INDEFINITE DESCRIPTIONS
Graham PRIEST
1) In tro d u ctio n
Unlike ma n y sentences t h a t a re independent o f Z. F. (f o r
example, t h e G. C. H. o r va rio u s la rg e ca rd in a l a xio ms) t h e
axiom o f ch o ice i s o b vio u sly t ru e . Co n sid e r t h e f o llo win g
reasoning: G ive n a n y non-empty se t o f non-empty sets x , i f
we map e ve ry member o f x, say y, onto a member o f y, then
this mapping w i l l be a choice function on x. (I t is t o t a lly irre levant that there is in general no «effective» wa y of doing this.
Mathematics is not concerned so le ly wit h the «effective»).
The p ro b le m n o w arises: ca n we fo rma lize th e above re a soning in a natural way? Clearly, to do this we must have some
formal equivalent of the phrase 'a (particular) x wit h such and
such a property'. In other words we need a theory of indefinite
descriptions, i. e . t h e in d e fin ite a rt icle . T h e purpose o f t h is
paper is t o establish such a theory. I w i l l attack the p ro b le m
by t ry in g t o establish th e co rre ct semantics f o r su ch te rms.
Inevitably th is w i l l in vo lve us t o a ce rta in extent in th e o ld
problem of h o w to handle denotationless singular terms.
2) I n f o rma l Analysis
Perhaps th e f irs t p o in t t o co n sid e r is wh e th e r su ch t e rms
need special semantics. A f t e r a ll Russell's t h e o ry o f descriptions ma y be seen as a cla im that definite descriptions need no
special semantics since sentences containing definite d e scrip tions cen be paraphrased a wa y in terms o f ones not containing
them. I s i t possible t o f in d su ch a paraphrase f o r in d e fin ite
descriptions ? The answer to this is «Almost certainly, no». The
reason is as follows.
As i s w e l l kn o wn , i f w e a d d Hilb e rt 's E -o p e ra to r t o t h e
language o f set theory and add the a xio m
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(3y)yEx—> ( Ey ) ( y x ) G x
(
1
)
to Z. F., then providing we a llo w e -terms to occur in the a xio m
schemes o f Z. F., th e A x io m o f Choice is provable. (See e.g.
Leisenring [1969] p . 105-107). N o w Hilb e rt's E-symbol behaves
in ma n y wa ys lik e th e in d e fin ite a rticle . Ce rt a in ly (1) i s i n tu itive ly co rre ct if we do in te rp re t i t th is wa y. Suppose then
that f o r e ve ry sentence containing E-terms, w e co u ld g ive a
lo g ica lly equivalent sentence not containing them; and in such
a wa y that the paraphrase o f (1) we re a lo g ica l truth. Then if
the paraphrase is a t a ll reasonable we wo u ld be able to paraphrase the proof o f the a xio m o f choice in to a corresponding
proof i n Z . F. B u t t h is i s imp o ssib le . I n p a rt icu la r re a d in g
'( 3 x) (P x A O r f o r 'P(Ex0)' w i l l not do.
We are forced then to lo o k f o r semantics o f indefinite description. I f 0 is a sentence o f some language, le t us read ',;x0'
as so me (particular) x wh ich O's,' and le t D be the domain o f
existent objects.
i) A t o mic Sentences
Consider first the tru th value of an atomic sentence 'P(gx1)'.
If the set X of x's in D wh ich satisfy 0 is non-empty, then 'gx0'
denotes one such x and 'P (;x0 )' is tru e if f P is tru e o f that x.
This mu ch is straightforward. I f there is n o such x, (i.e. X is
empty) then we suggest that 'P(gx0)' is false. Fo r if there is no
man next door, then it can not be true that a man next door is
a good chess p la ye r o r a h o micid a l maniac. Hence i t is false.
Some h a ve o f course argued th a t sentences wh ich contain
non-denoting t e rms a re n e it h e r t ru e n o r fa lse , e .g . Fre g e ,
Strawson in [1950] and elsewhere. But apart fro m the fact that
patently, i f 'gx0' does n o t denote, then 'gx0 does n o t exist' is
true, i t has been we ll argued b y Du mme tt [1973] p p . 413-420
that ca llin g a sentence neither tru e n o r false, ra th e r than ju st
plain false is an empty gesture. To quote fro m p. 419.
'In order then t o explain the use o f sentences to make
assertions, a ll that has to be appealed to is a two fo ld clas-
I NDEFI NI TE DESCRIPTIONS
7
sification o f sentences ... in to those wh ich could be used
to ma ke a co rre ct [i.e. t ru e ] utterance, a n d those t h a t
would re su lt in an in co rre ct [i.e., false] one'.
No w i f gx1I) does n o t e xist then i t can n e ve r be co rre ct t o
assert 'P(gx(1:9'. Hence it is false.
It i s interesting th a t Du mme tt then goes o n (p. 420-429) t o
argue th a t we d o need t o distinguish t wo d iffe re n t types o f
falsity (loosely, false because non-existent and false because
existent b u t wro n g ly applied) i n o rd e r t o p ro vid e adequate
truth functional semantics f o r compound sentences. Th is cla im
I ta ke t o b e false. I hope i t w i l l be cle a r b y th e end o f the
paper that satisfactory tru th value semantics f o r a ll sentences
can be obtained with o u t such a distinction.
Some people h a ve a ct u a lly cla ime d t h a t a to mic sentences
containing non-denoting te rms ma y a ctu a lly be true. Hilb e rt ,
for example, gave an a rb it ra ry denotation in cases o f natural
reference failure. This wo u ld o b vio u sly a llo w fo r some atomic
sentences to be true. Ho we ve r, this is o b vio u sly ad hoc.
More interestingly, t h e f o llo win g so rts o f examples wo u ld
seem to show that such sentences can be true.
1) Th e alchemists sought the philosopher's stone.
2) Sh e rlo ck Holmes live d in Baker Street.
However, examples like this always depend on non-denoting
terms occurring in non-extensional contexts. No w opaque contexts a re a n interesting b u t t o t a lly separate problem. (Th e y
give rise to problems such as failure of Leibniz' la w completely
independently o f whether o r n o t th e te rms in vo lve d denote),
and to run it into the problem of non-denotation is to in vite confusion.
It ma y not be clear that the terms in 2) (and others truths o f
fiction and mythology) a re in non-extensional contexts. Ho wever, this is not d ifficu lt to demonstrate. 2) is not lit e ra lly true,
though 'I n th e b o o k b y Conan Doyle, Sh e rlo ck Ho lme s live d
in B a ke r Stre e t' c e rt a in ly is . T h e wo rd s ' I n t h e b o o k b y
Conan Do yle ' are absolutely necessary, though often omitted.
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Take a n y wo rk o f fictio n F, wh e re the ma in character X, is
an actual historical figure who achieves in the book something,
Y, he failed to do in real life. Obviously, to suppress the phrase
'In t h e w o rk o f f ict io n F ' f ro m ' I n th e w o r k o f f ict io n F, X
achieved Y' is to in vite a contradiction that does not exist.
Further, such a phrase puts a n y t e rm in it s scope in t o a n
opaque context. Otherwise, we should have, b y Leibniz' la w:
in the wo rk o f fictio n F, the character wh o d id not achieve Y,
achieved Y. Such a p lo t wo u ld be d ifficu lt t o writ e !
We ma y su m u p th e discussion o f th is p o in t b y sa yin g i f
'polv does n o t exist, and 'P' is an ordinary, extensional p re d icate, 'P(gx(I))' is false.
ii) Tru t h Functions
Once we have settled the tru th va lu e o f a to mic sentences,
we ma y consider the tru th value o f sentences containing connectives to be determined b y o rd in a ry tru th functions. Th is is
certainly the simplest course. I t ensures tha t classical propositional logic holds, and does no great violence to o u r intuitions.
There is o n ly one p o in t wh e re i t ma y be argued th a t th is
gives, in tu itive ly, th e wro n g value, and th is is p ro b a b ly wh y
Dummett feels th a t o rd in a ry t ru t h tables a re inadequate.
According to our analysis, if gx(I) does not exist then is 1
true.
o f people have fe lt tha t if there is no
P ( gNxo(w1 a) )number
'
man next door then 'The man next door is a maniac' and
The man next door is not a maniac'
(
6
)
are both false. I f we a sk w h y th is v ie w is held, w e a re p ro bably to ld that f ro m 'The man n e xt d o o r is n o t a maniac' we
can lo g ica lly in fe r 'There is a man next door'. Thus (6) can not
be true if there is no man next door.
We can account f o r th is p o ssib ility in t wo wa ys. Firstly, i f
we accept that 'not' in o rd in a ry English can be used as p re d icate negation as we ll as sentence negation (as argued, e.g. b y
Jackendoff (1972) p . 325 ff.), th e n w e ca n in te rp re t (6 ) a s a n
atomic sentence wit h a co mp le x predicate. Th is wo u ld ma ke
I NDEFI NI TE DESCRIPTIONS
9
the inference correct. Predicate negation is e a sily formalisable
and I w i l l not consider this wa y further.
Alternatively, we can accept that the 'not' in (6) is sentence
negation and th a t
1 P (g x
4 ) exists
gxcl)
)
is not a lo g ica lly va lid inference. (A f t e r a ll the more general
V(gx(1))
;x(1) exists
is ce rta in ly not lo g ica lly valid).
However, this is not to deny that if someone utters 'The man
next door is not a maniac' there are grounds fo r believing that
there is a man n e xt door. Fo r people do not usually make such
utterance unless there is a man n e xt door. Thus
A utters ' 1
3 gx(1) exists
(g x(1 ))
.
would be a good inductive
inference.
Other cases o f wh a t Strawson [1952] pp. 175-179 ca ll's p re supposition could be handled in a simila r way.
Thus i t is possible t o defend o u r v ie w th a t sentences co n taining connectives ma y be evaluated wit h normal tru th tables
in either of these ways.
iii) Qu a n tifica tio n
The truth value of quantified sentences is simp ly determined
by the normal quantifier semantics o ve r the domain D o f existent entities. The temptation to a d mit the tru th o f
'There are things wh ich don't exist, e.g. Pegasus'
(7)
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(e.g. Resher [1959]) should be fought, if 'There are re a lly means
what it says. O f course, we ma y interpret the quantifier in (7)
substitutionally and this has additional p la u sib ility in v ie w o f
Quine's re ma rks on quantification in [1973]. Indeed it ma y be
argued t h a t Meinong's d e sire t o assert t h a t i n so me sense,
obects that do n o t e xist a re existent (Meinong [1907]) ca n be
laid a t the door o f substitutional quantification. B u t to f o llo w
this wo u ld take us too fa r a field.
Hence '( y ) y = gx1:19' w i l l be tru e if f 'gvtly denotes. I t ma y
therefore conveniently be read as 'gx(I) exists'.
iv) I d e n t it y
Finally, a fu rth e r problem arises wit h id e n tity and indefinite
descriptions. Wh e n is 'gx(1) c x 1 p (1 ) t ru e ? There are ma n y
possible answers. Hilb e rt's E operator makes (1) t ru e wh e n (1:0
and 11) are extensionally equivalent. Th e re seems n o in t u it ive
justification fo r this o r fo r any other vie w except that the o n ly
time we have a guarantee that (1) is true, is when (I) and II/ are
the same. This is the condition we w i l l adopt. Hence, wh ich o f
the objects satisfying (I) is to be denoted b y *oar"' w i l l have to
depend on 0:1) itself. Ho w to achieve th is w i l l become cle a r in
section 3. Perhaps i t is wo rt h pointing o u t that o u r semantics
can be modified in f a irly obvious ways to accomodate Hilb e rt's
vie w o r a number of others.
This concludes t h e in f o rma l a n a lysis o f th e semantics f o r
indefinite descriptions.
3) W e will now give the formal version of the heuristic semantics o f the previous section.
In what follows, I will use the fo llo win g schematic variables:
(1), p
sequence
of terms.
,
0 Let L be a first order language with variables v
f i
predicates
Pin(i<o)) and no constant o r function symbols. Fo r
simp
o , ilicity,
< o )we
, wni l l -le tpid le natityc be
e P.
r
Let L b e the language L wit h g-terms added, i.e. we add the
f
following clauses t o th e definitions o f 'te rm' a n d 'fo rmu la '.
o
r
m
u
l
a
s
;
x
INDEFINITE DESCRIPTIONS
i) g x0 is a term.
ii) P
i
Let
n A be a stru ctu re suitable f o r L. W e w i l l extend A t o a
( t
structure suitable f o r L
i
t
Notation
n
)Let the domain o f A be A , and le t V be the set o f variables o f
i g, h will be maps fro m V into A and g(x/a) w i l l denote
L.
s
a
g — (x, g (x)) U { (x,a)
f
o
A 0 [g] w i l l mean that 0 is true in A under the valuation o f
r
the free variables given b y g.
m
dp(0) (th e depth o f 0 ) is the length o f the longest chain o f
u
nested g-terms in 0 . (Hence if 0 E L, d p (0 ) 0 and i f dp(0)
l
n, d p ( 1 )
a
1
.
, a0 (it y) means 0 wit h a ll free occurrences o f ex» replaced
( gby
x 0gy».
) ) (with o u t any change o f bound variables). No w le t
= it be the set o f choice functions on A , (71 = { f ; f : P (A )—
n{A }
+1 > FA)be the set o f formulas o f L . Then if Q: V x F x Av
Let
(i.e.. ifa x V , g E A
n C2 is suitable iff
on
v TA),
h
d
i), Idf C
the variables o f 0 o ccu r amongst x,y1,...,y
1
<
e
fi E< n
r1
F
,t adh(n ed n f o r
S eB
g (y
- p)
i
2 It(f Eyx is, distinct
ii)
)
fro m z and y is free in 0(x/y) then
0 hB , =
g o}
) h
C2(z,0(x/y),g) C2 (z, 0 , g (xig (y))).
f
)
i
s (
a a . conditions
These
y
ensure t h a t Q(x,0,g) depends o n l y o n t h e
c g h
io
i - c
)e
t
f
u
tn
c e t
hi
o r n
e
m
n
o
Q
g
(
x
x
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members of g wh ich are relevant to 1 and not on the particular
variable used to denote a parameter.
Let Q be suitable and le t B < A , Q>
, AT. We
is
h e define
d oB (mN ga ] ib yn recursion
,
o ve r dp(1).
B dp(0) , = 0 , then B oO [ g ] iff f A ( D i g l
if
Suppose
the definition is made for all (I) of depth
B
The d e fin itio n f o r depth n + l i s it se lf b y re cu rsio n o ve r th e
formation of (I).
If i s atomic, it is of the form P
For
i
simp licity, le t us suppose that o n ly one t e rm t
n
sn and
i( st ithat
ot this
f is dpap.
e p t h
a
B
) .1
31
(t
fx (g ) is defined a n d B P
1
n
p
where
( t iz is a Zvariablet not occurring in lp
a
n rt (g) is the p a rtia l function defined as fo llo ws:
and
p fxp
)
[ g ( z i f x
—
, p
if
t {bEB; B l lp[g(x/b)] i s non e mp ty
(
g
)
)
]
n
fx (g) = 0 (x, v, g ) {bei3i B l p [ g ( ' i b ) ]
)
[
fx (g ) i s undefined otherwise.
g
]
i(Thus f x (g ) i s th e in te rp re ta tio n o f th e t e rm pcip u n d e r th e
1
f
1 g.) I f II) i s e v w, l i p o r ( x )t p , t h e clauses o f th e
evaluation
f
) are as usual. A lit t le thought shows th at apart fro m
definition
the need f o r E2 to be suitable, these are the fo rma l equivalent
of the h e u ristic semantics o f the previous section. Th e suitab ilit y conditions arise o n ly because we have to deal wit h open
sentences a n d th e parameters a re n o t p a rt o f th e language.
Thus we have the t wo fo llo win g lemmas:
I NDEFI NI TE DESCRIPTIONS
1
3
Lemma 1
If th e fre e variables o f (I) o ccu r amongst y,
h(y,) g ( y , ) I i
n then
n
and
B [ 9 1 if f B [ h i .
Proof
By induction over the formation of (I).
Co ro lla ry
If (I) is closed then B ( I ) [ g ] f o r a ll g o r f o r no g; i f f o r a ll
g then writ e B
Lemma 2
I f y is free in (I)(x/y),
B (1 4 g (xig (y))] if f B 1 1 )(xlyilg l.
Proof B y induction o ve r the formation o f O.
This le mma is necessary to validate a xio m vi) o f section four.
4. W e w i l l n o w characterize the semantics a xio ma tica lly. L e t
T be a first order theory in the language L. The fo llo win g theory
g
in the language L i s called T • The axioms o f T a r e specified
as fo llo ws:
i) A n y theorem of T is an axiom.
ii) A xio ms f o r the predicate calculus o f L
a) — * (1p ---> (I))
b) (4 ) --> (1p 0 ) ) --> (((1) 1 p ) c)
› ( 04 )
— *
(01 () )V x)
d)
) 1 --› (1)(x/y) ( y not bound in 0 (
e)
1 ( V x) (0 f)
) x )= x
›1
/ y () V
) (x 4
(0
g)
Vx
= y (c 1 ) (1 )(xt y)) ( y not bound in (I)(x/y))
x( )
l1 p )
(0
x
—
n
o
t>
i1
1
n
) )
1
)
0
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Axioms fo r ;-terms
iii) y = t -->(1
2
iv) P
7 (F)
1v) ( 3 x)(I) ( 3 y) y = g x 0
4 +
P
n
i y = gx
vi)
(gx
,n
0j)
( c yrules
r1
; of inference of T a r e modus ponens and generalizaThe
, ( 1 )
+
tion.
fo llo win g are theorems of T p ro vid e d that y does not
( x The
/
(occur in the scope o f a g-term in 8 , and a ll the variables free
y )
3
in
( gx11)y are free in 0(Y/gx4:1)).
x
n
a)
) ( 3 y) y = g x(I)-+ ( 3 x)D (Fro m a xio m iv))
o
t
b)
( y = gx(I) ( 0 -E+ 0 (
f3 (By
r in d u ctio n o ve r th e fo rma tio n o f O. A x i o m i i i ) i s t h e
1
e
1
x0 ))
0 g basis).
e
c) ( 3 x)(1) A ( y)C0 e (Y /g x(1 ))
i (From b) and axiom v))
n ( 3 x)0 e ( x / g x e )
d)
1 (From
: 1 b) and axioms v), vi)).
:The0fa ilu re o f b ) — d ) f o r general contexts 0 (y ) i s n o t su r(prising.
m /; is not an extensional operator and hence there is no
yreason
) )to suppose th a t gz0 is the same as gzO(Y/gx(D), even i f
y = gx(I). Fin a lly, we have the fo llo win g results.
Theorem I.
If A T , Q is suitable and B < A , Q > , B T
;
Proof
The proof is straightforward using Lemmas 1 and 2.
Co ro lla ry I.
If A is a n y first o rd e r structure and Q is suitable then axioms
ii) — vi) hold in < A , Q > .
INDEFINITE DESCRIPTIONS
1
5
Co ro lla ry 2.
If T is consistent then so is T
Co ro lla ry 3.
T i s a conservative extension of T.
Proof
Let (
ByI Lthe completeness of first order logic, we can fin d a model A
ofa{14:1)} U T. Let C2 be suitable B = < A , Q > . Since dp(11) = 0 ,
n
B d1s
5.
( 1uI n section f o u r we sa w th a t o u r a xio ms we re sound. Th e
axioms
cp
o
are also complete in the fo llo win g sense:
Hp
e For a given theory T, a sentence (I) of L i s va lid if fo r every
A o T and every suitable g2
ns
ce
B < A , Q >
et
Theorem
2.
bh
ya
For
T t a given T (130 is a theorem of T
g
i s
hT i f f 4 : 1 )
Proof
ve a l i d .
.
o4
As we saw in the previous section, a ll the provable sentences
r,
are
valid. We will prove that if E' is any set of closed sentences
eof
meL co n sist e n t wit h respect to T t h e n E has a model. Th is
implies
1l
that a ll va lid sentences are provable.
, )So le t E be such a set. Take a countable set A o f new consI
tant symbols and extend E to a set of closed sentences A o f L
S
U A such that
(
1i) A is ma xima lly consistent.
) A is satured, i.e. if ( 3 x)61)6 A then fo r some a E A, (Fria ) e A.
ii)
.
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This is done in the usual wa y (as in He n kin [1.949]).
Wit h e ve ry fo rmu la o f the f o rm 3 x0 in A we associate an
a E A as follows:
( x ) 0 ---> ( 3 y)y g x 0 E A.
so for some a E A
a — g x 0 s A.
We associate wit h 0, the a wit h the lowest index in some fixed
we ll ordering of A. We w i l l denote this b y ax
41)
Thus
ai g x 0 E A.
The following lemma is useful.
Lemma 3.
For e ve ry ax , 0(x/ax ) E A
(1)
Proof
( y ) (y ; x 0 ---> 0(x/y))E A.
Since
ai = g x 0 E A,
0
then
0(x/axel)) r A since A is ma xima lly consistent.
No w we can construct the model B A . Since A D f.„ t h is
gives re su lt. L e t A
— = ThAe n A
c-terms).
sentences
oLaf L
n i s
—
( ia .x ei m.
m
t l lh y e
a
cs oe nn s t i
se t n e c n e t
,s
so
a
t
f
u
r
a
tA
e
d
sw
i
I NDEFI NI TE DESCRIPTIONS
1
7
If0 , is 0 wit h e ve ry variable y free in 0 , replaced b y g(y),
then we can define in the usual wa y a model A wit h domain A
for A
—
s u
A O [ g ] if f e A
c h
—
t(see He n kin [1949]).
h We can n o w define a W e d o th is b y defining a series o f
a
functions
C2„(n e (.0) such that R, is a map f ro m V x F„ x Av in to
t set of choice functions on A, (where F
the
of
r depth n ) such that
, i s
t h e
s e t
o i) Onf , I Q . f o r m u l a s
ii) Q
n I f Q is a n y map f ro m V x F x Av in to a such th a t C2 D Q„,
iii)
i s then fo r e ve ry formula 0 o f L o f depth n, and e ve ry g,
s u
i t
< A , 0 > O [ g ] if f e A.
Depth
0
a
b l
Q
e . result holds b y the above.
the
o
i
Depth
n + I
s
t
If dp(0) n Q
h
If
n dp(0) = n + 1, then fo r x E V, g E AY, if
e
(x,0,g)
e
=
Q
X = {13 A ; O
m
, ,
g w
p
and X is not empty, then
o
t
A
}
y
Q„ 1
f
( x ,
u
where lp is 0 with every variable y, free in 0 except x, replaced
g )
byng(y). (This is possible since ( 3 x0)g E A)
X
cIf Y c A, Y X and
Y is non-empty then
=
t
a
i
Q,„
l
o
1
t
n
( x
o
.
,
,
I
0
f
, y
d
)
p
Y
(
r
0
18
G
R
A
H
A
M
PRIEST
(Say the least member of Y in some fixe d we ll ordering o f A.)
a) g"2„,
1 iXs = { b e A; ( D
a oThen
)), E A (b y the Lemma 3).
m (e3 x(1)), e A, so (10(x/ax
1P
c hA }
o iai.e.On d
V
V
g
c eX
b) i( x j
f ui
.c) f a
2x s
n when
c
o
" 21 n)
1
t in
,1
o ee
n
m
p
n A
1
X = {b ; O
. t
y
.
Q
iT s i
g (Ta )
srh N. o w if geand A
i)
h agree
on a ll the variables free in (11 except x,
)
, i then
u
a n
e e
to a .
d
rb l a
d p (log(xfb) is the same as 0
n
i .x
e
( 0 1) 1 ( x t b )
l
vHence
A
y E X = =RI ; O
ih ( xX/ b )
g
En
i
a
i
A
} +
n .
V
ila
1
n d
t
lii) S imila rly. Q(z,(1)(x/y),g)
n
2
= Q(z,(121,g(x/g(y)))
e
,yr
(
the same as 4 : 1 )
.t since (0(x/y)), is
e Fin
d)
g ( vaelly,
y ) )if g2 Dx(D
h
,
s , 1
u
e
c < A P > O [ g ] i f f 9 , E A.
,t
a n d
o
1
i
O
i
s
n
)
nThe proof
o
f
is b y induction
o ve r the formation o f O. I f 0 is Itp,
lxd
, e re su lt fo llo ws f ro m th e ma xima l consisg V lpe o r (p 3 x)1p,
t th
ytency
g
and saturation
o f A. Hence we need o n ly p ro ve i t f o r
c
h
in
atomic
O.
)
a
n
Suppose
B — < A ,X
C2 > 1 e [ g ]
s
+
t1
m
=
e
e
Q
iand 0 i s P
rsonly
i ( t gx(la is o f depth
( n ):
e
iw
x
sh. . . g x
,
te( 1 ) . . . t
(
in„ , )
1
n(
a
)
gn
d
,
cw
h
ae
)
sa
s
X
es
u
=
I NDEFI NI TE DESCRIPTIONS
19
No w B P r
i
a (g x4 );
P
o
but txt )(g ) (= t
[4) g ]
g
So
u
1 ( xb ,y4in) d
,g
) n hypothesis t x4)(g)I = Pa x w h e re lp i s 4) w i t h
) ctio
{ 1 )variable
;i
every
y free in 4) replaced b y g(y), except x.
B
s
4 ) [ gd
( x Hence
/
eB P a ' (z;T) [g(x/axIP)]
b ) ]
f
So
1 by induction
hypothesis (P
i
i.
But
B ( 3n x)(1)[g]. Hence 3 x4)g E
InP ( a ; fe
)A , a n dd
(gx4)), ----• a x L A .
IP
s
a
.
c n
A
.
Hence
d
B
(PT (gx4);1)
1
g s
:
A ,
since axiom iii) e A.
1
1
Conversely,
if not B O [ g ] then if 8 is 1
1
3
is n o t defined,
o r it is defined and B 1 P 7 ' (z ; t ) [ g ( z / l x
7 ' ( g x &( i ) e i t h e r
f x
o (g ) is not
tx
defined
then B 1 ( 3 x)4)[g]. Thus (
z
4)o
-( g ) ) ] .
I; f
( g ) hypothesis.
induction
1( 3
x[ ) 4 ) ) ,
e
A
b
y g
Since a xio m iv) E A, then
(
z
/
11
f
3
x
i.e.
7 'P7'(gx4);t1
o
g ( g0
A .
If P
4) I P
(
x
4
o
g
);I
(g
)
)
)
)
g
i
]
L
s
A
d
e
f
i
20
G
R
A
H
A
M
PRIEST
and
So
Hence the result.
1P
r
i
a
(P
g n
Xi
(1
Ia)
g( g
E
0
Ax
U Q
.:flEw
t)
No w if we put Q =
2
g
and ,
,
A
B < A , Q >
.
then it is easily checked that Q, is suitable and
B 0 [ g ] i f f 0 , E A.
So if 0 is closed, 0 is 0 , and since Q is suitable,
B i f f
0 E A.
Thus the semantics are complete.
We have the fo llo win g two corollaries.
Co ro lla ry 1.
If 0 is tru e in a ll structures o f the f o rm < A,Q> th e n 0 is
provable from axioms ii) — vi).
Co ro lla ry 2.
Compactness: i f e ve ry fin ite subset o f E has a mo d e l, then
so does E.
6) Conclusion
Thus we conclude the paper. The above system o f indefinite
INDEFINITE DESCRIPTIONS
2
1
descriptions is, I think, an adequate formalisation of our notion
of the indefinite article. There are t wo fin a l comments.
i) W e have treated id e n tity as an o rd in a ry predicate. Thus
if •gs x lor does n o t denote, 'g x (1) g x (I)' is false. Some
people have argued that this is n o t so. (e.g. Leblanc and
Halperin [1959]). Ho we ve r, th e arguments a re n o t v e ry
convincing and b y fa r the simp le r course is th e one we
have taken. Nevertheless, th e above semantics co u ld be
adapted t o a llo w f o r t h is b y s imp ly a d d in g a sp e cia l
clause f o r id e n tity statements in the d e fin itio n o f '
ii) I t ma y e a sily be checked th a t if we add the description
axioms t o Z.F. (and a llo w d e scrip tive te rms t o o ccu r in
the Z.F. schemes) then the axiom o f choice is indeed provable.
REFERENCES
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HENKIN L. [1949] ' Th e Completeness o f t he Firs t O rder Func t ional Calc ulus '
in J ournal of Sy mbolic Logic 14 pp. 159-166.
JACKENDOBFF R. [1972] Semantic Interpretation in Generativ e Grammar, M.I.T.
Press.
LEBLANC H. & HALPERIN T., [1959] 'Non-des ignat ing S in g u la r Terms ' i n P h i l
Rev iew 68.
LEISENRINC A. [19691 Mat hemat ic al Logic a n d Hilb e rt s c -s imbol, Mac donald
& Co.
MEINONG A419071 Ub e r d ie St ellung der Gegans tand t heorie i m Sy s tem d e r
Wissenschaf ten, Leipz ig.
QUINE W. [1973] Roots o f Reference, O pen Court .
RESHER N. [1959] ' O n t h e L o g ic o f Ex is tenc e a n d Q uant if ic at ion' i n P h i l
Rev iew 69 pp. 157-180.
STRAWSON P. [1950] ' O n Ref ering' i n M in d L I X pp. 320-344.
STRAWSON P. [1952] I nt roduc t ion t o Logic al Theory , Methuen.