COMPLETENESS O F S OME Q UA NTI FI E D MO D A L L OGICS
James W. GARSON
O. In tro d u ctio n
We g ive here simple completeness proofs f o r a wid e range
of quantified modal logics. The systems in question have the
axioms a n d ru le s o f some standard syste m o f p ro p o sitio n a l
modal lo g ic, a xio ms f o r intensional id e n tity, a n d q u a n tifica tional principles d ra wn fro m free logic. The semantics we g ive
is more general than usual, and perhaps that explains the simp licit y of the completeness results.
Usually a semantics f o r q u a n tifie d mo d a l lo g ic in vo lve s
the introduction of a function D wh ich assigns to each possible
wo rld d the set D(d) o f individuals wh ich «exist» in that wo rld .
We ma y th in k o f D as the intension o f a corresponding e xistence predicate 'E' in the object language.
In o u r semantics, we introduce a function Ill, wh ich assigns
to each possible wo rld d a set W(d) o f in d ivid u a l concepts, o r
wo rld lin e s (i.e., functions f ro m the set o f possible wo rld s t o
the set o f individuals). I n this wa y we agree wit h Montague's
[5] and Bressans [1] mo re general treatment of the intension of
a predicate in o u r description o f the intension o f 'E'. (
result,
th e t e rm position in 'E' becomes opaque, a n d n o t sub1
ject
of identites. (
) Ato ssubstitution
a
2
)
(
and1 by Bressan [1] on p. 24 and f ollowing.
()
semantical
2
treatment o f 'E' t o t he systems o f this paper. Th e o n ly changes
T
required
)h
inv olv e res t ric t ion o f t he s ubs titution o f identities t o t e rm -plac es
of Iipredic ates wh ic h a re n o t giv en t h e mo re general int erpret at ion.
st
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1. F o rma l Mo tiva tio n
Wh y should we wa n t to generalize the treatment o f the in tension o f 'E' in this wa y ? The f irst set o f reasons are formal.
When D is used, there are several ways of introducing the truth
definition f o r t h e quantifiers, a n d i t i s n o t e a sy t o d e cid e
wh ich is best. () A t least one o f these results i n a d e fin itio n
of va lid ity wh ich is not axiomitizable (system Q2). On another
definition (syste m Q 3 ) t h e completeness re su lt s a re h ig h ly
sensitive t o t h e stre n g th o f t h e mo d a l operator. Strategies
which w o r k (sa y) t o sh o w t h e completeness o f S4 strength
quantified modal logics (fo r example those used to show completeness o f Q3-S4) a re n o t applicable t o the we a ke r systems
such as Q 3 -M o r Q3-K. That d ifficu lty does not arise wit h the
more g e n e ra l interpretation o f th e quantifiers w e g ive here.
One simple strategy serves to provide results for any quantified
modal lo g ic whose modal fragment is determined b y some nonempty class o f K rip ke frames, and f o r wh ich a standard He n kin style completeness proof is available. In short, the methods
of th is paper yie ld simp le completeness proofs f o r quantified
versions o f t h e va st ma jo rit y o f stu d ie d systems o f mo d a l
logic.
2. I n f o rma l Mo tiva tio n
Are there a n y reasons o th e r than fo rma l f o r accepting th e
approach o f th is paper ? I believe there are philosophical in tuitions wh ich recommend it, p a rticu la rly when the quantified
modal lo g ics a t issue are tense logics. I t is g e n e ra lly fe lt that
it is not part of the province of logic to determine wh ich of the
competing ontological theories is co rre ct. I t f o llo ws t h a t se mantics f o r quantified mo d a l lo g ics should n o t ru le o u t a n y
reasonable ontological theory.
(
discusses
s imila r matters f o r anot her int ens ional logic .
3
)
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COMPLETENESS O F SO ME Q UA NTI FI E D M O D A L LOGICS 1 5 5
There is a venerable and pervasive ontology, wh ich we might
call o rd in a ry onotology, o r Aristotelian ontology, wh ich seems
to require the approach we have taken to the intension o f the
existence predicate. I t is basic t o th is p o sitio n th a t there a re
objects, and that objects change. So the fo rma l counterpart o f
an object in a tense lo g ic semantics mu st be a t ime -wo rm o r
individual concept, a n d n o t a slice o f a t ime wo rm, sin ce i t
makes no sense to say o f the la tte r that it changes. I n a tense
logic then, a n o b je ct sh o u ld b e treated a s a fu n ctio n f ro m
times in t o te mp o ra l slice s o f objects ( o r point-events), a n d
hence objects a re t h e intensions, n o t t h e extensions o f t h e
terms.
No w what does this have to do wit h whether the intension of
'E' should yie ld a t time t a sot o f in d ivid u a ls o r a set o f in d ividual concepts ? W e kn o w th a t there a re ma n y predicats P
wh ich are extensional, wh ich means that the calculation of the
truth-value o f 'Pn' (sa y a t t ) depends so le y o n wh e th e r th e
time slice (individual) refered to b y 'n' a t t fa ils in the extension o f 'P' (a t t), and does not otherwise in vo lve the intension
of 'n'. Couldn't it tu rn out that 'E' is one o f these extensional
predicates ? I th in k there are stong intuitions wh ich rule against
this a n d wh ich sh o w th a t a t least one ra th e r basic treatment
of existence i n te mp o ra l situations re q u ire s t h e in te n sio n a l
approach.
To see this, imagine wh a t we wo u ld say (at t) i f a temporal
slice (o f fin ite , b u t sma ll duration, i f yo u like ) o f a n o b je ct
(say a slice o f Gerald Ford) we re to be present a t t, in a context wh e re th e preceding and subsequent te mp o ra l slice s o f
Gerald Fo rd were absent. No w it is tru e that we mig h t be in clined to sa y that something existed a t t, n a me ly a temporal
slice o f Gerald Ford, o r a fleeting «apparition» o f Gerald Ford.
It wo u ld be strange, however, to cla im that Gerald Ford existed a t t under these circumstances, f o r p a rt o f wh a t makes us
want to sa y th a t Gerald Fo rd e xists a t t, is th e existence o f
previous and subsequent stages o f Gerald Fo rd arranged in a
certain coherent fashion.
This re su lt should not be surprizing. The o rd in a ry ontology
takes objects as basic, so questions about existence and non-
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existence o f objects should be questions about objects and not
about ce rta in o f t h e ir te mp o ra l p a rts. S o i t f o llo ws t h a t i f
objects are to be treated as functions fro m times to times slices,
then th e referent o f 'Er a t a t ime t should be a se t o f these
sorts o f functions, and not a set o f time slices.
3. T h e predicate 'E'
The predicate 'E' is included as a p rimit ive o f o u r systems.
Its p rimit iv e in clu sio n i s imp o rta n t t o th e g e n e ra lity o f t h e
results. I n free logc, and in modal logics wh e re the domain o f
quantification shifts fro m possible wo rld to possible wo rld , the
usual a xio m o f u n ive rsa l instantiation i s in va lid , sin ce i t is
possible f o r V xA t o b e t ru e a t wo rld d , a n d f o r A n i x t o
be fa lse wh e n ' n ' denotes something outside t h e d o ma in o f
quantification fo r d. I f we conditionalize the consequent o f this
axiom, a v a lid f o rmu la results, wh e n t h e rig h t co n d itio n i s
chosen. The condition we wa n t is a fo rmu la wh ich expresses
that 'n' refers to some «object» in the domain o f quantification
for d. I n fre e lo g ic ' 3 xx—n' serves th a t purpose. Ho we ve r,
this fo rmu la w i l l not wo rk in an intensional lo g ic wh ich gives
' --' t h e intensional o r we a k in te rp re ta tio n o f id e n titiy, a n d
which includes o u r approach to existence. That is because we
now need a co n d itio n wh ich says th a t the function refered to
by 'n ' fa lls in to the domain 1
1But
' ( d' )3 xx—n'
o f says something weaker, n a me ly th a t there is a
f u n cg tsuch
function
i o that
n sg(d) is the extension o f n at d. I t does not
f
improve
o matters
r
to replace ' 3 xx—n' wit h ' x x = - n ' , unless
t
the
u n hd e rlyineg mo d a l syst e m i s a s st ro n g a s Szl. (
w3 xElx—n'
o
r
d there is a function g T ( d ) wh ich
'4
says
oln ly that
d
.
)agrees
F o wit
r h the intension of n a t those arguments in the range
of the Kripke relation R at d.
(
n n4 F r i ) bec omes v a l i d o n o u r s emantic s , i n s p it e o f t h e f a c t t h a t
x) 0 x n does not express t hat t he f unc t ion ref ered t o b y ' n ' a t d f alls
in W
.411(d).
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COMPLETENESS O F SO ME Q UA NTI FI E D M O D A L LOGICS 1 5 7
4. Rig id it y of variables
One o f t h e ma j o r simp lif yin g assumptions ma d e i n Q 3 ,
Thomason's [6] approach t o quantified mo d a l lo g ic, i s t o a ssume th a t th e intensions o f th e variables a re constant fu n ctions, that is to say, th a t th e ir extensions do n o t change f ro m
possible wo rld to possible wo rld . On the other hand, he a llo ws
the te rms t o b e non-rigid. Th is leads t o f o rma l a n d p h ilo so phical akwardness.
The fo rma l akwardness is th a t the domain o f quantification
becomes the set of constant functions, o r as Thomason believes
he should ca ll them, the substances. This invalidates the usual
axiom fo r universal generalization. B y adding an instantiation
condition H x x —n', a valid axiom
A4' V x A D ( 3 x D x — n D A n x )
results, b u t o n ly wh e n th e u n d e rlyin g mo d a l lo g ic i s S4 o r
stronger. I n lo g ics o f we a ke r mo d a lity, A 4 i s in va lid , a n d i t
is d ifficu lt to specify the instantiation condition that is needed
for a sound and complete system. Hin t ikka [4] describes t h is
matter in detail and gives the complicated axioms wh ich must
be used. Thomason's re fu sa l t o co n sid e r we a ke r mo d a litie s
saves h im f ro m considerable complications, a n d even so, t h e
rules f o r his system Q3 are elaborate.
Insisting on the rig id it y of variables leads to fo rma l co mp lications, b u t I d o u b t t h a t i t i s acceptable o n p h ilo so p h ica l
grounds either. Thomason motivates h is decision to « rig id ify»
the variables b y cla imin g that this a llo ws us to incorporate a
concept o f substance in to modal logic. I n choosing o u r va ria bles to be rig id designators, we are supposed to get the effect
of quantifying ju st o ve r th e substances. B u t w h y should w e
conclude t h a t t h e substances a re e xa ct ly t h e intensions o f
rig id designators. I t is at the ve ry least inconvenient to do so.
First, the insistence that a te rm fo r a substance refers to the
same in d ivid u a l in each possible wo rld imme d ia te ly poses the
problem of providing id e n tity conditions fo r individuals across
possible wo rld s. I s i t sensible t o assume th a t Plato (wh ich I
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take to be a substance fo r the present discussion) in this wo rld
is e xa ctly identical to the Plato of some other possible wo rld ?
We ll, n o t unless Plato's id e n tity does n o t depen don h is a ccidental properties. But if trans-world identification is to depend
only o n essential properties, h o w can we be sure th a t there
aren't possible wo rld s wh e re t w o substances (s a y A rist o t le
and Plato) have exactly the some essence, and so are identical
in that wo rld , wit h the result that they must be ru le d identical
in a ll worlds ?
The inconvenience becomes mo re apparent wh e n we g ive
modal logics a tense interpretation. We wo u ld like to q u a n tify
over Gerald Ford as a member o f our ontology, ye t the extension o f 'Gerald Ford' is n o t constant across times, since each
time slice o f his has its o wn peculiar properties. Insistence on
the myth that 'Gerald Ford' is after a ll a rig id designator p ro vokes a fru itle ss search f o r «something» wh ich remains co n stant in all these time slices.
The difference between substances a n d o t h e r «things» i s
undoubtedly too subtle t o b e captured v ia th e simp le n o tio n
of th e rig id it y o f designators. I n fact, i n o rd e r t o b e f a ir t o
any syste m o f ontology, t h e b e st t h in g t o d o wo u ld b e t o
make no assumptions a t a ll about the semantical properties o f
the terms that happen t o re fe r to substances.
We now find another reason f o r choosing the general approach o f th is paper. O u r existence predicate E has a set o f in dividual concepts as its extension, and th is set can serve admira b ly as the boundary between substances and non-substances fo r those ontologies wh ich bother to make the distinction.
Since there are no conditions on this set in o u r definition o f a
model, any sort of ontology of substances may be accomodated.
Since the substances so identified are not necessarily constant
functions, t h e p ro b le m o f tra n s-wo rld id e n tifica tio n does n o t
need t o b o th e r u s unless t h e o n to lo g y i n question re q u ire s
such a concept. In this wa y we leave the problem to the ontologist, not the logician.
Special lo g ics designed t o capture assumptions about substance o r any other interesting sorts f o r quantification can be
constructed b y p u ttin g appropriate conditions o n the seman-
COMPLETENESS O F SO ME Q UA NTI FI E D M O DA L LOGICS 1 5 9
tical behavior of E in o u r definition of a model. So Thomason's
approach can be reconstructed wit h in ours. I f we wa n t a lo g ic
which does n o t close ontological issues, however, w e should
choose the generality o f the approach o f this paper.
4. A x io ma t ic Features of MQ
We w i l l demonstrate t h e completeness o f a system M Q ,
writte n i n a mo rp h o lo g y L f o r quantified mo d a l lo g ic w i t h
identity, wh ich includes th e p rima t ive predicate constant 'E '
for 'exists'. M Q contains th e p rin cip le s o f some propositonal
modal logic M for wh ich a standard Henkin-style completeness
proof is available. (B y standard, we mean to include th a t the
definition o f the K rip ke re la tio n R in th e canonical mo d e l is
given in the standard wa y: Rdc if f {A : 0 A E d }c c . ) I t includes
as we ll axioms for intensional identity:
(A = ) t = t
(AS) H ( s = t D (F(s) F ( t ) ) , where F(s) is atomic and F(t) is
the result o f replacing t f o r some occurrence o f s in F(s).
A formula is atomic just in case it contains no logical constants.
We consider 'E' t o be a lo g ica l constant, so substitution guaranteed b y (AS) does n o t a p p ly t o 'E'. M Q also contains the
following quantificational principles d ra wn f ro m free logic:
(VG)
H
( A D (Ex DB))
H ( A D V xB)
(V I )
(
V xA D (EtDAt/x)).
A t/x i s t h e re su lt o f re p la cin g a n y t e rm t p ro p e rly f o r x
wherever x appears in A.
'When l'' is a set o f formulas, ' F H A' means th a t either A is
provable in MQ , o r there is some conjunction G o f members
of I ' such th a t H 31Q
(G D A ) .
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contradiction o f propositional logic.
Let T be a n y set o f te rms o f L, and le t F(T) b e th e set o f
formulas o f th e mo rp h o lo g y lik e L save th a t i t has it s o n ly
terms the members of T. Let T b e the set of terms that appear
in the set of formulas
A se t o f formulas A i s a MQ -se t i f f f o r e ve ry f o rmu la A
of F(TA) j. A 4
,— V x A
-E
1, A
, MQ-se t is ju st a saturated set wit h respect to the set
L sA.
H
F(TA)
.
t h o
e f formulas of the morphology wh ich contains o n ly terms
in f in A , wh e re saturation is defined as is appropriate f o r
found
free
At hlogic.
Lemma. I f F is a set o f formulas such th a t r
(e MQ-set
r
and
Ae there
, is an in fin ite se t o f terms T o f L, none o f wh ich is
tin
i Th
ers
n
F
(
T
A
a
1
rU
tt Proof: S imila r t o t h e p ro o f o f t h e Lindenbaum Lemma.
U
fhe MQ-sets o b e y th e usual properties o f saturated sets wh e n
T
A
r
e
attention
is re stricte d t o th e fo rmu la s o f t h e ir corresponding
)lnm
morphologies.
In particular
.1tt P. Wh e n A i s a MQ-set, t h e f o llo win g properties h o ld f o r
—
all
hi A, B F ( T
,en
A
1. i f A
)arT: 2.
H —
A A, E A if f A E A .
nea
t h( AeD B ) E A if f A(4 A o r BE A.
3.
dis
n A s
isu 4. V x A L i f f for all terms tETt,, ( E t D A t x ) L .
.
iac
E
iM
h
5. Semantics for MQ
.Q
t
i-h
fsa An M-mo d e l U f o r L is a quintuple < D , R, I, u > , wh e re
<t D, R > is a member of the set of K rip ke frames wh ich detere
t(
(5) Fo llo win g Hans s on a n d G ardenf ors [31 a K r i p k e f r a m e i s a p a i r
S
A
< D, R > c ons is ting o f a non-empt y domain D (o f pos s ible worlds ) a n d a
st
binary
relat ion R o n D. A propos it ional mo d a l lo g ic M is det ermined b y
u&
c—
hA
tt
hi
ax
t)
F
c
COMPLETENESS O F SO ME Q UA NTI FI E D M O D A L LOGICS 1 6 1
mine M, (
which
assigns t o each d D a subset o f th e se t o f functions
5
) I D into I, and u is an interpretation function wh ich assigns
from
i seach te rm t a function u(t) f ro m D into I, and wh ich assigns
to
a each j-a ry predicate letter Pi (other than E) a function f ro m
to
D
n inotonthe p o we r set o f V. (We le t P
3
null
set, so th a t i n t h e case o f a propositional va ria b le 1 :
=
1
u
({1m 0p
1 ,
e ()
,w
1
e r 'eU d
0
t We
y h define
)is A (s r e a d
0) U d Et iff u(t) (1 )(d )
(te) '
A
h
e
1) U d Pit
)t i
s
2)
] .U. . t s = t iff u(s) (d) is u(t)(d)
( d t3)
)
r
u
i Ui fd f — A iff not U d A
io 4)
e U ( A B ) if f not U A or U B
< u ( t
sf o
5)
i U 1 d V x A iffUf/x or fix A fo r all f U ( d )
{i nn
) (Ufix the model identical to U save th a t its interpretation
0
d U i function assigns f to x.)
( d )
}v 6)
a i U D A iff for all e I D , if Rde, then U e
, . .
=
t u.
d A
,
u
1
d l t o f formulas F o f mo rp h o lo g y L is MQ-sa tisfia b le i n L
a
A se
( t
.sjust
' )in case
) there is a n M-mo d e l f o r L en wh ich each o f the
i
o
a
,
members
o f F is true. V a lid it y is defined f ro m satisfaction in
)
rithes usual way.
(
0
. fIt is a osimplel matter to show by induction that
d
(i l
o
)
ts Sw. U " ( t)s/
>
ra x: d
A
u
u
f i f f
(
e
u Ud
P
6.
T h e completeness of MQ
o
n A
ti
fc x
)
ft .Let us define a canonical model U
(
quintuple
< D , R, I, 1 U
a
i can ( o r
d is
1
there
an
in fin ite set o f terms o f L n o t in T
f
o
r
lo
)s: Dh A co d} rc , t and
1
{d
A
) iii. u(t)(d) is { s : s = t c.1}, and iv. u(Pl)(d)
sn
,e aua >n ,d s i i .
R
d
c
iw
h
f
e
f
r
e
t
h
e
)a class K o f K rip k e frames jus t i n case f o r ev ery member < D, R > o f K,
i.every model
.
f o r < D, R > satisfies a l l and o n ly t he theorems o f M.
d
D
i
f
f
d
i
s
a
M
Q
s
e
t
,
a
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is t < u ( t 1 )
(is i f o r some te rm t o f L and d E D, and vi. f k l f ( d ) i f f u(t) is f
and
c 1 ) ,EtE d for some term t of L.
• •If .fo, llo ws from this definition that
u ( t i )
( dE. )E t E d iff u (t)EU(d )
> :
P Proof:
i
Th e p ro o f f ro m le f t t o rig h t is t rivia l. N o w suppose
tthat u (t )E U(d ). Th e n f o r some t', u (t) i s u (t') a n d Et' Ed. B u t
iu(t) is u (t ) o n ly when t is t', fo r when t is not t' there is always
t MQ -se t ei•D wh e re t E T , a n d t ' OT,, w i t h t h e re su lt t h a t
a
iu(t)(e) is not 0 , wh ile u(t')(e) is. I t fo llo ws then that Et E d .
E We are not ready to prove
d
} The Model Lemma F o r e ve ry fo rmu la A and each d E D, i f
,
A E F (T
a
d
n
The
proof is b y induction
), on the fo rm of A. Cases fo r formulas
d
having
f o rms o th e r th atn hOeB a re e a sy t o p ro vid e g ive n P.
vS., a n d E . . So w e t u rnn t o t h e case f o r fo rmu la s h a vin g th e
.
form
OB. Suppose d E D Uandc E I B E F (T
Then
i
d
a n
e N o w
).
a s sd u m e
f (I) f o r all cE D, if Rdc,
U
d Athen U B .
i
D
B
.
i
fNo w le t d b e { A : Df A E d I , a n d suppose f o r re d u ctio th a t
f
f
ud . 1
A
(,set T o f terms fo re ig n to
E c, and since th e members o f d U
t•
d
){B—B} are a ll in F (T
.
El
(.d
No
T
), wn f oorm t w o in f in it e d is jo in t se ts T
d
T
1
h e m b e r
)1m
,o
that
e T d 1=
f1U { —B} c e and e c F ( T
U
2
d
U
T O .
n
T
T
,d
T
u pr c eeh
t
h
a
t
a (9 hTspo edo
this, order Ti then let T
n
iof
T
s
i
s
U the
a
1 crordering.
o n ss i s t
o
f
.a
iI o d n d ,
(i—a n nf i d n i t e
n
as1 T
e2
t
do
f
)3
Ue
v
e
n
B
) m
—
e
m
b
e
yI r
s
B
tS
hi
en
COMPLETENESS O F SOME Q UA NTI FI E D M O D A L LOGICS 1 6 3
of terms (n a me ly T
2—Bee. Sin ce E ) B e F ( T
)d n oByt the hypothesis o f the induction and B e F ( T
clude
i) , nB th
de aFtTn( oTt U , 1 3 . So there is a member e o f D such th a t
Rde
and
(1). O u r reductio is comed
) not
w e1 1 , Bc, wh
o ich
n contradicts
plete,
and
so
we
kn
o
w
that
d
1—B.
I
t
fo llo ws th a t e ith e r
,) ,
s
o
sb
oy
or
eP th e re a. re fo rmu la s D A
B. I n the f irst case D B , hence D B e d b y P., a n d
lei & A „)n
at
in
th
e
second
case i t fo llo ws b y th e p rin cip le s o f M (wh ich
,F
f Duo Ar l t lh
is
as K) that ( D A , & & DA „) D DB . So d 1— DB ,
„eo asrstrong
w m e so
and
drt eby P.,h OBe d .
No u
w let ce be any
member of D and suppose that O B e d and
sR
h
a
t
Rde.
Then
Be
e
,
and
td
h
a
t so B e F ( T
e
the
induction
and
th
is la st re su lt t o obtain U h
(e
A
that
U
(
1
.
)l . W e
e
a
1 .2Theorem.
B I. pt l MO
e
B
m
fo oyisl strongly
l o w complete.
s
&
n Proof. We show th a t a n y consistent set o f formulas is MCIt
h
e
d
h y p Let
satisfiable.
o It' beh a ney consistent
s
set o f formulas in morphology
L.
Let
L
b
e
the
mo
rp
h
o
lo
g
y
th
a t results f ro m adding to
i s
othe te rms o f L a n in f in it e se t T o f terms fo re ig n t o L . N o w
divide T into two d isjo in t in fin ite sets T
1
Lemma,
there is an extension A o f F such that A F ( T U TO.
, 1
No
. w consider U
it s K rip ke fra me < D , R > i s a me mb e r o f th e cla ss o f
cthat
9 a• n (
.Kripke
e
wh ich determines M, b y re vie win g th e appro6 W frames
priate
portion
o f the completeness proof fo r M, since the d e fim
a
y
)
B
y
t
h
e
nition
g iveen fo r R in U c
uM
s
O
s
e
t
a
No
t n w ith
hse re i s a n in f in it e s e t (n a me ly T
s2
which
n dnot
a in
r dT .
e t a are
A
Lemma,
)S
a ifr f UiU a B
b l y e fo
s rmin g the model like U
p or f oi A ven A
c,o
a ana
n spfinterpretation
da l v e
Lfunction' is restricted to the te rms o f L,
cthat
i its
an M-mo d e l f o r L wh ich satisfies th e se t o f f o rs„ewe ,produce
os
iomulas
t o fsA wh ich are writ t e n in L. Since F is a subset o f this
set,
fa
l lbe
n oMO-satisfiable in L.
f oF ust
M
w
s -have been somewhat detailed in o u r exposition o f th is
M We
proof.
w e h a ve ru n a considerable r i s k o f
tm
hoTh a tadi s because
e
t
tlofailure
. in introducing saturated sets wh ich do n o t contain a ll
A
s
D
h
.o
B
w
y
t
h
e
164
J
A
M
E
S
W. GARSON
the terms o f our language. I t was therefore important to show
that the proof wo rks in spite o f the use o f such improvershed
sets, a n d so w e p a id p a rticu la r attention t o h o w me n tio n o f
the language plays a ro le in this proof.
Un ive rsity o f No tre Dame
J
a
m
e
s
W. Garson
BIBLIOGRAPHY
[1] A . BRESSAN, A G e n e ra l I nt erpret ed M o d a l Calc ulus , Y a l e Un iv e rs it y
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vol. 2. (1973), pp. 102-118.
[3] B . HANSSON a n d P . aiRDENFORS, A g u id e t o int ens ional s emantic s , i n
Modalit y , M o r a l i t y a n d O t h e r P roblems o f Sens e a n d Referenc e,
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151 R. MONTAGUE, T h e p r o p e r t reat ment o f quant if ic at ion i n englis h, i n
Formal Philos ophy (ed. R.H. Thomas on) Y a le Univ ers it y Press (1974),
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[6] R. THOMASON, M o d a l lo g ic a n d metaphy s ic s , i n T h e L o g ic a l W a y o f
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