex pressions. W e shall w rite ‘Cxyz’ to mean that y and z are composed of w hole characters of the language , in normalorientation to one another , and contain neither s plit- off frag ments o f characters

ñor anything extraneous, and that the inscription x consists of y followed b y z. The characters comprising y and z may be irregularly spaced; furthermore theinscription x w ill be considered to consist of y followed by z no matter w hat the

spatial interval between y and z, provided that x contains no character that oc-curs in that interval.

The two remaining primitives of our syntax language are abbreviations of thefa m iliar predicates ‘is pa r t of’ and ‘ís bigge r tha n.’ ‘P ar t x y ’ means that x ,

T h e i de a o f d e a l in g w i t h t he l a n g ua g e o f c l as s i ca l m a t he m a t i c s i n t e r m s o f a n uc l e a r

s y n t a x l a ng u a g e t h a t w o u l d r ae e t n o m i n a l is t i c d e m a n d s w a s s u g g e s t ed in 1 940 b y T a r s k i .

I n t h e c our s e o f t h a t y e a r t h e p r o j e c t w a s d is c us s e d a m o n g T a r s k i , C a r n a p , a n d t h e p re s e n t

w r it e r s , b u t s o lut io n s w e r e n o t f o u n d a t t h a t t im e f o r t h e te c hn ic a l pr o b le m s in v o lv e d .



w hethe r or no t it is ide ntical w ith y , is contained entirely w ithin y . ‘Bgr x y ’

means that x is spatially big ger than y .

O ur sy ntax lang uag e, then, contains the nine predicates ‘Vee’, ‘A c’, ‘L P a r ’,

‘R P a r ’, ‘S tr ’, ‘E p ’, ‘C ’, ‘P a r t’, an d ‘B g r’, tog ether w ith variables, quantifier sand the usual truth- functional notations ‘v ’, V , etc. T he var iables take as

valúes any concrete objects.

6. S ome aux iliary def initions . W e now proceed to define cer tain useful aux il-

iary predicates. F irs t, it is conv enient to have four- , five- , and six - place predi-cates of concate nation. T he definitions are obvious:

D I . Cxyzw = ( 3 t ) ( C x y t. C tzw)fli

D2. Cx y zw u = ( 3 t )( C x y t . Ctzivu),

D3. Cxyzums = ( 3 t ) ( C x y t. Ctzwus).

A ls o, la ter de finitions w ill be shorte ned cons ide rably if we can say br ie fly tha t

a given individual is a charaeter of our object languag e. Since a character is

any concrete object t ha t is either a vee or an accent or a lef t parenthesis or etc.,the definition runs:

D4. Char x = . V ee i v A c i v L Pa r x v R P a r x v S t r x v E p x .

Convenience iss imilar ly served by the definition of an inscription as an object

composed of w hole characters in norm al orienta tion to one another . In view of the inter pre tation of ‘C ’ in §5, the def inition is easy :

D6. Insc x = . C ha r x v ( 3 y ) ( 3 z ) C x y z .

A n ins cr iption x is said to be an in itia l segment of another, y, i f x is identical

w ith y or there is some inscription z such tha t y consists of x followed by z.

D6. InitSeg x y = . Insc x . x = y .v ( 3z)Cyxz.

The definition of final segment is strictly parallel :

D7. FinSeg x y = . Insc x . x = y .v (3z)Cyzx.

A n ins cript ion x is said to be a segment of y if x is an initial segment of some

final segment of y.

D8. Seg x y = (3z) ( InitSe g x z . FinSeg zy).

A seg ment x —whether initial, final, or interior—of an inscription y w ill becontinuous relative to y , in the sense that if x contains two characters of y then

11 T h e s i g n *= ’, w h e n i t o c c ur s a a t h e n i a i n c o n n e c t i v e i n d e f i n it i o n a i n t h i s p a p e r , isn o t to b e t h o u g h t o f a s e x p re s s in g i d e n t i t y . I t i s t o be r e g a r de d r a t h e r a s c o n s t i t ut in g ,

i n c o m b i n a t i o n w i t h t he ‘D ’ w h i c h p re c e de s e a c h d e f i ni ti o n- n um b e r , a m a r k o f d e f í n it io n a l

a b b r e v i a t io n ; a n d i t m a y o c c ur b e tw e e n n a m e- m a tr ic e s a n d s t a t e m e nt - m a t r ic e s indif fe r -

e n t ly . T h e d e f i n it io n D I i s t o be u nd e r s t o o d a s a c o n v e n t i o n t o t h i s e f fe c t: ‘C x y zw * i s t o

be understood as an abbreviation o f ,‘{3 t)(C xy t . C tziv)’] and a similar underslandir.g is to

obtain when a ny other variables are used in place o f (x \ ly \ V , a n d ‘u>\ pro vid ed th at a va ri-

able dist inc t from them is used i n place o f ‘t \ O t h e r d e f i n i ti o n s a r e t o b e c o n s t r ue d

a n a l o g o u s l y .



x mus t contain all the characters w hich occur in y betwe en those two. T he char-

acters of a segment x of y may still be irregularly spaced, but only because of

irreg ular s pacing in y itself.

W e sha ll la te r w ant to be able to say th a t t w o ins cr iptions ar e e qua lly lo ng , not

in the sense that their ends are equally far apart but in the sense that each in-

sc ription has as m any c haracters as the other . Since the characters in any in-

sc ription are discrete from one another , this numer ical comparison can be handled

in a w ay ex plained in §4. W e begin by so defining ‘B it’ for our present purposes

that ‘B it x 1means tha t x is jus t as big as every smallest character.

D 9 . B i t x = . (*/)(Char y 3 ~ B g r x y ) . ( 3 z ) ( C h a r z . ~ B g r zx).

It mus t no t be supposed th a t, because accents are in general the s malles t charac

ters of our object lang uage , every accent w ill be a b it . For accents may v ary insize, and o nly the smallest characters, along w ith eve ry thing tha t is jus t as big,

w ill be bits . A n ins cr iption x is longer than another, y, i f x contains more characters tha n y .

Us ing the same method as for the ex ample of cats and dogs in §4— where a verbal

explanation is given—we define:

D IO. L ngr x y = . In s c x - Insc y . (z) {(w )[ Char w . Part w x . 3 (3 u )(B it u .P a r t uw . P a r t uz)] 3 (3 ¿ ) [ ( r )(C h ar r . Part ry . 3 (3 s ) (B it s . P a r t s r . Part

st)) . B g r zt]\ .

T wo inscriptions are equally long if ne ither is long er th an the other.

D l l . E qL ng x y = . In s c x . Insc y . ~ L n g r x y . ~ L n g r yx .

W e can now de fine w ha t we s ha ll mean by say ing th a t tw o ins cr iptions ar e

like one another. T w o characters are alike if bo th are vees, or both are accents,or etc. T w o inscriptions x and y are alike if they are equally long and if, for

every two e qually long inscriptions z and w such that z is an initial segment of x

and w is an initia l seg ment of ?/, the s eg ments z and w end in like characters.

D12. L ike x y = . E q L n g x y . (z)(w) ¡ E qL n g zw . Ini tSeg zx . Ini tSeg wy . 3(3s) (3¿)(FinSeg s z . FinSeg tw : Vee s . Vee t . v . A c s . Ac t . v . L P a r s . L P a r t . v .

R P a r s . R P a r t . v . S t r s . S tr t . v . E p s . Ep¿) j .

No te tha t only inscriptions can be “a like ” , in the sense here defined, since only

inscr iptions can be equa lly long ; and furthe r, tha t likeness depends solely upon

the component characters and their order of occurrence, not upon identicalspacing.

7. V ariables an d quantification. A v ariable of our object lang uage is a vee,

or a vee tog ether w ith a str ing of one or more accents follow ing it. We first

define a string o f accents as an y inscription of w hich every par t tha t is a character

is an accent.

D13. AcStr ing x = . Ins c x . (z) ( Pa r t zx . C ha r 2 . 3 A c z).

The definit ion of a variable is then readily formulated.



D 1 4 . T O x = . V e e x v ( 3 y ) (3 z ) (V e e y . A c S t r in g z . C xyz).

A v a r ia b le is a v e e o r t he r e s ult o f c o n c a t e n a t in g a v e e w i t h a s t r in g o f a c c e nts .

A q u a n t if ie r w i l l b e s im p ly a v a r ia b le i n pa r e nthe s e s . B u t i t is m o r e us e f ul

t o d e f in e a s tring o f ( o ne o r m o r e ) quanti f iers d ir e c t l y . A m e t h o d f o r d o i n g th is

b ec o me s e v id e n t w h e n w e r e f le ct t h a t a n y i n s c r i pt io n w i ll b e a s t r i n g o f q u a n t i

f ie r s if i t b e g i ns a n d e n ds w i th f a c i ng p ar e n th e s e s a n d is s u c h t h a t e v e r y p a i r o f

f a c i ng p ar e n th e s e s w i t h in i t f r a m e s a n in s c r i p tio n t h a t i s e i th e r a v a r i a b le o r c o n

t a i n s p a r e n t h e s e s b a c k t o b a c k .

D I B . Q f r S t r i n g x = ( 3 y ) ( 3 z ) { L P a r y . R P a r z . ( 3 w ) G n /w z . ( s ) ( 0 ( M) (k )

[ L P a r t . R P a r k . C s luk . S e g s x . 3 . V b l u v ( 3 p ) ( 3 g ) ( 3 r ) ( R P a r q . L P a r r .

C pqr . S e g p u)] } .

T h e n le t u s c a l i x a quan tif ication o f y i f x c o n s i s t s o f a s t r i n g o f q u a n t i f i e r s

f o l l o w e d b y y .

D 1 6 . Q f n x y = ( 3 2 ) (Q f r S t r in g z . Cxz y) .

8. Formulas. A n a t o m ic f o r m u l a o f t h e o b je c t la n g ua g e co ns is ts o f tw o

v a r i ab le s w i t h a n e p s il o n b e tw e e n t h e m .

D 1 7 . A t F m l a x = ( 3 u O ( 3 y ) (3 z ) (V b l w . E p y . V b l z . C xw yz).

W e a r e s u pp o s in g t h a t t h e c la s s lo g ic t o b e d e v e lo pe d i n t h e o b je c t la n g ua g e w ill

us e o n e o r a n o t h e r o f t h e a lt e r n a t iv e s t o t h e t he o r y o f ty p e s , s o t h a t e ps ilo ns m a yg r a m m a t ic a l l y o c c u r be t w e e n a n y v a r i a b le s w i t h o u t r e s t r ic t io n .

T h e no n - a t o m ic f o r m u la s o f t h e o b je c t la n g u a g e a r e c o n s t r uc t e d f r o m t h e

a t o m i c f o r m u l a s b y q u a n t if i c a t io n a n d a l t e m a t i v e de n i a l . I n o r d e r t o d e fin e a n

al tema t ive denial w e f ir s t ne e d t o b e a b l e t o s a y t h a t a g i v e n in s c r i p tio n x con-

tains exaei ly as m an y lef t a s r ight parentheses. T h i s w i ll b e t h e c as e i f x l a c k s

p a r e n th e s e s a l t o g e t h e r ; a n d i t w i l l b e t he c a s e al s o if t h e i n s c r i pt io n w h i c h c o n

s is t s o f a l l t h e l e f t p a r e nt he s e s i n x a n d t h e i n s c r i p t io n w h ic h c o ns i s t s o f a l l t h e

r i g h t p a r e n th e s e s i n x a r e e q u a ll y l o n g in th e s e ns e o f D l l . I n s y m b o ls :

D 1 8 . E q P a r x = . ( u )( L P a r u v R P a r u .Z > ~ S e g ux) v ( 3 y ) (3 z ) { E q L n g

y z . ( w ) ( C h a r w 3 : L P a r w . S e g w x . = S eg w y : R P a r w , S e g w x . = S e g wz) | .

N o w f o r a n in s c r i pt io n x t o b e t h e altemative. de nia l o f y a n d z i t is ne c e s s a r y t h a t

x c o ns is t o f a l e f t p a r e n th e s i s f o llo w e d b y y , t h e n a s t r o k e , t h e n z , a n d f i n a l ly a

r i g h t p ar e n th e s is . B u t t h i s is n o t e n o ug h . W e m u s t m a k e s u r e t h a t t he be g in-

n i n g a n d e n d in g pa r e n th e s e s ar e “ m a t e s ” — t h a t is , t h a t t h e y a r e p a ir e d w i th

e a ch o t h e r a n d n o t w i th o t h e r p ar e n th e s e s t h a t o c c ur b e tw e e n t he m . A l s o w e

m u s t m a k e s u re t h a t t he s tr o k e b e tw e e n y a n d z i s t h e m a i n c o n n e c t i v e i n x .

W e c a n a c c o m plis h a l l t h is b y r e q u ir in g t h a t y c o n t a i n a n e q u a l n u m b e r o f l e f ta n d r i g h t p a r e n th e s e s , a n d s im i l a r ly f o r z , b u t t h a t t h is b e t r u e o f n o i n i t ia l s eg-

m e n t o f x ( e x c e p t x i tse l f ) .

D 1 9 . A D xy z = . E q P a r y . E q P a r z . ( r ) ( s ) (O r s 3 . ~ E q P a r r ) . (3 í) (3 m )(3 u> )

( L P a r t . S t r u . R P a r w . Cxtyuzw ) .

T h e fo rm u la s o f t h e o b je c t l a n g u a g e c o m pr is e t h e a t o m i c f o r m u la s a n d e v e r y

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



denial. Some ways in which one m ig ht natur ally seek to reduce this to a formal

definition are not feasible in a nominalistic sy ntax .14 O ur me thod is to begin by

defining a quasi- formula as any thing w hich is an atomic form ula, an al tem ative

denial, or a quantif ication of an a tomic for mula or altem ative denial.

D20. QuasiFmla x = (3?/)(x = y . v Q f n x y : A t F m la y v ( 3 w ) ( 3 z ) Á D y w z ) .

A quas i- formula w il l nót necessar ily be a fo r m ula , since the compo ne nts of the

altem ative denial are no t required to be formulas. B ut in terms of this notion

of quasi- formula we can now easily define formula:

D21. Fmla x = . Q ua s iF m l a x . ( w ) ( y ) ( z) ( A Rw y z . Seg w x . 3 . Q u a s iF m la

y . QuasiFmla z).

In other w ords, a for mula is a quasi- formula such tha t every alte m ativ e denialin i t is an altem ative denial of quasi- formulas.

B y r equiring.eve n the shortest altema tiv e denials in a form ula x to be alterna-

tiv e denials of quasi- formulas, the def inition requires them t o be altem ativ e

denials of atomic formulas or of quantif ications of atomic formulas, and this

make s them g enuine f orm ulas in the intu itive ly in tended sense of the word.

A ccording ly , by re quir ing als o the ne x t mo re comple x a lt e m a tiv e de nials in x

to be alte m ativ e denials of quasi- formulas, the def inition guarantees tha t these

also will be formulas in the intuitively intended sense; and so on, to x itself.

9. A x ioms and rules . Now tha t we have specified the characters and formulas of the object language w ithin our nom inalistic sy ntax languag e, the nex t

problem is to describe the sorts of notational operations which pass for logical

proof amo ng the users of tha t object languag e. A f ull solution of this problem

w ould cons is t in the fo rm ula tio n, in our s y nta x la ng ua g e , of a co ndit io n w hich isnecessary and sufficient in order that an inscription a; be a theorem of the object

logic.The theorems are those formulas of the object language which follow from

cer tain axioms by cer tain rules of inference. T he ax ioms should be so chosen

that we can obtain from them, by the rules of inference, every formula which isvalid according to the logic of altemative denial and quantification and, in addi-

14 U s i ng e s s e n t i a ll y t h e m e t h o d o f F r e g e ’s d e f i n it i o n o f t h e a n c e s t r a l o f a r e l a t i o n , w e

m i g h t s a y t h a t x i s a f o r m u l a if i t b e lo n g s t o e v e r y c la s s w h ic h c o n t a i ns a l l a t o m i c f o r m u l a s

a n d a l l q u a n t if i c a t i o n s a n d a l t e m a t i v e d e n ia l s o f i t s m e m b e r s . B u t t h is d e f i n it io n is

u n a l l o w a b l e b e c a us e o f i t s us e o f q u a n t i f i c a t i o n o v e r c l as s e s ; c f . §4 . — T h e r e is i nd e e d a

c o m p l e t e l y g e n e r al m e t h o d , in s y n t a x , o f d e r i v i n g a n c e s t r al s a n d k i nd r e d c o n s t r u c ti o ns

w i t h o u t a p p e a l t o cla s s e s o f e x pr e s s io ns . T h is is t h e m e t h o d o f “ f r a m e d in g r e d ie n t s ”

w h ic h a p pe a r s i n Q u in e , M a th e m a tic a l lo g ic , §5 6 . T h e m e t h o d c o n s i s t s e s s e n t ia l ly o f

t he s e t w o s t e p s : (1 ) t h e F r e g e f o r m o f d e f i n it i o n is s o r e v i s e d t h a t t h e c la s s e s t o w h i c h

i t a p p e a l s c a n b e l im i t e d t o f i n it e c la s s e s w i t h o u t i m p a i r i n g t h e r e s u l t ; ( 2 ) f i n it e c la s s es

o f e x p r e s s io n s a r e t h e n i d e n t i f ie d w i t h i nd iv idua l e x pr e s s io ns w h e r e in t h e “ m e m b e r ” -

e x p r e s s i o n s o c c u r m e r e l y a s p arts m a r k e d o f f i n c e r t a i n r e c o g n iz a b le w a y s . H o w e v e r ,

w he n a s n o m in a l is t a w e c o n c e iv e o f e x pr e s s io ns s t r ic t l y a s c o nc r e te in s c r ip t io n s , w e f in d

t h e m e t h o d o f f r a m e d in g r e d i e n t s u n s a t i s f a c t o r y , b e ca us e i t s s uc c es s d e p e n d s t o o m u c h

o n w h a t in s c r i p t io n s h a p p e n t o e x is t in t h e w o r l d . A c t u a l l y , t h o u g h , t h e n o m i na l is t i c

d e f i n it io n o f p r o o f i n t h e p r e s e n t p a pe r w i ll b e s i m p l e r t h a n t h a t i n t e r m s o f f r a m e d i n g r e

d ie n t s ; f o r i t w i ll n o t r e q uir e t h e l i ne s o f a p r o o f t o b e c o n c a te n a t e d , ño r t o b e m a r k e d o ff

b y i n t e r v e n i n g s i g n s .



tion, a goodly array of formulas whose alleged validity is supposed to proceedfrom spccial properties of class- membership. We cannot aspire to completenessin this last regard, in vievv of Gódel’s result.

T h e re a r e m a n y e s s e n t ia l l y e q u iv a l e n t s e ts o f a x i o m s s u i ta b l e t o t h e a b o v e

p ur p o s e s . T h e a x i o m s w h ic h w e s h a l l a d o p t f a l l u n d e r t h r e e h e a d s : a x i o m s o f

al ternat ive den ia l , ax io m s o f q i ian ii fi ca t ión , a n d a x i o m s o f m e m b e r s h ip . I n s e t t in g

t h e m f o r t h l e t u s u n de r s t a n d • 9 a s s h o r t f o r ‘ ( •••| ••■) ’•

A x io m s o f a lt ernati ve d en ia l: A l l fo r m u la s o f t he f o r m :

( ( p i (Q i « ) ) i a s i ~ s ) i a s i q ) i ~ < p i s ) ) ) ) “

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

A x io m s o f q ua n tifica tio n : A l l f o r m u la s o f t he f o r m s :

(1) (W(P I~ Q ) I~ « v ) P |~«©),

( 2 ) ( R | ~ ( t>)R) (w h e r e V is n o t f r ee in

( 3 ) ( (v ) P | ( w h e r e is t h e r e s u lt o f s u b s t it u t in g s o m e v a r i a b l e f o r

V i n lP ' ) .

I f t he r e a d e r r e fle cts t h a t t h e s i g n - c o m b in a tio n ‘ | a m o u n t s t o ‘ 3 ’, h e

w i l l r e c o g n iz e i n t h e f o r m s (1 ) — (3 ) a f a m i l i a r s e t o f a x io m - s c h e m a t a f o r q u a n t i

f i c a t i o n t h e o r y . 16 L i k e c a p i ta l s i n ( 1 ) - ( 3 ) a r e o f c o ur s e t o b e u n d e r s t o o d a sr e p la c e d b y l ik e f o r m u la s , a n d t h e v e es b y l ik e v a r i a b l e s . T h e tw o br ie f

p r o v i s o s a p p e n d e d t o (2 ) a n d ( 3 ) , a b o v e , m a y b e s t a t e d m o r e p r e c is e ly a s f ol-

l o w s : ( i ) t h e f o r m u la s s u p p l a n t i n g t he ‘R ys c o n t a i n n o f r e e v a r i a b le s l ik e t he

v a r i a b l e s s u p p l a n t i n g t h e v e e s , a n d ( ii ) th e f o r m u l a s u p p l a n t i n g t h e ‘S * i s l ik e

t h e f o r m u l a s u p p l a n t i n g t he ‘P ’ e x c e p t p e r h a ps f o r c o n t a i n in g o t he r f r e e v a r i

a b le s , l ik e o n e a n o t h e r , in p l a c e o f a l l f r e e v a r i a b le s l ik e t h e v a r i a b l e s u p p l a n t in g

t h e v e e .

A x io m s o f m em bersh ip : H e r e i t h a p p e n s t h a t a l i m i t e d l i s t o f s p ec if ic e x p r e s s io ns

is a d e q u a t e ; e .g . , H a i l p e r i n ’s . 17 L e t u s s u pp o s e s u c h a l is t p u t o v e r in t o t h e

p r i m i ti v e n o t a t io n o f o u r o b je c t la n g u a g e a n d s e t d o w n h er e ; t h e n o u r a x io m s

o f m e m b e r s h ip a r e a l l i n s c r i pt io n s l ik e t ho s e i n t h e l is t .

In addition to the ax ioms, we need two ru les o f in ference :(1) From any formula, together with the result of putting a formula like it

for *P’ and any formulas for lQ y and ‘ R 1 in ‘(P | (Q | R ) ) \ infer any formula likethe one w hich was put for ‘Q ’.18

,6 T his is L ukas iewic z’e s im p l i f ic at ion of Nic o d ’s ax iom sc hema. See J a n L ukasiew ic z ,

XJwagi o aksyom acie Nicod 'a i o “ded ukcyt uogóln ia jqce j”, K sig ga pam iq tkowa Polskiego Towarzystwa Filozoficznego we Lwowie , 1931, pp.2- 7; als o Je an N ico d, A reduction in the

number oj primitive propositions o f logic, Proc eedings o f th e Cam bridge Philosophical

Society, v ol. 19 (1917- 20), pp. 32-41.‘•T he y answ er to 4.4.4, 4.4.5, and 4.4.6 of F . B . F itch , The consistency of the ramified

Prin cip ia , this J o u r n a l , v ol . 3 (1 938) , p p . 1 40- 1 49; a l so to *1 02- *104 of W . V . Q uine,

M ath em atical logic, p. 88.17 Theodore Hailperin, A set o f axioms for logic , this J o u r n a l , vol. 9 (1944), pp. 1-19.

18 T his is N icod’s g ene ra l ization of modus ponens; seo footnote 15.



(2) Fr om any for m ula infer any quantif ication thereof.

T o reach a def inition of ‘A x iom' we m ust f irs t be able to define w hat i t means

to be an axiom of any given one of the f ive kinds abo ve described. A s imple

aux iliary definition w il l be use ful:

D22. T)xy = (3 z ) (L ik e y z . A Dx y z);

i .e., that x is a denial o f y means that x is the altemative denial of y and someother inscription exactly like y .

Definition of ‘AADa;’, meaning that x is an axiom of altemative denial, is

achieved by stating formally what we can observe from the general schema al-

ready g iven: that every axiom of a l temative denial is an al temative denial of

two formulas; one of these two main components is an altemative denial of

formulas of which one is an altemative denial of formulas; the other of the twomain components is an a ltema tiv e denial of formulas o f w hich one is a n alter-

native dental of a formula with a formula l ike the denial of that formula, while

the other is . . . etc., etc. In symbols:

D 23 . A A D a; = ( 3 /) (3 ^) (3 « ( 3 t )( 3 j )( 3 f c ) (3 0 ( 3 m ) (3 n ) (3 p ) (3 7 ) (3 r ) ( 3 S) (3 0

( 3u) (3w ) (32 /) (3 z) (F m l a / . F m l a g . F m l a h . F m l a i . L i ke k i . L i ke Ig - Likem f . Like n i . A D pg h . A D qfp . D r i . A Ds ir . A D I k l . A D um n . D w u . A Dytw .

A D zsy . A Dxqz).

Formulation of ‘AQ1 x\ meaning that x is an axiom of quantif ication of kind(1), proceeds in the same w a y ; we sha ll omit the def inition.

Formulat ion of ‘AQ2 x ’ offers the one additional difficulty that in order to

express stipulation (i), appearing in the above description of the axioms of

quantification, we must have a definition of free var iable. A v ar iable a; is a free

variable in an inscription y if x is a segment of y not fo llowed by any addit ional

accents in y , and if furthermore x is not a segment of any seg ment of y tha t co n

sists of a for mula preceded by a quantif ier consisting of a va riable l ike x framed

in parentheses.

D24. Free x y = . V bl x . Seg xy . (z)(w )(A c w . Czxw . ~ S e g zy ) .

(#)M(s)(¿)(w)(LPar q . L i ke rx . R P a r s . F m l a t . C u q r s t. Seg u y . 1> ~ S e g x u).

T he definition of ‘A Q2 x y is then quite straightforw ard and m ay be omitted here.

Formulation of ‘AQ3 x ’ offers a further complication for nominalistic s yntax .

T he problem lies in the notion of substitution, involv ed in s tipulation (ii). L et

z and w be the respective form ulas s upplanting the iP f and l S * of (3 ), let y be the

va riable supplanting the V , and let x be like the free variables which are to

appear in w in place of the free variables like y in z. W e have to f ind a way

w ithin nom in a lis tic s y nt ax of de f ining ‘S ub s t wx yz, ’ meaning that the fo rm ula w

is like the formula z except for having free variables like x wherever z contains free

variables like y . O ur method of definition depends upon the fac t that the condi-

tion in the foreg oing italics is equiv ale nt to the follow ing one: W ha t r emains w hen

all free variables like y are omitted from the formula z is like what remains when

some free variables like x are omitted from the formula w. T he form al definition

is as follows:



D 2 6 . S u b s t w x y z = . F in ia w . F m i a z . (3 /) (3 w ) | L ik e tu . ( s ) [ C h a r s Z>:

( r ) ( L i k e ry . F r e e rz . 3 ~ S c g sr) . S e g sz . = S e g s u : ( r ) ( L i k e rx . F r e e rw . 3

~ S e g s r ) . 3 . S e g sw = S e g s /J } .

It . w a s la r g e l y f o r t h e pu r p o s e o f t h i s d e f i n it i o n t h a t w e s o d e f i n e d l i k e n e s s o f

i n s c r i p ti o n s a s t o a l lo w t h e i r c ha r a c t e r s t o b e d i f ie r e n t ly s p a c e d.

N o w t ha t. t h is d e f i n it i o n i s a c c o m p li s h e d , t h e d e f i n it i o n o f 4A Q 3 x ' o f f e r s n o

f u r t h e r d i f f i c u l t y ( a n d i s o m i t t e d h e r e ) .

D e f i n it i o n o f a x i o m s o f t h e f i f t h a n d f in a l k i n d — t h e a x i o m s o f m e ml> e r s hip,

" A M " — p rc s e nt s n o pr o b le m ; w e c a n s p ec if y t h e m in o u r s y n t a x s i m p l y b y s pell-

i n g t h e m o u t e x p li c it l y w i t h t he h e l p o f o u r p r i m i t i v o p r e dic a t e s .

W e ar e t h e n r e a d y f o r a g e ne r a l d e f in it io n o f w h a t it. m e a ns f o r x t o b e a n axiom

o f o u r o b je c t l a n g u a g e . Tt, m e a n s s i m p l y t h a t .t i s a n a x i o m o f o ne o f t h e fiv ok i n d s s p e c if ie d .

D 2 6 . A x i o m x = . A A Da: v A Q l x v A Q 2 x v A Q 3 x v A Mar.

A n in s c r ip t io n x i s c a l le d a n immediate conseqwmce o f i n s c r i p t io n s y a n d z j u s t

in c as e x f o ll ow s f r o m y a n d z b y o n e a p p li c a t .i o n o f r u le o f i nf e r e n c e ( 1 ) , o r f r o m y

b y r u le o f in f e r e nc e (2 ) .

D 2 7 . I C x y z = . ( 3 w ) ( 3 u> ) (A D uo : uf . A D y zu v A D ¿ y u ) v Q f n xy .

1 0 . P r o o f s a n d t he o r e m s . A n i n s c r i p t io n is a t h e o r e m i f i t h a s a p r o o f ; a n da p r o o f i s c o n s t r u c t e d b y a s e r i e s o f s t e p s o f i m m e d i a t e c o n s e q u e n c e , s t a r t i n g

f r o m a x i om s . R o u g h l y , a p r o o f i s d e s c r i ba b le a s c o m p os e d o f o ne o r m o r e line s

s u c h th a t. e a c h i s e i th e r a n a x i o m o r a n i m m e d i a t e c o n s e q ue n c e o f p r e c e d in g l in e s .

A c t u a l l y w e nc e d n o t r e q uir e t h a t t h e s o- call e d "l in e s ” o f a pr o o f be a t dif f e r e nt

le v e l s o n a p ag e , o r b e s e g r e g a t e d f r o m o ne a n o t h e r b y a n y o t h e r d e v i c c . T h e y

c o u ld e v e n b e w r i t t e n e n d t o e n d w i t h o u t i nt e r v c n in K p u n c t u a t i o n , a n d w e c ou ld

s t i ll s i ng l e th e m o u t u ni q ue l y a s s e p a r a t e “ l in e s . ” F o r , th e g r a m m a r o f t he

o b j e c t l a ng u a g e is s uc h t h a t t h e r e s u l t o f d i r e c t ly c o n c a t e n a t in g t w o f o r m u la s

2 a n d w w i ll n e v e r b e a s e g m e n t o f a l a r g o r f o r m u l a , ño r w i ll i t c o n t a i n a s s e g m e nt ea n y f o r m u la s o t h o r t h a n t ho s e w h ic h a r e s e g m e n t s o f 2 a l o n e o r w a l o n e . A c e or d -

i n g l y i t w i ll b e c o n v o n i e n t i n g e n e r a l t o s p e a k o f x a s a Une o f y ( w h o r o y m a y o r

m a y n o t be a p r o of ) i f x is a f o r m u la w h ic h i s p a r t o f y b u t n o t p a r t o f a n y o the r

f o r m u la i n y.

D 2 8 . L i n o x y = ( z )( F m l a 2 . P a r t x z . P a r t zy . = . z = * ) .

I f a t h e o r e m is t o b e d e f in e d a s a f o r m u l a f o r w h i c h a p r o o f o x i s t s , it. is im p o r

t a n t n ot . t o d e m a n d t h a t a l l U n e s o f t h e p r o o f Ij e a s s e m b l e d i n p ro j e r o r d e r in

a n y o n e p la c e a n d t im e . A c c o r d i n g l y w e s h a ll s o d e f in e a p r o o f a s t o a l lo w i t toc ons is t o f l ine s u h e r o v e r t h e y m a y b e— p e r h a ps s c a t te r o d a t r a n d o m t h r o u g h o u t

t h e un iv e r s e , : m d p e r h a p s n o t e v e n a l l e x i s t in g a t. a n y o ne m o m e n t o r w i t hi n

a n y o ne c e n t ur y .

A c c o n l in g t o t h e r o ng h c h a r a c t e r iz a t io n o f p r o o f pr o po s e d tw o pa r a g r a phs

b a c k , e a c h li ne m u s t b e e i t h e r a n a x i om o r a n i m m e d i a t e c o ns e q ue n ce o f pre

ccding l in e s . T h e r e a s o n f o r t h e w o r d ‘p r e c e d i n g ' h e r e is t o r u l e o ut . c a s os w h o r o



ever y line is deducible from other Iines, in circular fas hion, w hile not all l ines are

deducible ultima tely from axioms. Howev er, we must now resort to some other

expedient for excluding such circulanty; for we have chosen to dispense with

the ordering of lines of a proof, and this deprives us of the notion of a “pre-ceding” line.

A n e x pe die nt w hich v ill be s ho w n to m eet the requir ements is this : W e stipu-

late that i f any indiv idual y contains as parts some lines of a proof x but none

w hich ar e ax ioms , the n som e l in e of x w hich lies in y m us t be an imm ediate conse-

quence of lines of x which lie outside y . T he follow ing, then, is our de finition:

D29. Proof x = { y) | (3 z )(L i ne zx . P a r t zy ) . (u>) (A x iom w . L ine w x . 3~ P a r t wy ) - 3 ( 3 s )(3 ¿ )(3 w )(L i ne sx mp a r t Sy mL jne _ ^ p a r t ty . Line ux -

~ P a r t uy . ICs¿w)).

In order to establish that this definition is adequate to our purposes, we shall

now show (1) tha t i f x is a “ proof” in the sense of D29 , then w e can specify an

order of “ precedence” am ong the lines o f x such th a t every l ine is either a n ax iom

or an imme diate consequence of “ ea rlier” lines ; and we shall also show convers ely

th a t (2) if a: is such th a t an order of precedence of the a bove k ind can be specified

amo ng its lines, the n a; is a “ proof” in the sense of D29 .

(1) is es tablishe d as follow s. Suppose x is a “proof” in the sense of D29.

W e can be g in our s pe cif ication o f an orde r of prece dence am ong the lines of x

by picking o ut, in an ar bitrary order L y , L * , •••, L * , al l thos e lines o f x whichare ax ioms. Ne x t, from among the rem aining lines of x , we pick one, cali it

L k + i, w hich is an immediate consequence of l ines from among L i L * .

(There wil l be such a l ine; for, by D29, that individual y which consists of all

lines of x except L\ , L n , •••, L * m us t co nta in a line w hich is an imm ed iate

consequence of lines of x outside y .) Ne x t, from among the rem aining l ines of x,

we pic k one — cali it L * +2— w hi^h is a n im m e dia te co ns eque nce of lines from

a m o n g L \ , L * , •••, L k+1 . (T here w ill be such a one, for the same reason as

before.) C ontinuing thus, we ev entually specify an order of precedence of the

required kind.(2) is es tablished as follow s. Suppose the lines of x can be counted off in

some order such that each line is an axiom or an immediate consequence of

earl ier l ines. Now consider any thing y w hich co ntains some lines of x bu t none

w hich are ax ioms . F rom among thos e lines of x which are parts of t/, pick out

the one w hich is earlies t according to the assumed order. It mus t be either an

ax iom or an imme diate consequence of ear lier lines o f x . B ut i t is not an ax iom,

for y contains none of the lines of x w hich are ax ioms. Henee it is an immediate

consequence of earlier lines of x\ and those earlier lines are not in y . W e see

therefore that y contains a line of x which is an immediate consequence of linesof x outside y . Since y 'vas taken as any individual containing some lines of x

but none which are axioms, it follows that x is a proof in the sense of D29.

So it is now clear tha t D2 9 , w ithout stipulating any order among l ines, gives

us an adequate versión of ‘proof.’

Note incidentally that D29 abstains even from any requirement that a proof

consist wholly of formulas; the “l ines” of a proof x are indeed formulas, but x



ma y contain also any manner of additional debris w ithout ill effect. Proofs are

no t in general “ inscr iptions,” in the sense of D 5 .If a theorem is any inscription for which there is a proof, then an inscription

is a theorem if and only if it is a l ine of some proof. B ut this form ulation is alittle too narrow. G ive n any inscr iption y for which a proof x exists, it will be

true that for each inscription z that is l ike y , and that lies outside of x , a proof w ill also ex is t, cons is ting for ex ample of z together w ith those lines of x th a t are

not identical w ith y . Henee if y is a theorem a ll such inscriptions lik e i t w ill alsobe theorems. B ut suppose th a t some inscription w which is like y lies e mbedded

w it hin some line t in the proof x, and suppose that no other line like t exists; in

this case there m ay be no proof for w , so tha t some inscriptions like the theore m y ma y not be theorems. T o prev ent this anom aly , we construct our def inition so

tha t a n inscription w ill be a theorem if and only if i t isUke some line of some proof.(‘L ik e ’ has o f course been so defined as to be reflexive.)

D30. Thm x = (3y )(3z ) (P roo f y . Line zy . L ike xz).

W it h th e de finit ion so cons tr ucte d, it follow s tha t a ll im m ediate cons equences

of theorems are theorems. B ut some form ulas ma y s till fa il to qualify as theorems solely because no inscription exists anywhere at any time to stand as a

needed inter me díate line in an otherwise v a lid proof. S uch limita tions w ouldprove aw kw ard if w e had to depend upon the ac cidental ex istence of insc riptions

that are perceptibly marked out against a contrasting backgr ound. B ut wemay rather, as suggested earlier (§2), construe inscriptions as all appropriately

shaped portions of ma tter . T hen the only sy ntactical descriptions that w ill failto hav e ac tual inscriptions answ ering to them w ill be those tha t describe inscrip

tions too long to fit into the whole spatio- temporally ex tended universe. T hislimita tion is hardly l ike ly to prove embarrassing. (If we ever should be handi-

capped by gaps in the proof of an inscription wanted as a theorem, however, we

can strengthen our rules of inference to bridge such gaps; for, the number ofsteps required in a proof depends upon the rules, and the rules we have adopted

can be altered or supplemented considerably w ithout v iolation of nominalisticstandards.)

It may be interesting to observe in passing that the theoretical limitations justconsidered obtain under platonistic syntax as well, if that syntax construes ex-

pressions as shape- classes of inscriptions; for, shapes having no inscriptions asinstances reduce to the nuil class and are thus identical.19 T he platonist ma y

indeed escape the limitations of concrete reality by hypostatizing an infinite

realm of abstract entities—the series of numbers—and then arithmetizing hissy ntax ; the nom inalist, on the other hand, holds t ha t any recourse to platonism

is both intolerable and unnecessary.11. Co nclusión. In our earlier sections we s tudied the p'- oblem of tra ns lating

* •A c c o r d i ng t o t h e c la s s i ca l p r i nc i p ie s o f s y n t a x , a n y t w o e x p r e ss i on s x a n d y h a v e

c o n c a t é n a t e a f y ; a n d m o r e o v e r x~y i s a lw a y s d i s t i n c t f r o m z ' x d , un l e s s t h e ch a r a ct e r s o ccur -

r i n g i n x a n d i n y a r e s u c c e s s i v e l y t h e s a m e a s t h o s e i n z a n d i n w. T h is c o m b i n a t i o n o f

p r i n c i p i e s i s a s u n t e n a b l e f r o m t h e p o i n t o f v i e w o f a p l a t o n i s t i c s y n t a x o f s h a p e - c l a s s e s

a s f r o m t he p o i n t o f v i e w o f n o m i n a li s m .



into nominalistic language certain nonsyntactical sentences which had appeared

to be ex plicable only in platonis tic terms. In §§5- 10 we hav e been concerned w ith g iv ing such a tr ans la tion for sy ntax . T his sy ntax enables us to des cr ibe

and deal with many formulas (of the object language) for which we have nodire ct nominalistic tra nsla tion. For ex ample, the for mula w hich is the fullex pansión in our object language of ‘(n)(n + n = 2n )’ will contain variables

calling for abstract entities as valúes; and if it cannot be translated into nominalistic languag e, it w ill in one sense be meaningless for us. B ut,.tak ing thatformula as a string of marks, we can determine whether it is indeed a proper

for mula of our object lang uage , and w hat consequence- relationships it has toother formulas. W e can thus handle much of classical logic and m athematics

w itho ut in any fur ther sense unders ta nding , or g rant ing the tr uth of , t he formulas we are de aling w ith.

T he ga ins which seem to have accrued to na tura l science fro m the use of mathe-matical formulas do not im ply th a t those formulas are true statements. N o one,

not even the hardiest pragmatist, is likely to regard the beads of an abacus astrue; and our position is that the formulas of platonistic mathematics are, like

the beads of an abacus, convenient computational aids which need involve noquestion of tr uth. W ha t is me aningful and true in the case of platonistic ma thematics as in the case of the abacus is not the appara tus its elf , bu t only the descrip-

tion of it: the rules by w hich it is constructed and run. T hese rules we do under-stand, in the strict sense that we can express them in purely nominalistic language. T he idea tha t classical mathema tics can be regarded as mere apparatus

is not a novel one among nominalistically minded thinke rs ; bu t it can be main-tained only if one can produce, as we have attempted to above, a syntax which is

itself free from platonistic commitments. A t the same time, ever y adv ance we can mak e in f inding dir ect tr ans la tions

for familiar strings of marks will increase the range of the meaningful languageat our command.

Ü N I V E R S I T Y O F P E N N S Y L V A N I A

H A R V A R I J Ü N I V E R S I T Y

Documents

Steps Toward a Constructive Nominalism - Nelson Goodman and W. V. Quine