REFERENCE, VARIABLES A ND THE EMPTY UNIVERSE
Howard G. CALLAWAY
This paper is concerned wit h two imp o rta n t lo g ica l notions.
First wit h 'reference i n Quine's usage, wh e re the variables o f
quantification are said to refer to, o r take as values, objects in
a universe o f discourse. I w i l l p o in t o u t some imp o rta n t i mplications of this notion of reference. I will depend upon exposition to support my interpretation. Second, I w i l l also be concerned wit h 'the empty universe' and the relation between this
notion and 'reference'. Fin a lly, I w i l l argue that this notion o f
reference a llo ws us to dispense wit h the notion o f the e mp ty
universe and that there is a considerable theoretic advantage
in doing so wh ich affects the definition o f va lid ity.
On Quine's vie w, «...the va ria b le o f quantification i s u lt imately to ca rry f u ll re sp o n sib ility f o r objective reference.» (
Throughout
his wo rks Quine maintains that variables o f quan1
tification
re fe r to objects. In «On Wh a t There Is», f o r instance
)
he mentions reference v i a t h e va ria b le s a s a v irt u e o f t h e
theory of descriptions:
the burden o f objective reference wh ich had been p u t
upon the descriptive phrase is n o w taken o ve r b y wo rd s
of the kin d that logicians ca ll bound variables, variables
of qualification, n a me ly words like 'something', 'nothing',
'everything'. Th e se wo rd s, f a r f r o m p u rp o rt in g t o b e
(
editors,
1
Words and Objec tions , rev is ed edit ion (Dordrec ht, Ho lla n d : Reidel,
1975)
) p . 320. Thank s a re due t o Professors Q uine, Leblanc and Nels on f o r
helpful
w
c omments o n t his paper and a near ancestor. Errors remaining a re
my. own.
y
.
Q
U
I
N
E
,
(
(
R
e
p
86
H
O
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A
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D
G. CALLA WAY
names... refer to entities generally, wit h a kin d of studied
ambiguity peculiar to themselves (').
This «studied ambiguity» consists in the variables re fe rrin g to
objects (taking objects as values) although they do not purport
to name, p ick out o r specify objects
. Since the variable is to ca rry «full responsibility» fo r reference, o n ly this reference with o u t specification w i l l be required.
Expressions p u rp o rtin g t o sp e cify p a rticu la r objects (d e fin ite
descriptions and names) re ce ive definitions wh ich a llo w them
to be eliminated f o r the purposes o f th e o ry and reintroduced
for the sake o f practically. O n ly variables re ma in as sin g u la r
terms i n p rimit ive vocabulary (
3 In Th e Roots o f Reference, Qu in e holds th a t quantification
encapsulates
a ll references:
).
Quantification i s a we lco me encapsulation o f th e re ferential apparatus. Once a theory is formulated in quantificational style, its objects of reference can be said simp ly
to be the values of its quantified variables. This o f course
is e xp licit in the intended readings o f the u n ive rsa l and
existential quantifiers: 'e ve ryth in g x is such that', 'something x is su ch that'. A n d th e convenience o f t h is e n capsulation becomes e vid e n t wh e n y o u t r y t o s a y i n
some other wa y wh a t the objects o f a th e o ry are (
4
Not o).n ly are the variables o f quantification re fe rrin g expressions, there is a single answer to the question «what wa s re ferred to ?» That answer is, «the values o f the quantified va ria bles.» Th is a n swe r ma y seen d istu rb in g ly uninformative. But,
if so, w e mu st lo o k a lit t le further. Qu in e re g u la rly answers
questions concerning the referents o f the variables o f a p a rt i(
V ie
2 w (Cambridge, Ha rv a rd Univ ers it y Press, 1961) P. 6 Cl. ibid. , p. 13, a n d
Quine,
)
Th e Roots o f Referenc e (LaSalle, O p e n Court , 1973) pp. 98-101.
Q QUINE, Ma t h e ma t ic a l L o g ic , r e v is e d e d i t i o n (Ca mb rid g e , H a r v a r d
Univ
U ers ity Press, 1951) pp. 146-152.
I QuiNE, Roots o f Reference, p. 100.
N
E
,
«
O
n
W
h
a
t
REFERENCE VARI ABLES A N D THE E MP TY UNIVERSE
8
7
cular theory o r sentence. He gives a more specific a n swe r on
the assumption that the th e o ry o r sentence is true, i f th is assumption is plausible. Reading o ff the ontological commitments
of a theory we determine wh a t the values o f the variables are
purported to be. I f the th e o ry is true the values are wh a t they
are purported to be.
A central argument for this notion of reference without specification is that it provides fo r reference where specification is
not to be had. Sometimes, Quine argues, the objects of a theory
cannot a ll be specified:
For instance, under classical set theory there are, given
any interpreted notation, some re a l numbers that are n o t
separately specifiable in that notation. The existence sentence «there are unspecifiable re a l numbers)) is true, and
expressible as an existential quantification; but the values
of the variable that account fo r the tru th o f this quantification are emphatically not objects wit h names (
5
In context
).
this argument is designed to show the advantage of
the objectual reading of the quantifiers o ve r the substitutional
interpretation. However, this passage also suits the present purpose since it is p re cise ly the unspecified reference o f the (objectual) quantified variable which is required to account for the
truth o f Q u in e s e xa mp le . T h e corresponding su b stitu tio n a l
quantification will not be true since an appropriate substitution
instance cannot be produced.
Reference without specification will seem a ve ry weak notion
to those who bind reference and specification together and assume that a referring expression is one whose essential function
is to p ick out o r specify objects. Howe' ver, as we have already
seen in outline, Qu in e seeks both to p ro vid e f o r specification
where it is wanted and also to provide fo r cases where specification is beyond the power of our notation.
Quines tre a tme n t o f puzzles g iv e n ris e b y non-denoting
QUINE, «Ex is tenc e a n d Quantific ation» i n O nt ologic al Re la t iv it y a n d
Other Essays (Ne w York , Columbia Univ ers it y Press, 1969) p. 95.
88
H
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G. CALLA WAY
names and descriptions is i n th e sp irit o f Russell's t h e o ry o f
descriptions and has a prominent place in his wo rk. We sh a ll
look at this in more detail shortly. His treatment make use o f
the n o tio n o f reference v ia th e va ria b le s and th e d istin ctio n
between reference and purported reference. The importance o f
Quine's procedure wit h these puzzles is t h a t i t accomodates
the in tu itio n that reference is a re la tio n so as to meet the demands o f an extensionalistic interpretation o f relations. Fo r i f
a re la tio n is to be interpreted as a set o f ordered pairs there
can be n o re la tio n t o th e non-existent. Th is is imp o rta n t because emphasis u p o n reference a s a re la tio n has sometimes
brought philosophers t o in tro d u ce special status o n to lo g y o r
complete realms o f being in o rd e r t o deal wit h non-denoting
names and descriptions (
is
6 required me re ly to meet the needs o f a th e o ry o f reference.
Rather
). O th
n e grounds and epistemic status o f an a n swe r to th e
question
referred to ?» are the same as the grounds
Q u i n«what
e ' was
s
vand epistemic
i
e
wstatus o f a n a n swe r t o t h e question « wh a t i s
there
?» Th
n
o is re su lt is t o be welcomed and expected i f a tru e
answer
t
o
s
p
ethe cfo rme
i r question gives p a rt o f the answer to the
latter
question;
i.e.,
if reference is a relation.
a
l
I
will
offer
no
argument
with the broad lines of Quine's vie w
o
n
t
o
of
l reference.
o
g They question I want to raise is this: Does the purported reference of a theory always in vo lve reference ? Quine's
answer is: yes, b u t o n ly so lo n g as th e universe o f discourse
is non-empty. A th e o ry o r sentence ma y p u rp o rt reference to
particular o b je cts o r o b je cts o f so me p a rt icu la r k i n d wh e n
there are no such objects. Th is is to sa y that a th e o ry o r sentence m a y in v o lv e fa lse o n to lo g ica l co mmitme n ts. B u t a re
there cases o f t h is s o rt wh e re t h e re a re n o va lu e s o f t h e
variables ? A re false sentences sometimes false due to reference failure on the part of the quantified variables ? To hold that
(
King
I o f Franc e i s wis e ' denotes o r ref ers t o a n o b je c t wh ic h does n o t
exist,
) p a rt ly o n t he grounds t hat «...there ju s t c annot b e s uc h a t h in g as
a grammat
F
ic al s ubjec t wh ic h does n o t denote (o r ref er) t o any t hing; t h a t
(byo definition) i s what t he s ubjec t t erm o f a sentence does.» See L. Linsky,
Referring
(Ne w Y ork : Humanit ies Press, 1967) p. 87.
r
i
n
s
t
a
n
c
e
,
L
REFERENCE VARI ABLES A N D THE E MP TY UNI VERSE
8
9
they are sometimes false in this wa y interrupts wh a t is o th e rwise an interesting and elegant u n ifo rmity o f semantic theory.
Otherwise we may hold that every sentence has some referents
(those o f the variables in it) and that it is in virtu e o f whether
or n o t the re le va n t predicates a re t ru e o f these objects t h a t
every sentence has it s truth-value. I t is th is u n if o rmit y I w i l l
attempt to establish.
Quines answer to th is la st question is evident in h is tre a tment o f the notion of the e mp ty universe. Wh e re Quine treats
of va lid it y in terms o f set-theoretic semantics, h e fo llo ws th e
standard practice. Schemata are counted va lid if they are tru e
under a ll interpretations in a ll non-empty universes
seem
of the e mp ty universe a ma tte r of
. T htoi make
s
mthe aexclusion
y
logical tru th and thus insure th a t there a re a lwa ys values o f
the variables. Bu t th is is n o t quite Quine's vie w. He regards
the e xclu sio n o f th e e mp t y u n ive rse a ma tte r o f p ra ct ica lly
governed b y the fa ct th a t i n th e usual case «...the u n ive rse
relative to wh ich an argument is ca rrie d out is already kn o wn
or confidently believed not to be empty, so that the fa ilu re o f
a schema in the e mp ty universe is u su a lly nothing against i t
from a practical point of view.» (
7 In Methods o f Logic, a fte r g ivin g th is la st argument, Qu in e
)
provides
a separate test wh ich a llo ws a supplementary check
for va lid it y in the e mp ty universe. Su re ly there is no practical
d ifficu lty i n su ch a n account o f va lid it y. B u t le a vin g aside
questions o f p ra ctica lly we fin d , f ro m a th e o re tica l perspective that the notion of va lid it y is given two distinct definitions
wit h separate tests f o r each. I f we ca n e limin a te th e e mp t y
universe, t h e n o tio n o f v a lid it y and te stin g f o r v a lid it y w i l l
be thereby simplified. Th is objective, together wit h the above
mentioned u n if o rmit y o f reference serves t o mo tiva te th e a ttempt to e limin a te the e mp ty universe. I w i l l urge mo re conclusive considerations i n the f in a l section. O t h e r points mu st
first be developed.
A s imila r simp lifica tio n o f the n o tio n o f va lid it y mig h t b e
(
Wins
7 ton, 1972) p.98.
)
Q
U
I
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E
,
M
e
t
90
H
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A
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G. CA L L A WA Y
achieved b y considering va lid o n ly those schemata wh ich a re
true under all interpretations in e ve ry universe, whether empty
or n o t (')• Schemata excluded o n such a n account o f va lid it y
depend f o r t h e ir tru th upon there being something (o r other)
and mig h t be construed as a p ra ctica l supplement to q u a n tification th e o ry f o r use in considering a n y non-empty universe.
Some w i l l fin d such a proposal mo re appealing than my own.
Perhaps the most appealing argument fo r such an account depends, however, upon holding that lo g ica l tru th s a re a ll analytic, t h a t sentences represented b y schemata su ch a s ( R x)
(Fx y —Fx) are not analytic and thus that our notion of va lid ity
should b e fashioned i n re co g n itio n o f t h is, in clu d in g o n l y
schemata wh ich e q u a lly hold in the e mp ty universe. Th is so rt
of argument has been criticize d b y Qu in e a t length. I d o n o t
propose to repeat his argument here (
this
a alternative is that Quine's crit icism o f this argument fro m
a).n a lyticity
o n ly th e vig o ro u s assertion th a t
M y appears
p o i t on leave
t
logic
i
ndoes n o t legislative regarding existence as a defence o f
this
m ae ltenrn attive
i simp
o nlifica
i tio
n n go f t h e n o tio n o f va lid it y. B u t
from th e present perspective such a n argument is ma n ife stly
question begging. "Whether o r n o t lo g ic legislates re g a rd in g
existence t o th e extent o f te llin g u s th a t something o r o th e r
exists i s th e issue a t stake i n deciding upon a d e fin itio n o f
validity.
Intuitively, a u n ive rse o f discourse co mp rise s t h e o b je cts
with which a discussion o r theory is concerned
sort
. I otf definition
h i n k ist noth quite
a tas cle a r as it is sometimes taken
to
t be.
h Thi e la
s ck o f c la rit y is d if f icu lt t o a vo id . W e h a ve n o
alternative to construing the universe o f discourse o f a th e o ry
(
Domain»
8
i n Selec t ed L o g ic a l Papers ( N e w Y o r k : Ra n d o m Hous e, 1966)
pp.) 220-223.
(Q
of 9
uV ie w, op. cit., pp. 160-65.
i)
Q
N
U
E
Id
N
e
E
s
,c
«
r
M
i
e
b
REFERENCE VARI ABLES A N D THE E MP TY UNI VERSE
9
1
except fro m the perspective provided b y o u r o wn best theory.
But this is a limita tio n since the objects of our theory ma y turn
out t o b e q u ite d iffe re n t f ro m wh a t w e ta ke th e m t o be. A
theory mig h t in vo lve reference t o Venus wit h o u t recognition
of the fa ct th a t Venus is a v e ry massive p h ysica l o b je ct o rbiting the sun. Such a th e o ry mig h t even deny that there a re
such objects. Those wh o kn e w nothing o f modern astronomy
could o n l y d e n y t h a t t h e ir u n ive rse o f discourse contained
such a massive o b je ct although t h e ir th e o ry d id in vo lve re ference t o Venus. Th e o n ly co mmitme n t o f t h e ir th e o ry wa s
perhaps to mo vin g lights in the sky. The d if f icu lt y is that we
cannot g ive th e f in a l w o rd o n wh a t th e objects o f o u r co n cern ma y tu rn out to be. But we should not expect to g ive the
final wo rd here a n y mo re than we should expect t o g ive th e
final wo rd on what there is.
Interpreting a th e o ry involves much concern wit h purported
reference and less concern wit h reference. De cid in g wh e th e r
there are objects as purported b y a theory T ma y in vo lve first
determining its ontological commitments and comparing these
to the ontological commitments of T', deciding between the two
p a rtly on grounds o f economy. We need n o special semantic
provisions i n o rd e r t o in te rp re t a t h e o ry wh ic h re ma in s i n
doubt. To interpret a theory is to say what values its variables
have if the th e o ry is true. S till, since we sometimes decide to
concern ourselves wit h a universe o f ten objects, o r five , o r
one, there ma y seem no reason to exclude consideration of the
empty universe. The proposal to eliminate the e mp ty universe
will meet wit h the fo llo win g so rt o f objection.
To accept t h is p ro p o sa l o n e mu st re je ct consideration o f
discussions w h i c h a re p u re ly hypothetical, e.g., discussions
concerning th e hypothetical inhabitants o f another so la r system and generally any discussion where the non-emptiness o f
the universe of discourse is open to question. I t will be argued
that lo g ic must provide rules o f reasoning wit h application in
all cases whether the universe is e mp ty o r not (1 .
(
0
example,
(Enc ino Ca lif o rn ia : Dic k ens on P ublis hing Co, 1972) p . 199. «Free
) is a n ex tens ion o f c las s ic al quant if ic at ion t h e o ry t hat prov ides p rin logic
S
e
e
,
K
.
L
A
M
B
92
H
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A
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G. CALLA WAY
Wh a t I wa n t t o sh o w f irs t i s th a t th is so rt o f objection is
misconceived. I f ( x ) ( F x x—Fx)' is a lo g ica l t ru t h th is does
not exclude a n y discussion, wh a t e ve r it s sp e cia l vo ca b u la ry
and regardless of our assumptions concerning the denotation of
the non-logical vocabulary. Descriptions and names ma y s t ill
be in te rp re te d a s non-denoting a n d a n y p re d ica te m a y b e
assigned t o t h e n u ll se t except i n th e o b vio u s cases wh e re
logical tru th dictates otherwise. Th e lo g ica l constraints in t ro duced b y e xclu sio n o f t h e e mp t y u n ive rse a re (1 ) t h a t f o r
each predicate e ith e r th e se t t o wh ich i t is assigned is n o n empty o r the complement o f the set is non-empty and (2) ce rtain predicates are assigned to non-empty sets as a ma tte r o f
logic, e.g., ....is self-identical' and .
these
. . . i s are
i dassigned
e n t itoc the
a l universe as a whole.
w I iwill
t consider
h . . some
. ' cases
,
wh ich might appear to demand in s
terpretation
i
n in the
c e mp
e ty universe. Beside answering the above
objection wh a t f o llo ws w i l l a lso se rve t o illu st ra t e t h e a d vantages o f the notion o f reference with o u t specification. Th e
assumption in force is that we want to make use of the relevant
expressions with o u t commitment to objects named o r denoted.
The sentence 'unicorns h a ve ma g ica l p o we rs' ma y b e re n dered as a universal conditional:
(1
)
The sentence
(
is n o t a lo g ica l tru th . B u t i t is true, g ive n th a t
thereVa re n o unicorns. Th e re is n o p ro b le m in te rp re tin g t h is
sentence
in a non-empty universe simp ly because the sentence
D
does M
n o t lo g ica lly imp ly th a t there a re unicorns. Th e re is a
schema
x true fo r all non-empty universes wh ich ma y seem relevant )here.
x)Fx (
Fx
x)
ciples f o r reas oning i n s ituations wh e re t h e objec t s o f o u r dis c ours e a re
either non-ex is tent o r hav e o n ly put at iv e ex is tenc e. S o i t enables o n e t o
measure t h e w o r t h o f reas oning i n f ic t io n a l dis c ours e a s w e l l a s i n
discourse about, say, t he hy pot het ic al ent it ies o f science.»
REFERENCE VARI ABLES A N D THE E MP TY UNI VERSE
9
3
But this me re ly leads us from (1) to an existential quantification
which is no mo re onerous than (1) itself.
(2) ( 3 x)(Ux M X )
Holding (2) t ru e me re ly amounts t o cla imin g (wh a t is e q u ivalent) that there is something x wh ich is e ith e r not a unicorn
or has magical powers. This example sets the pattern fo r other
cases wh ich are sometimes thought more problematic. Wh a t it
serves t o emphasize is th a t accepting 'something exists' as a
logical t ru t h amounts t o 'so me th in g o r o t h e r e xist s ( ( x )
(Fx v ----Fx)). The ontological commitment o f (2) cannot be tie d
down fu rth e r than a disjunction. Since the quantified variables
of (2) do not purport to specify any values and the tru th o f (2)
does not guarantee that a n y o f the values are unicorns o r are
objects w i t h magical powers, we are fre e to assume th a t th e
values o f the variables of (2) are a ll non-unicorns. This is merely t o recapitulate th e reason f o r accepting (1) a s t ru e i n th e
first place.
No w suppose we have a sentence ma kin g use o f a definite
description wh ich ma y seem t o re q u ire in te rp re ta tio n i n th e
empty u n ive rse . ' T h e p re se n t K i n g o f Fra n ce h a s ma g ica l
powers', may, f o r instance be renedered as (3),
(3) M((lx )K x )
This has in turn the fo llo win g expansion provided by the theory
of descriptions.
(4) ( 3 x)(Kx & (( y ) ( ( y = x ) -It is -here th a t the «studied ambiguity» o f the quantified v a riables- provides
K y ) answers
&
t o puzzles in the th e o ry o f reference.
Wh ile
M (4)xpurports
)
reference to the present Kin g of France, we
need postulate n o su ch mo n a rch o r someone w i t h ma g ica l
powers t o in te rp re t (4) o n a n y non-empty universe. Ho ld in g
(4) fa lse , t h e q u a n tifie d va ria b le s re f e r t o va lu e s in clu d in g
something wh ich is not the K in g o f France. Assuming there is
94
H
O
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A
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G. CA L L A W A Y
no present King of France the fo llo win g schema (true fo r e ve ry
non-empty universe) provides th a t answer:
—( 3 x)Fx ( 3 x)—Fx
Reference with o u t specification together wit h the exclusion
of the e mp ty universe also provides a simple, stra ig h tfo rwa rd
answer to the problem of negative existentials. Wh a t the puzzle
rests o n is a semantic in t u it io n th a t a t ru e sentence a lwa ys
involves reference to something. Th is in tu itio n is provided f o r
here. The values of the quantified variables of negative existentials are the same objects as the values o f the variables o f the
positive existentials wh ich fo llo ws fro m them. So the variables
of a sentence of the fo rm —( J x)Fx, if true, re fe r to something
(i.e. e ve ry o b je ct i n th e universe o f discourse) o f wh ich th e
predicate is n o t true. Th is elegant solution is o n ly available if
the empty universe is excluded.
Individual constants re ce ive a s imila r treatment. Wh e t h e r
a constant purports t o name a unique o b je ct o r does not, i t
receives the fo llo win g sort of definition:
'a f o r '(1x)Ax'
There are a va rie ty of ways to understand this Quinean assimilation o f names t o d e fin ite descriptions. W e ma y t a ke t h e
non-denoting names 'Pegasus', fo r instance, to name the unique
object wh ic h pegasizes, i f a n y. A lt e rn a t ive ly w e m a y t a ke
'Pegasus' as naming the unique object o f wh ich the predicate
identical-with-Pegasus' is true, i f any ("). Wh e th e r o r not
a name names anything we ma y st ill make use o f it, reflecting
this consideration in the interpretation. I f a name fails to name
anything then the apparent reference is resolved in to the nonspecific reference of the quantified variables.
What has been shown is that a n y vocabulary ma y be in t e rpreted i n a n o n -e mp ty universe. Ho ld in g a u n ive rse o f d iscourse non-empty introduces n o substantial re stra in ts o n a n
(II) Th is f ormulat ion was s ugges ted t o m e b y s ome remark s o f G ilb e rt
Harman's, i n priv at e correspondence.
REFERENCE VARI ABLES A N D THE EMPTY UNI VERSE
9
5
interpretation. Therefore, excluding the non-empty universe in
an account o f va lid it y puts n o re strictio n o n the language o r
topic o f discussion,
Quine's treatment of names is quite similar, in this respect, to
the treatment rendered in so called (dree logics». This being so,
there is no need here to lo o k in to the comparative advantage
of those free logics which fo llo w standard practice by excluding
the e mp ty universe in th e account o f va lid ity. "Whatever o u r
treatment o f in d ivid u a l constants, n o to p ic o r vo ca b u la ry re quires t h e e mp t y u n ive rse . Wh e t h e r o b je cts p u rp o rte d b y
given expressions e xist o r not, the facts o f the matter ma y be
reflected in an interpretation in a non-empty universe.
An objection ma y be ra ise d a t th is p o in t wh ich is o f some
importance for present purposes. It will be noted that the above
account does a llo w fo r employment o f expressions o f the sorts
illustrated. B u t the interpretation proceeds i n th e knowledge
that there are no unicorns, that Pegasus does n o t e xist and so
forth. B u t we are not a lwa ys in a position to be so confident.
Physics ma y give up on the hypothesis of quasars o r the hypothesis o f neutrinos n e xt we e k (fo r a ll the lo g icia n is required
to kn o w) b u t we cannot ta ke th e course p ro vid e d above i n
interpreting theories wh ich make use o f 'quasar' o r 'neutrino'.
For such theories t e ll us th a t there a re such objects and th e
physicists w i l l continue to insist on the tru th of such cla ims so
long as th e theories re ma in liv e and via b le options. B u t f o llowing the procedures outlined above t o e limin a te the e mp ty
universe sentences such as 'a is a Quasar o r 'the unique object wh ich caused tra ck T is a neutrino' must be counted false.
The elimination o f the e mp ty universe is thus much too co stly
since one also eliminates the lo g ic needed fo r important areas
of scientific discourse. Simila r points could be made concerning
the evaluation o f inferences made in fictio n a l narrative.
Again I want to urge that this criticism is misconceived. The
point o f the procedures outlined above is to show that the use
of a predicate, description or name in a theory need not in vo lve
the theory in commitment to objects described o r named b y the
expression i n question. Th is is so even wh e n the universe o f
discourse is held non-empty. No w the question concerns theo-
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ries w h i c h d o in v o lv e e xiste n tia l co mmitme n ts, th o u g h t h e
existence o f objects as called fo r b y the theory is le ft in doubt.
There is n o special p ro b le m here. W e need o n ly distinguish
reference and purported reference. A theory ma y have existential implications whether o r not the existentials are true. Constructing a model f o r such a th e o ry these existential co mmit ments w i l l be reflected
. W eo n ly afo rrtheories
models
e
wh ich are in fact true. A model does
n co
o mmit
t us t o a n y cla ims regarding referents o f a th e o ry;
not
crather
o nit exhibits
s t r wh
a a t the referents are, i f the th e o ry is true.
i n e in d
Claims
v o lv in g su ch predicates a s '...names...', t r u e
t
of...'
and so
o fo rth are me re ly hypothetical.
c Though
o
na th eso ry T is n o t true, o n e s t ill ma ke use o f th e
t
r
u o f cthe th e o ry o f reference in wo rkin g through a n
vocabulary
tinterpretation. Su ch cla ims se rve t o in d ica te t h e p u rp o rte d
reference o f expressions o f the object language on a given in terpretation. A schema is va lid if it turns out tru e on each in terpretation is every non-empty universe. Thus on this account
a schema is va lid no ma tte r wh a t exists. Excluding the e mp ty
universe involves no assumptions concerning what exists. The
va lid schemata are those wh ich remain true regardless o f what
we take the purported reference o f the object language to be.
This is a powerful pragmatic consideration fa vo rin g a u n ifo rm
construal o f va lid it y excluding the e mp ty universe.
It st ill mig h t be urged that 'something exists' is n o t a lo g ica l
truth, even if this does n o t introduce substantial restraints o n
the interpretation of non-logical vocabulary. The in tu itio n ma y
yet remain. But this is not a sufficient objection in this context.
Even i f elaborated b y expressions o f concern f o r the «purity»
of logic, such an objection me re ly begs the question. Fo r wh a t
is in question is wh a t to count as a logical truth. On the other
hand th e re i s n o p u re ly lo g ica l o b je ctio n t o t h e v i e w t h a t
'something exists' is not a logical tru th wh ich w i l l not e q u a lly
beg the question. Such a standoff on lo g ica l grounds serves to
REFERENCE VARI ABLES A N D THE E MP TY UNI VERSE
9
7
direct a tte n tio n t o p ra g ma tic considerations a n d syste ma tic
philosophical considerations. I f the question is to be settled it
will n o t b e settled o n p u re ly lo g ica l grounds. Some readers
may th in k that the question cannot be settled. In vie w o f these
points I will advance some fu rth e r considerations o f a different
character.
Let us suppose a th e o ry T has been sh o wn false. No n e o f
the predicates a re t ru e o f anything; a l l t h e in d ivid u a l co n stants and definite descriptions are non-denoting. M y argument
is th a t even in th is extreme case w e do n o t kn o w enough t o
identify the universe o f discourse wit h the n u ll set. Rather we
must a llo w that the variables o f T do have e xistin g objects as
values.
In some such cases i t ma y be quite e vid e n t that th e quantified va ria b le s d o n o t f a il o f reference e ve n i f none o f th e
non-logical predicates a re t ru e o f anything. W e m a y k n o w
that th e variables d o n o t f a il o f reference i n virt u e o f b a ckground information. Concerning a ll cases generally, m y p o in t
is that one can never show that new information w i l l never be
had sufficient t o convince us th a t the variables o f T take as
values th e objects o f another th e o ry T' (wh ich mig h t o n ly be
formulated long after T has been disregarded). Here it is cru cia l
to keep in mind that two theories T and T' ma y in vo lve reference t o th e same objects although t h e p u rp o rte d reference o r
ontological commitments o f th e t wo theories a re t o t a lly d if ferent. Thus even if the non-logical predicates o f T are a ll n u ll
this never serves to guarantee the universe of discourse empty.
The exclusion o f the e mp ty universe is then ju stifie d i n th a t
this procedure a llo ws f o r su ch e p iste mic possibilities. I f w e
assume the universe o f discourse e mp ty on the evidence th a t
the non-logical predicates o f T are a ll n u ll, th is is t o ru le o u t
such possibilities and h o ld T irre le va n t t o a n y th e o ry whose
universe is not empty. But logical procedures are not sufficient
for judgments regarding relevance.
T mig h t be a th e o ry whose non-logical vo ca b u la ry includes
only predicates such as
a
witch ',
a
sorcerer', '...has
magical p o we rs' a n d o th e rs w i t h n u l l extension. A lt h o u g h
there a re n o witches, th e re we re (and i n some cu ltu re s s t ill
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are) people held to be witches. Alth o u g h no me re analysis o f
the t h e o ry T w i l l sh o w th a t ce rta in people a re regarded a s
witches a n d t h a t these people a re among th e va lu e s o f th e
variables o f T, we mig h t kn o w th a t T wa s formulated t o a ccount for the activities of a certain group of people. No analysis
of T will show that these people are not values of its quantified
variables unless we assume th a t th e universe o f discourse is
empty on the grounds th a t there a re no witch e s o r sorcerers
and so forth. I n vie w o f this and given the notion o f reference
without specification th e re seems n o reason t o h o ld th e u n iverse o f T e mp ty and thus f l y i n the face o f common sense.
From o u r perspective i t is evident that the th e o ry T does in volve reference to ce rta in people regardless o f the purported
reference o f T and regardless of what the witch-theorists mig h t
hold.
In general if 'F' is a predicate o f T wh ich is tru e o f nothing,
there is no guarantee th a t background in fo rma tio n w i l l n e ve r
be p ro vid e d su fficie n t t o sh o w t h a t F s a re n o th in g b u t G s
(where ' G i s a predicate o f n o n -e mp ty extension o f a t ru e
theory T'). But in order to make sense of this so rt o f cla im we
must suppose th a t T and T' in vo lve reference to the same objects although the purported reference o r ontological co mmitments of T are to ta lly different fro m those of T. To assume the
universe o f discourse o f T e mp ty on the evidence that its predicates are a ll n u ll is to assume that T is re fe re n tia lly incommensurate a n d th u s t o t a lly irre le va n t t o a n y t ru e t h e o ry T '
with a non-empty universe o f discourse. Such an assumption
is never justified.
M y conclusion then is th a t purported reference ca n n e ve r
safely b e regarded a s in vo lvin g n o reference a n d t h a t o u r
definition o f v a lid it y should e xclu d e t h e e mp t y u n ive rse i n
recognition o f th is point. I a m therefore q u ite willin g t o re gard 'something exists' as a lo g ica l tru th .
Drexel Un ive rsit y Flo rid a State Un ive rsit y
H . G. CALLAWAY