INTENSIONAL ISOMORPHISM
David E. COOPER
Several philosophers h a v e suggested in te n sio n a l is o mo rphism as a necessary and sufficient condition o f synonymy. (
IOne mo t ive b e h in d t h is suggestion h a s been t h e d e sire t o
)
provide
a co n d itio n stro n g enough t o guarantee interchanga b ility salva veritate in opaque contexts, e.g. belief-sentences.
The co n ce p t h a s b e e n discussed a lmo st e n t ire ly f ro m t h is
angle. Does it , o r does i t not, guarantee in te rch a n g a b ility i n
opaque contexts ? L e wis, Carnap, a n d Pu tn a m h a ve sa id i t
can. Mates, Scheffler, and Church have said that it cannot. (
2 i s unfortunate, I fe e l, t h a t in te n sio n a l iso mo rp h ism h a s
It
been
discussed o n ly f ro m th is angle. Fo r i t b lin d s u s t o th e
)
facts (a) th a t there are o th e r reasons f o r in sistin g upon such
a co ndition f o r syn o n ymy, and (b ) t h a t there ma y b e o b je ctions to it other than that it fails to guarantee interchangability
in opaque contexts. A t a n y rate, I a m n o t going t o discuss i t
from th is angle, b o th f o r th e reasons I h a ve mentioned a n d
others. Fo r one thing, I doubt if much more can be said about
intensional iso mo rp h ism v is -a -v is o p a q u e co n te xts. M o r e
important, I believe that interchangability in opaque contexts
is irre le va n t t o t h e question o f syn o n ymy. Th a t is, I t h in k
it possible t o p ro vid e a crite rio n o f syn o n ymy wh ich d istin guishes it from both extensional equivalence and L-equivalence
without appealing t o in te rch a n g a b ility i n opaque contexts.
(1) s ee C. I. LEWIS, T h e Modes o f Meaning". P h il. a n d Phen. Research.
1943-44., a n d R . CARNAP, M e a n i n g a n d Nec es s it y . (Ch ic a g o . 1967. 5 t h
impression),
(
ysis
7 1954: B . MATES, "S y n o n y mit y " i n Semant ic s a n d t h e P hilos ophy o f
Language.
)
e d . L in s k y ( I llin o is 1952). 1 . SCHEFFLER, O n S y n o n y my a n d
Indirec
t Dis c ours e". P h il. o f Sc ienc e. 1955: A . CHURCH, "I n t e n s io n a l I s o H
morphis
m and Identity of Belief". Phil. Studies. 1954.
.
P
U
T
N
A
M
,
"
S
y
n
o
n
748
D
A
V
I
D
E. COOPER
I d o n o t h e re defend t h is contentious cla im. M y a ims h e re
are purely negative.
Apart f ro m the in trin sic in te re st o f discovering a crit e rio n
of syn o n ymy, t h e re i s a f u rt h e r reason w h y discussion o f
intensional is o mo rp h is m i s v a lu a b le . P h ilo so p h e rs h a v e
claimed t h a t lo g ica l t ru t h s a re t ru e i n v irt u e o f syn o n ymy
relations h o ld in g between constants. O n t h is v i e w " S " i s a
logical tru th if it can be reduced to an instance o f the la w o f
non-contradiction o n ce syn o n yms re p la ce syn o n yms. (
I
example,
(p Vq) ( — p & — q) w i l l be a logical tru th because
antecedent
) F o r and consequent are synonymous. Th is vie w, h o wever, mu s t b e fa lse i f in te n sio n a l iso mo rp h ism i s ma d e a
condition f o r syn o n ymy, s i n c e t h e f o rmu la e ( p Vq) a n d
—— p & — q) a re n o t iso mo rp h ic. T o le a ve o p e n t h e v i e w
that lo g ica l tru th s a re such in virt u e o f synonyms, then, i t is
necessary t o re p u d ia te t h e suggestion t h a t in te n sio n a l is o morphism be made a condition f o r synonymy.
I a m g o in g t o argue t h a t intensional iso mo rp h ism sh o u ld
not be made a condition f o r syn o n ymy. Th e results o f doing
so wo u ld b e t o o co u n te r-in tu itive t o swa llo w. Wh a t , q u ite ,
is intentional isomorphism ? I shall consider Carnap's account,
and, although other accounts d iffe r fro m his, I th in k my objections t o h is account wo u ld also be objections to these others.
Carnap's f o rma l d e fin itio n o f in te n sio n a l iso mo rp h ism ru n s
as follows:
"a. L e t t wo designator ma trice s b e g ive n , e it h e r i n t h e
same o r i n t wo d iffe re n t semantical systems, su ch t h a t
neither o f t h e m contains a n o th e r designator ma t rix a s
proper part. Th e y a re in te n sio n a lly iso mo rp h ic — Df t h e y
are L-equivalent.
b. L e t t wo compound designator matrices be given, each
of them consisting o f one ma in su b ma trix (o f the typ e o f
(I) I t s hould n o t b e t hought a k noc k down objec t ion t hat , o n t h is v ie w,
the l a w o f non-c ontradic tion c annot it s elf b e t ru e i n v irt u e o f s y nony my
relations. I t m ig h t b e argued t h a t t h is lo g ic a l t r u t h i s q u it e s pec ial o n
v arious o t h e r grounds . See, f o r ex ample, P . STRAWSON, I n t ro . t o L o g ic a l
Theory.
I NTENSI O NAL I SO MO RPHI SM
7
4
9
a p re d ica to r, f u n ct o r, o r co n n e ctive ) a n d n a rg u me n t
expressions (and possibly a u xilia ry signs like parentheses
commas, e tc.). T h e t w o ma trice s a re in te n sio n a lly is o morphic = Df (1) th e t wo ma in submatrices are intensiona lly isomorphic, and (2) f o r a n y m f ro m 1 t o n , th e mt h
argument e xp re ssio n w i t h i n t h e f i r s t m a t r i x i s in t e n sionally isomorphic to the mt h in the second ma t rix ('the
mth' refers to the order in wh ich the argument expressions
occur in the matrix.)
c. L e t t wo compound designator matrices be given, each
of them consisting of an operator (universal o r existential
quantifier, abstraction operator, o r d e scrip tio n operator)
and i t s scope, wh ic h i s a designator ma t rix. T h e t w o
matrices a re in te n sio n a lly iso mo rp h ic = Df (1 ) t h e t w o
scopes a n d in te n sio n a lly iso mo rp h ic w i t h re sp e ct t o a
certain co rre la tio n o f variables o ccu rin g in them, (2 ) t h e
two o p e ra to rs a re L -e q u iva le n t a n d co n ta in co rre la te d
variables." (')
In le ss co mp lica te d te rms, a n d ig n o rin g t h e question o f
quantification, wh ic h need n o t concern us, Ca rn a p is sa yin g
this: F o r t wo expressions — wo rd s, phrases, o r wh o le se n tences — t o b e in te n sio n a lly isomorphic, i t is necessary (a )
that the t wo be L-equivalent, and (b) tha t th e y be stru ctu ra lly
similar, both grammatically, and in that f o r e ve ry component
expression i n the f irst th e re e xists a corresponding L --e q u ivalent component expression i n t h e second. T h e co n d itio n s
are jo in t ly sufficient. He says, f o r example, th a t " 2 4
-" Il 5su" m Va " n
a redin te n sio n a lly iso mo rp h ic "because t h e y n o t
only are L-equivalent as a whole, both being L-equivalent t o
7", but consist of three parts in such a wa y that corresponding
parts a re L-equivalent t o o n e another, a n d hence h a ve t h e
same intension". (
2(II, V ) I I I ) " a r e n o t iso mo rp h ic sin ce , a lth o u g h t h e y a r e
L-equivalent
as a whole, th e second contains expressions f o r
) O n
t h e
o (') tMe ahn in geandrNecessity. op. cit. p. 59.
h( a
n
2
d
,
" )
i
7b
>i
3d
" .
ap
.
n5
d6
750
D
A
V
I
D
E. COOPER
which there are no L-equivalent corresponding components in
the first.
According to Carnap, t wo expressions wh ich are isomorphic
in the above wa y are synonymous, and there are no synonyms
which are not isomorphic. He says
" It h a s o f t e n b e e n n o t ice d b y lo g icia n s t h a t f o r t h e
explication o f ce rt a in cu st o ma ry co n ce p ts a st ro n g e r
meaning re la tio n th a n id e n t it y o f intension seems t o b e
required. But usually this stronger relation is n o t defined.
It seems t h a t i n ma n y cases t h e re la tio n o f intensional
isomorphism could be used." (')
"Synonymity i s e xp lica t e d b y in te n sio n a l is o mo rphism". (
2
) reason fo r this cla im, as we have seen, is th a t Carnap
One
is searching f o r a condition wh ich w i l l guarantee interchanga b ility in opaque contexts — for, according to him, synonyms
must b e so interchangable. W i t h t h is aspect I a m n o t co n cerned. Ho we ve r, f o llo win g L e wis, Ca rn a p e mp lo ys a q u ite
separate a rg u me n t. T h e expressions ' ' ro u n d e xcisio n " a n d
circu la r hole" are L-equivalent. So are the expressions " e q u ilateral tria n g le " and "equiangular triangle". In tu itive ly, h o wever, t h e la t t e r p a ir a re n o t synonymous i n t h e ma n n e r o f
the f irst p a ir. W h y ? Because although " e q u ila te ra l tria n g le "
and "equiangular tria n g le " a re L-equivalent as a wh o le , t h e
component corresponding expressions, "equilateral" and "equiangular" a re not. Th a t is, th e t wo compound expressions a re
not in te n sio n a lly iso mo rp h ic. I n general, i t i s cla ime d , w e
may o f t e n co me across L -e q u iva le n t compound expressions
which, in t u it ive ly, a re n o t synonymous. Wh e re t h is i s so ,
it w i l l b e fo u n d t h a t t h e expressions a re n o t in te n sio n a lly
isomorphic. N o w n o d o u b t Carnap, a n d L e wis, a re rig h t t o
say that " circu la r hole" and "round excision" are synonymous
in a w a y i n wh ic h " e q u ila te ra l t ria n g le " a n d " e q u ia n g u la r
(
(1
)
2
)o
ip
.
b
ic
i
d
.t
.
p
.p
.
6
I NTENSI O NAL I S O MO RP HI S M
7
5
1
triangle" a re n o t. B u t t h e y misin te rp re t t h e sig n ifica n ce o f
such examples. Such examples d o n o t, despite appearances,
point towards a need fo r intensional isomorphism as a general
condition f o r syn o n ymy. I s h a ll re t u rn t o t h is p o in t a f t e r
levelling some objections against th e suggested condition.
The f irst objection is as fo llo ws: the suggested crite rio n f o r
synonymy cannot cover the favourite paradigms o f synonymy.
If one looks closely at Carnap's fo rma l definition of intensional
isomorphism, i t can be seen th a t no place is p ro vid e d f o r the
synonymy o f a n e le me n t a ry e xp re ssio n w i t h a c o mp le x
expression. Clause (a.) o f th a t d e fin itio n deals o n ly wit h the
synonymy o f t wo elementary expressions (i.e. those ma trice s
having n o p ro p e r p a rt). Clauses (b.) a n d (c.) d e a l o n ly wit h
the syn o n ymy o f two complex expressions. Yet the paradigms
of syn o n ymy are those cases wh e re an elementary expression
is said t o be synonymous wit h a co mp le x one. Fo r example,
"bachelor" — " u n ma rrie d ma n " , " lio n e ss" — " fe ma le lio n " ,
"brother" — " ma le sibling" . A n y account wh ich can have no
place f o r su ch cases o f syn o n ymy i s s u re ly unacceptable.
C. I. Lewis' account o f intensional isomorphism does a llo w f o r
the syn o n ymy o f a n e le me n ta ry expression w i t h a co mp le x
one, a n d so h e a d mits th e re is a n exception t o t h e general
condition o f isomorphism. Th e tro u b le h e re is: h o w a re w e
to re a ct t o a generalization a b o u t s y n o n y my w h i c h mu s t
be provided wit h an escape clause f o r the most common typ e
of synonyms ? First, i t makes the th e o ry lo o k ad hoc. Second,
it ma ke s o n e suspect t h a t in te n sio n a l iso mo rp h ism i s it s e lf
an e xce p tio n a l co n d itio n wh ic h syn o n yms mu s t me e t o n l y
in sp e cia l cases. I sh a ll sh o rt ly t r y t o sh o w t h a t examples
such as those concerning the triangles are indeed special, and
should not tempt us into any general demand fo r isomorphism.
A second o b je ctio n i s t h is; i t i s o fte n t h e case, i n mo st
languages, t h a t t h e re e x is t d if f e re n t g ra mma t ica l w a y s o f
saying the same thing: alternative grammatical devices wh ich
do not, in tu itive ly, a lte r the meaning o f wh a t is said. A good
example o f this occurs, q u ite commonly, in French. I t is v e ry
often th e p ra ctice i n t h a t language t o f o rm a d je ctive s f ro m
place-names, e.g. "alsacien", "marseillaise", 'lyo n n a is" . Wh e re
752
D
A
V
I
D
E. COOPER
this is done, i t seems imma te ria l wh e th e r we describe something b y using th e adjective, o r describe i t as b e in g " f ro m"
or " o f " the place in question. Fo r example, we can say e ith e r
" il e st alsacien" o r " i l e st d'Alsace"; e it h e r " u n e mè re ma rseillaise" o r "une mè re de Ma rse ille " . (Sometimes, o f course,
the g ra mma tica l d iffe re n ce ma rks a d iffe re n ce i n meaning.
"Pommes d e t e rre lyonnaises" a re n o t simp ly potatoes f ro m
Lyon). I n t u it iv e ly , I a m su re , t h e a b o ve p a irs w o u l d b e
marked a s synonymous. Ye t, cle a rly, t h e y a re n o t in te n sio nally iso mo rp h ic. " I l e s t d 'A lsa ce " co n ta in s a co mp o n e n t
expression, " d (e )" , wh ic h has n o corresponding L-equivalent
component i n " i l e s t a lsa cie n " . T o t a k e a n o t h e r t y p e o f
example th o u g h th is t ime a mo re contentious one: i t seems
often the case th a t passive forms are synonymous wit h a ctive
forms. Yet, cle a rly, actives and passives a re n o t in te n sio n a lly
isomorphic by Carnap's criterion.
A t h ird o b je ctio n i s t h is: th e re a re examples o f co mp le x
expressions wh ich a re L-equivalent, synonymous, stru ctu ra lly
simila r in grammatical terms, b u t wh ich are n o t intensionally
isomorphic. F o r example, t h e p a i r " illu st ra t ive sch e ma " —
"schematic illu st ra t io n " a re s u re ly synonymous. Mo re o ve r,
they a re stru ctu ra lly simila r in grammatical terms, since each
contains a n a d je ctive a n d a n o u n . Ho we ve r, t h e y a re n o t
intensionally iso mo rp h ic, f o r ( l ) t h e t w o adjectives, w h i l s t
grammatically simila r, are not L-equivalent, and (2) th e adjective in the first, a n d the noun in the second, wh ils t th e y a re
related i n meaning, a re n o t g ra mma tica lly iso mo rp h ic. Th a t
is, " illu stra tive " and "schematic" a re n o t even a like in meaning; a n d " illu st ra t ive " a n d " illu stra tio n " a re n o t st ru ct u ra lly
similar. Th is is b y no means an isolated example. A common
device i n ma n y languages i s t o t a ke a co mp le x expression
"a b " a n d t o tra n sfo rm th is in t o a n expression synonymous
with it b y transforming the adjective, " a " , in to its noun form,
and t h e n o u n , " b " , i n t o i t s a d je ct iva l f o rm. F o r e xa mp le ,
"professorial dean" " d e c a n a l professor", " fo o lish ma le " —
"male fool". (Th is is not to suggest that e ve ry such transformation ratains the meaning o f the o rig in a l. " A t h le t ic fo o l" and
"foolish a th le te " d o n o t me a n t h e same. A n a th le te i s n o t
I NTENSI O NAL I SO MO RPHI SM
7
5
3
simp ly o n e wh o i s a th le tic. M y a th le tic d o ct o r i s a d o cto r
and not an athlete).
No d o u b t t h is l i s t o f counter-examples, o f va ryin g types,
could be extended. But we need not do so. We can see that any
general demand f o r intensional iso mo rp h ism a s a co n d itio n
for syn o n ymy i s unacceptable. Wh a t , though, c a n w e s a y
of t h e e xa mp le s h o win g t h e d if f e re n ce b e t we e n " ro u n d
excision" — •
'''equiangular
c i r c u l a r triangle", wh ich tempted us in to demanding the
h o l e "crit e rio n ? We ll, Carnap a n d L e wis misconstrue
isomorphism
what
should fo llo w fro m the non-synonymy o f the la tte r pair.
,
Wh
a
t
does
n o t f o llo w i s e it h e r (1 ) t h a t co mp le x syn o n yms
a
n
must
be stru ctu ra lly identical, o r (2) th a t corresponding comd
ponents
" e q i n uco mp le x syn o n yms mu s t th e mse lve s b e s y n o inymous.
l a At l l t h a t d o e s f o l l o w i s a mu c h mo re re st rict e d
conclusion,
e r a l namely: f o r t wo co mp le x expressions o f id e n tica l
st ru ct u re t o b e synonymous, i t i s n e ce ssa ry
t(grammatical)
r
i
for
a t wo
n corresponding g ra mma tica l components t o b e syn o nymous,
g
l i f a n d o n ly i f a l l o t h e r corresponding components
are
synonymous.
Th a t is: suppose " a b " and " c d " a re syn o e
"
nymous.
A
n
d
suppose,
to o , t h a t " a " a n d " c' ' a re synonyms.
—
In th a t case, b u t o n ly in that case, " b a n d " d " mu st also be
synonymous. "Equilateral triangle" and "equiangular triangle"
are st ru ct u ra lly (g ra mma tica lly) simila r, a n d t h e t w o nouns
are, o f course, synonymous. Therefore, i f th e t wo complexes
are to be synonymous, then the t wo adjectives mu st be synonyms — which, in th is case, th e y are not. Cle a rly, then, Ca rnap's e xa mp le w a s a sp e cia l o n e . Wh e re t w o co mp le xe s
are n o t g ra mma tica lly simila r, o r wh e re t h e y a re g ra mma tica lly simila r but the corresponding components are not L-equivalent, th e re i s n o reason t o in sist u p o n intensional iso mo rphism. " Un e mè re marseillaise" and "une mè re de Ma rse ille "
are syn o n ymo u s b u t n o t st ru ct u ra lly s imila r. " P ro fe sso ria l
dean" a n d "decanal professor" a re st ru ct u ra lly simila r, b u t
none o f th e corresponding components a re L-equivalent; y e t
they a re synonymous. So, i n these cases, t h e conditions i n
which intensional isomorphism can be demanded do not exist.
Those conditions a re special, and should p o in t t o n o general
754
D
A
V
I
D
E. COOPER
demand f o r isomorphism. Let me g ive an analogy. Two maps,
in o rd e r to be accurate maps o f an area, need n o t be d ra wn
to the same scale, o r e mp lo y the same symbols. Ho we ve r, i f
you t a ke t wo maps b o th d ra wn t o th e same scale, a n d e mploying t h e sa me symb o ls, th e n , n o doubt, a g ive n b i t o f
the one ma p mu st be id e n tica l wit h a g ive n b it o f the other
map i f both a re t o b e accurate representations o f th e area.
Similarly, t wo expressions need n o t be in te n sio n a lly iso mo rphic to mean the same. However, i f yo u take t wo expressions
"a...n" and "a'...n' " such that each component in the sequence
"a...n — 1" i s synonymous w i t h a corresponding component
in t h e sequence " a '. . . n — 1
—
must
b e synonymous i f th e t wo complexes a re t o b e syn o nymous
, t h easna ,whole.n
o
d Let
o ususeebh o w
t these
,
considerations a p p ly to the syn o n ymy
or
" o th enrwise o" f lo g ica l constants. (
Ia
that
constants
ma
n
d y be synonymous wit h one another is th is:
)" gTicahlly
lo
r ee q ui iva le n t fo rmu la e co n ta in in g d iffe re n t constants
o
are
j e
ctucra lly
t i do
issimila
n
r, th a t is, t h e y a re n o t in te n sio n a lly
— bstru
isomorphic.
I n fact, Carnap does a llo w f o r a ce rta in typ e o f
t
o
synonymy
o ld between constants — b u t th is is
s
a
yre la itio n nt o h g
only where we are dealing wit h a p a ir of elementary expressions, o r a p a ir o f co mp le x ones. Th is, i n effect, means t h a t
Carnap o n l y a d mits o f syn o n ymy re la tio n s t o e x is t across
different notations. He says, f o r example, t h a t we ca n t re a t
V " and " A " as synonymous. Presumably we ca n also tre a t
Apq a n d p V — q a s synonymous. Cle a rly, though, Ca rn a p 's
criterion ca n n o t a llo w f o r t h e syn o n ymy o f " V" , i n so me
construction, w i t h " " a n d " & " i n so me construction. F o r
p V q and — (—p & q ) a re not isomorphic. Th e la tte r fo rmu la
contains components w h i c h a re n o t L -e q u iva le n t t o co rre sponding components i n th e fo rme r. H i s reason f o r d e n yin g
synonymy here wo u ld be the same as f o r denying syn o n ymy
in the case of "7 > 3" and " Gr(su m(II,V)III)" .
Surely, h o we ve r, t o d e n y t h a t t w o st ru ct u ra lly d issimila r
(I) S t ric t ly s peak ing, i t is n o t c ons tants t h a t c an b e s y nony mous , but ,
rather, f ormulae c ont aining them. This is a res ult o f it being impos s ible t o
give other than c ontex tual definitions o f the constants.
I NTENSI O NAL I SO MO RPHI SM
7
5
5
formulae c a n e v e r b e syn o n ymo u s me e t s w i t h a g la rin g
exception. T h e co n sta n t "<--->" w a s in ve n t e d t o a b b re via te
certain fo rmu la e o th e rwise re q u irin g th e constants " -->" a n d
"&" f o r th e ir expression. Fe w logicians, wh a te ve r t h e ir vie ws
on syn o n ymy i n general, wo u ld d e n y th a t p q means t h e
same as (p --> q) & (q -> p). I a m sure Carnap wo u ld n o t d e n y
it — yet, i f we take h im lite ra lly, h is crite rio n f o r syn o n ymy
implies su ch a denial. L e t u s wa iv e t h is p o in t, a n d assume
that such abbreviations ca n b e d e a lt w i t h a s sp e cia l cases.
The ma in point is this: we have seen that, in general, there
is n o reason t o demand intensional iso mo rp h ism as a co n d ition f o r the syn o n ymy o f co mp le x expressions. I can see n o
reason w h y t h is co n clu sio n needs amendment because t h e
expressions in v o lv e d h a p p e n t o in clu d e lo g ic a l constants.
Now, w e d id a d mit th a t th e re we re ce rta in cases i n wh ich
corresponding components mu st themselves b e syn o n ymo u s
if the expressions as a wh o le we re t o be synonymous. Ho wever, no such cases can arise wit h in a single lo g ica l notation.
They ca n o n ly a rise across notations. F o r example, i f " A "
and " V " a re synonymous, a n d i f ApE)
nymous,
. a n d t h e n p" - " a n dV " - " mu s t b e synonymous like wise .
This
is e xa ctly
q parallel to the demand we made in the case o f
"equilateral
t
a
r
e ria n g le " a n d " e q u ia n g u la r tria n g le " . B u t, a s I
said,
cases
s
yn o such
n
o
-ca n a rise wit h in a notation, p - p a n d
( - p V q) a re L-equivalent, b u t sin ce t h e y a re st ru ct u ra lly
dissimilar, i t is n o t a case wh e re intensional isomorphism can
be demanded a s a co n d itio n f o r syn o n ymy. Th e syn o n ymy
of these t wo formulae, i f th e re b e one, wo u ld p a ra lle l t h a t
between " i l est alsacien" and " i l est d'Alsace".
We c a n see, th e n , t h a t t h e d e ma n d f o r in te n sio n a l is o morphism i n th e f ie ld o f lo g ica l constants, i f t h e y a re t o b e
synonymous, is e ith e r mista ke n o r irre le va n t. I t is mista ke n
if the demand is f o r stru ctu ra l id e n t it y between formulae: i t
is irre le va n t i f t h e d e ma n d i s f o r L-equivalence b e twe e n
corresponding components o f stru ctu ra lly simila r formulae —
since, wit h in a notation, th e L-equivalent formulae n e ve r w i l l
be stru ctu ra lly simila r. (Except in th e t riv ia l case wh e re th e
two fo rmu la e are identical). I am not saying that L-equivalent
756
D
A
V
I
D
E. COOPER
formulae i n lo g ic a re synonymous. I cla im, o n ly , t h a t t h e
demand f o r intensional iso mo rp h ism ca n n o t sh o w t h e y a re
not, f o r that is a mistaken demand. I t is mistaken both in the
field o f lo g ic, and, mo re wid e ly i n th e f ie ld o f language a t
large.
Un ive rsity o f Mi a mi
D
a
v
i
d
E. COOPER