COMPACTNESS A N D LO WENHEIM-SKO LEM PROOFS
IN M O D AL LO G I C (t)
Bas C. VAN FRAASSEN
1. Introduction
In this article we prove a number of theorems in the semantics of the modal systems M (von Wright), B (Becker's "Brouwersche" system), S
quantification
t
theory. I n section 2. we describe the topological
method
used
(employed
elsewhere to prove similar theorems for
, a n d
free
and identity theory) i n a general way. I n
S quantification
5
section
3.
the
method
is applied to propositional modal logic,
( L e w i s
and
in
section
4.
(via
a
theorem on substitution in infinite sets)
) ,
to modal quantification theory.
a
n
d
2. Semantic entailment, compactness, and ultrafilters
t
h
e
i Ther present section concerns the semantics of arbitrary formal
languages,
e
x
tand enot jus t the languages o f modal logic . By a
language
L
n
s
i weomean a mathematical structure comprising at
least
a
set
of sentences W
n
V
L
t
, F}.
definitions are standard.
1set
,o a{ I ' n
d The following
a
,s e t
Definition
,o
f 1. A sentence A of L is v alid (1— A) in L if f v(A)
ea d m i =s Ts fori every admissible valuation v of L.
ab l e
Definition 2. A set o f sentences X of L semantically entails a
cv
a l u a t
sentence A of L ( X 1—A) in L if f every admissible
hi o n s
valuation v of L such that v(B) = T for all B in
m
X is such that v(A) T .
e
m
(I) Th e research f o r t his paper was supported under NS F grant s
b
GS-1566
and GS-1567. The author wishes t o acknowledge his debt to his
colleague
e
R. H. Thomason.
r
o
f
V
I
,
m
a
p
168
B
A
S
C. V A N FRAASSEN
Henceforth we shall adopt the abbreviations
1- 1
1
H
we shall,I use "member [subset] of H
,( A)
V
L " f o
r "
H
Im e =m b e r
{ b s e t ]
I,[ s u
<
The
following
vV f
definitions and theorem introduce our basic
,"o
topic
of
concern.
E
(,
1
V
A
,
o
Definition
)m
,1 3. The valuation space H of L is compact if f any set
of sentences X of L such that n H(B) = A has
,{
ie
BEX
lt
I:
a finite subset Y such that n H(B) = A
vte
BEY
(
m
1
h
A
1 4. A language L has unitary semantic entailment if f
e
Definition
n
,)
s
for any sentence A of L, any set of sentences X
=
such that X 1—A in L has a finite subset Y such
tu
(
T
that Y E
a
A
b
}
-A
)
sr
Definition
n
L has exclusion negation iff for every
yc
:, 5. Ai language
sentence
A
of
L there is a sentence A of L such
L
.
A
rc
that H(— A) = H — H(A).
li
E
a
W
p
Theorem I . I f L has exclusion negation then L has finitary
st
L
semantic entailment if f the valuation space of L is
sL
)
compact.
e
>
w
The
sh proof
; is immediate from the consideration that in a language
with exclusion negation, X1-- A if and only if n H(B) n H(— A)
a
e
BEX
n
= A . This theorem shows that in the usual context one may
d
cconcentrate on compactness, although for applications in comH
o
pleteness proofs, one needs in general a proof that the language
In
has finitary entailment. For this reason, we turn now to a general
,v
method
for proving this.
te
h
n
e
i
ve
a
n
lt
u
,
a
tn
L6WENHEIM-SKOLEM PROOFS I N MO DA L L O G I C
169
Definition 6. A filter F on a set X is a non-empty family of
non-empty subsets of X such that if Y, Z are in F,
so is Y n Z and any superset of Y.
Definition 7. An ultrafilter on a set X is a filter o n X not
properly contained in any other filter on X.
Definition 8. A filter F on a valuation space H converges to a
member v of H iff for every elementary class H(A),
vEH(A) i f f H ( A) E F.
We say that a filter converges if f there is some valuation t o
which i t converges (
an
2 ultrafilter on the same set (e.g. Gaal, p. 265, Th. 1); and if
on X and Y, Z subsets of X, then Z n YEE' iff
)F. is an
W ultrafilter
e
and either X- Y or Y is in F (e.g. Gaal, p. 265,
nZEFoand
t YEF,
e
Th. 2).
t h a t
a
n
y
Theorem 2. I f every ultrafilter o n the valuation space H o f
f
i
l
t
language L converges, then L has finitary semantic
e
r
entailment.
i
Proof. Let X be a non-empty set of L; { X
s
empty
subsets of X; A a sentence of L; and let A be false
i
cfor allo ic i.nThent define
}, i c i
t h e
a
i
n
e
f i n= {NTH
i t ev ( A ) T and v(B) = T for all B e X
d
n
mcl;
let
o1 131 n= { -} , mc i. By hypothesis,
A
f o r any mEI.
i
Moreover
7I
is
closed
under
finite
intersection:
x
1 f o r
n
n= '
,? n F f Y
n c H D ' e ) „ , for some m c il
1
is a filter on H. By a well-known theorem on filters noted above,
„there is an ultrafilter F on H such that F c F
w
1 Let us assume that all ultrafilters on H converge. Then there
is
h. a valuation v such that vEH(B) if f H(B)Er, for all sentences
B
e of L.
r (a) Let BEX; then { v a t : v(B) T and v(A) * T1 61, and
e
X ( t o it .
similar
2
, )
, O
Xu
„ r
1n
I o
t
t i
i o
170
B
A
S
C. V A N FRAASSEN
this is a subset of H(B), so H(B) EF H e n c e vEH(B) that is,
v(B) = T, for all B in X.
(b) Let BEX; then tv EH : v(B) = T and v ( A) *T 1 C
thise set has an empty intersection w ith H(A). Hence i f H( A)
were
F' would contain A ; so H(A) gr . Hence v a l( A)
) ; ina rn, d
v(A) T .
Therefore, X does not semantically entail A.
QED.
3. Application to propositional modal logic
The languages corresponding to the modal systems M, B, So S5
(see e.g. Kripke) shall be called L„,, L , L
over
4 the index set { m, b, 4, 5}.
, L
Definition
9. A T-model structure (T-ms) is a couple M =
5
< K , R >, where K is a non-empty set and R a
;
w e
u s e
dyadic reflexive relation, and such that R is symt
t metrico if t = b, transive if t = 4, and both tranr
a
n
gsitiveeand symmetric if t = 5.
Definition 10. The syntactic system Synt is a triple < A , S,
where A is a denumerable set (atomic sentences);
S is the set {A, , , ) , 0
W is the least set containing A and such that if
A, B are in W, so are ( A A B), — (A), EA) .
Definition 11. A valuation over a T-ms M = < K , R > i s a
mapping v of K x W into { T , F} subject to the
conditions that for all a in K and A, B in W:
v
va
e
.v
,(that
o aRf3, where we designate y relatiyized t o
a
,—in K as v
venient.
,,A
( a n d
o
()E Mm < i K , R > and member a of K such that
If there is a Tins
tA
Y'
,
we call v' a r-valuation.
=
p
a
r
e
A
)
T
n
B
i= t h e
s)fT e s
w
T
fi
h
e
ivf
rf.
e
cf,v
o
v$
L6WENHEIM-SKOLEM PRO O FS I N MO DA L L O G I C
1 7 1
Definition 12. The language L , is the couple <Sy n t , V C >
where V is the set of all T-valuations.
We call W the set of sentences of L
sible
t a valuations
n d
Vof ,L
t h e
s
e
t
Theorem
,o a n f d3. Every ultrafilter on H , converges.
Proof.
F(T)
o f a ll ultrafilters o n H , . W e
w
i tm
e ibe the
a r d Let
s family
define
the relation R(T) on F(T) as follows:
H
if F, reF(T) then FR(T) F' if f for all AeW such that H ,
r
f
o ( D Ar)
F I a1. Isit(T) = <F ( T ) ,R( T ) > is a r
Lemma
,
L
Lemma
, s .. H2. ,The mapping v o f F(T) x W into { T , F} such that
m
= T iff H
( A
F
in
F(T),
is a valuation over M
r
)
It is clear
that
(
each
ultrafilter
F on H , converges to the T-valua(A)
E F
tion v ' .
t). E
F
Proof
of lemma 1. I nFL,,, El A A ; hence H
r.ce R(T) is reflexive. In
f Lb,
o A 0 0 (cf. Kripke); let FR(b)F
H a
eE nAd )
(1
H ,r ( A ) ;
nb c on
hfilter
e K contains
n
- either
a
lH
)e(
contain
111,(0 B ) =
—
B ) . But K
l
A,
Hence R(b) is also symmetric.
r(iB
B )so this
o ris impossible.
H A
1
(It) InBL4,) C A 1—nELIcA; Fhence
l
if
FR(4)F,
and
W
O
A)
E
F,
then
H( LIA) E F'. Hence R(4) is
t,re
1 W
also
transitive.
I
n
L5
we
prove
similarly
that R(5) is both sym,-eP (1
—
,
metric
and
transitive.
Q
E
D
.
1
B
)
=
. 4
m
f
Proof of lemma 2. Because an ultrafilter F must contain either
1
aI ( 1 1 1 A ) . F o1
( B) or H,-H,(B) for any sentence B, vp( B ) = T iff
,iH
f (4 B ) ) ,
r
T.
contains H , ( B A C) H
l
snH (Because
o Ean ultrafilter
a
iff
it
contains
both
H,(B)
and
H,(C), v
T
.n
tsV
L E h A e)
l
i( BT ) h n e r He ( f Co )
F
'
(
o
l
v
,r That
r
e
w
o in two
u steps.
l
n
we
((BBFprove
A
C )
=
d
) ( E I' fBH,(111A E F then H ( A ) e F' for all F' such that F il( r ) r ;
lC
T (i)
i
f
f
yfollows
)=F )
by the deminition of R(T).
t= (ii)
= If H
h rT
o
T
p
.e (i 1 2 A )
rn g
f
o1 F
f
v- t= h
e1 e
T n
t( F
f
hB
172
B
A
S
C. V A N FRAASSEN
E11 is a family of sets such that each of its finite subfamilies
has a non-empty intersection. For if it were not so, then there
would be sentences B
L
1 , and H , ( 0 1 3
1
,1—D'A
1 3would hold in L , (let v ( El A) = F and v ,( EIB
for
i)„ , i 1sH, ui, nc ; hthen there is a 13 such that aR13 in the relevant
model
(t) E=h structure
l akT t) and v
F);
so
then
o
(
B
,
) H (=LIA) would be in F. We conclude that the family
E
B
of
supersets
of members of F * is a filterbase on 1-1,, included
T
F
.
I
in
an
ultrafilter
F'. Clearly FIZ(r)r and H,( A) g
fB
o u rB
,
i
QED.
it=
Finitary
tA
h entailment and compactness theorems for M, B, S4, and
S
5
now
follow
by theorems I . and 2.
1
e
n
h
o,
n
O
l
d
4.
Application
to quantificational modal logic
b
B
Iu
s
t,i Compactness vproofs can be extended to quantification theory
via
f•n a theorem on variable substitution in infinite sets of sentences
,• and a device due to Beth and Hasenjaeger (Beth, pp. 264
sections I I I and IV). We begin by extending
-2(• 65); van Fraassen
A
the
language
of
modal
logic to quantification theory along the
),
lines
o
f
Thomason's
semantics
f or the system Q
=
O
section
of identity, names, and definite
1 ( T 5),
h oomitting
m a s the
o ntheory
,
B
descriptions. This is probably the simplest way in which modal
n
logics
c an be extended to quantification theory, but the application of a general method may appropriately be shown in a
simple case.
Definition 13. The syntactic system GoSynt is a quadruple < V ,
P, S
,V is a denumerable set (the variables);
P
1 is a non-empty set, at most denumerable (the
predicates)
o f whic h each member has associa,
ted
with
it
an integral degree n > o ;
W
S„
e iis the set — , A, EL) , (I;
W
>
of
degree n and x
u
w
(i Pox __ x
h
n,s—(A),
x ( A A B), 0 ( A ) , and (x)(A).
e
)„t E a r e
r
W
vh a r i a b
e
„le e sa ,
:
ntl d h
ie f n
Aa
,s
L6WENHEI M-SKO LEM PROOFS I N MO DA L L O G I C
Definition 14.
ms ) i s
q u a d r
1 7 3
A T q-model structure ( T q
M = < K , R, D, f > where < K , R.> is a T-ms, D
a non-empty
set (the domain), and f a function
a
uwhich
p lassigns
e t to each n-ary predicate P
na set
o fJP")
f C of
) n-tuples
S y n of
t members of D, for each
member a of K.
Definition 15. A valuation over a tq-ms M = < K , R, D, f >
is a mapping v of the variables of C)Synt into D,
and o f K x W,, int o ( T , F) such that v is a
valuation over < K , R > and for all a in K,
( P"x
l
va.((x)A)
= T if f v ( A ) = T for every valuation
v
'
over
...x
with
.„ ) Mrespect to x.
w
= hM i =c < K , R, D, f > , member a of K, and
If there is a Tq-ms
valuation v overh
T M, we call the restriction v o f v to a a
valuation (over(overiM). s
lf i
Definition 16. The
1,
kf language
e
where
V
„
is
the
set
of
all
Tq-valuations.
1
v< .
We
(cf. van Fraassen, section I). A sub., turn now to e
x
vsubstitution
stitution
function
is
a
one-to-one
mapping of the set of variables
1 i s
t h c(e e
into
its
elf.
When
E
is
the
expression
e
c o u p l p
xe t
f(E)
=
e
l< e Q s y p
1 n et ,
e,*
= e, otherwise.
l2
V e
„
r) >, h
*nIt ise easy to see that such substitution cannot result in confusion
a
p
v
2
of
and that f( — A) = — f(A), f ( A A B) =
, bound
w variables,
e s(
f ( A ) , f((x)A) = (f(x))f(A). I f X is a
*(f(A)
d eA f(B)),
f i f ( EA)
n
x e
subset
of
W
e
„
q
„ * f(X) = { f( A) A e X} .
)
,w wh e
>
Convention:
d
e e r f i when f is the only substitution func tion being
discussed,
we
write
c E* for f(E). We shall say that v . satisfies a
n
e
e
set X W „ if f (f B ) T for all B c X.
e
„
,Theorem 4. For any substitution function f, any rq-ms M, and
(
*
any1set of sentences X is satisfied by a tq-valuation
=
3
f
1
(
1
e
)
,
,
)
i
f
174
B
A
S
C. V A N FRAASSEN
over M i f and only if f( X) is satisfied by a tg-valuation over M.
Proof. Let v be valuation over M; and k E D
the valuations v * and V
m
k t t for
;that
ho eallnbx ceV,w
e
e
dt he ov*(x)
f si =
n v(x*);
e
v a lvk(x)
u =
a v(y)
t i when x is y * and vk(x) = k when x is not
y*
for
any
yEV.
o n s
The
now
o theorem
v
e follows from the following two lemmas.
rLemma 3. For any tici-valuation v „ and any sentence A, v e
M
= v * ( A) .
t
s( A * )
u
cLemma 4.h For any -NI-valuation v , and member k of D
valuation ov er M , and any sentence A , 1
m
, v
a,
( A are
) proved
=
These lemmas
by strong induction on the length of A
exactly as are lemmas 1. and 2. in (van Fraassen, section I) except for the clauses concerning necessity which we prove below.
Proof o f lemma 3. Hypothesis o f induction: f o r all sentences
B of length less than A, and all valuations v
vo o v e r
M ,
v
o
Case
f
i
o
(
B * )
*'=all f3 such that c(1113, v
Therefore,
v,((11113)*) = v „ ( EJ(B*)) = T if f v,* ( ( DB) ) = T .
:(o
B
F
(Proof
Br* ) of lemma
=
4. Hypothesis of induction: for all sentences B
)T m
o
i
f A, and all valuations v
of length less than
.ta 1,1E09
n
d
,s o v e r
M ,
v
hCase
. n: From
o
l
y
he
hypothesis
of induction i t follows that for
o
eall
v( f3
such
that
aft(3,
v
B )
=
ho= (T
y Bi f)f v (111(B*))
= v,k((013)*) = T.
= *
pT
( o B i f ) f
QED.
tv= h
eo
s
T From
k on we shall need to consider only one substitution
here
ifunction
s
(.
B f*: let) us designate alphabetically the v a r i a b l e as x,
oand
= let f(-x
fiT
.
i)H = e
n
nxc
e
dv
,
u(
E
1
cB
)
t
i
L6WENHEI M-SKO LEM PRO O FS I N MO DA L L O G I C
1 7 5
A set X N V , in which no odd variables appear (bound or free)
we call a regular set; clearly f(Y) is a regular set for any Y W
We
q c all a Tq-valuation v , a regular valuation i f f v satisfies
the
. following conditions (
sen,
3 section I I I ) :
) ( c fWith
. each natural number m we associate a variable
B e ty„,h y ,
the first sentence of the for m (x)A; y
t .
p
p alphabetically
n
i 4sthe- first
2 6 is
2 odd
6 variable after y„ which does not appear
in
alphabetically
the first (n + 1) sentences of the form
,
5 ; t
(x)A.
1
ha
v
n
Then
v
,
is
regular if f v, ((x)A) = F only if v , ( ( y
F
rf i a
a
k= F, where (x)A is alphabetically the k t
s
-r
begins
quantifier.
n t ea universal
n c e
/shx )sA e)with
i cresult
h of replacing all free occurrences of x
Here (ytw
/x )Ahis the
in A byo occurrences o f y , after rewriting bound variables i f
necessary
d t o avoid confusion o f bound variables. We assume
withoutdproof the familiar result that v'(y/x)A = v ( A) if v' is
like v except
that v'(y) = v(x), for any sentence A of 1_, tg and
v
any Tq-valuations v ' and v.
a
The first
result we need is that to the satisfaction of regular
r
sets, only regular valuations are relevant.
i
a
Theorem 5. A regular set is satisfied by a valuation v
b
c
only i f i t is satisfied by a regular valuation v„'
l
, i f
a nover
d the same Tq-ms.
e
This is immediate from the following lemma:
w
Lemmah5: For any sentence A , i f vo, and v a r e valuations
i over the same Tq-ms and alik e w it h respect t o a ll
c variables which occur in A, then vo, (A) = v ( A) .
This is hproved by an easy induction on the length of A. We are
now in da position to prove that wtih respect to the questions of
o
e
(
mulations;
s t h e device i s d u e independently t o Bet h and Hasenjaeger
8
(Beth,
loc.
)
n cit.).
T
o
h
t
e
a
p
p
r
p
e
s
e
e
a
n
176
B
A
S
C. V A N FRAASSEN
finitary entailment ( and mutatis mutandis f o r compactness)
we need only consider the space of regular valuations.
Theorem 6. Xi— A in L
which
satisfy f( X) also assign T to f(A).
ei
, i f I f Xa1—n
d XI-I{ — Al is not satisfiable, so f(X1.1{
Proof.
A then
o
n
l
y
is not satisfiable (Theorem 4). A fortiori, f( XU { A } ) is not
satisfied
i
f by any regular valuation; hence every regular valuation
which
satisfies
a
l
l f( X) also satisfies f(A).
r If eX 1—gA does
u not
l hold, then X1.1{ A I is satisfiable, hence
f(XU{
a
r A } ) is satisfiable (Theorem 4) . But f(XLI{ A } ) is a
regular
v
a set,
l and
u must
a then also be satisfied by some regular
valuation (Theorem 5) . But then not every regular valuation
t
i satisfies
o
n f( X)
s also satisfies f(A).
which
QED.
Let R , be the set o f regular Tq-valuations and R , ( A) those
members of R w hic h assign T to A. Then 1-1
: Ac W„ ) > w ill be called the regular valuation space o f
1t
That
,
< h Ra s finitary
( R entailment
, ( A )now follows from Theorems
2, 6, and 7.
Theorem 7. Every ultrafilter on H
over a rq-ms with denumerable domain.
ii
, c oLet
n vF(T)
e rnow
g ebe
s the set of all ultrafilters on 1-1
Proof.
tdefine
o
a
the relation R(T), set D(T), and function f(T) as follows:
1t
t i o 3.n
,v aan R(T)
dl u: as ain Theorem
D
( the function assigning to each n-ary predicate P
f(T)
t tf (ht )e
n
set
each
F in F(T).
)
F
P "M(T)
)
Lemma (=6.
= <F ( T ) , R(T), D(T), f ( r ) > is a Tg-ms w ith
V denumerable domain.
=
{( <
Lemmaxt 7. The mapping v such that v(x) = x for all x in V,
and of F(T) x W„ into { T , F} such that V
lh
iff
F R.,( A) 0, f or a ll Ac W, for a ll F i n F(T), is a
,e regular valuation over M(T).
(A)
=
T
xs
„e
t
>
.o
:f
v
R
,a
(r
L6WENHEIM-SKOLEM PROOFS I N MODAL LOGIC 1 7 7
It is clear that each ultrafilter F on H
valuation
v
it
numerable
r, c o n domain
v e r gD(T)
e s V . Hence it remains only to prove
the
,t lemmas.
ao
t
h
e
rT
e
g
u
q
Proof of lemma 6. That <F(T), R( T) > is a T-ms is proved as
lforalemma
r
1. That D(T) is a denumerable set, follows from the
v a l o f QSynt. That the function f(T) is as required by
definition
u
a t 14. also follows immediately from its definition.
definition
i o n
QED.
o
v
Proof
of lemma 7. That v is a valuation over <F(T), R(T)>- is
e
proved
as i n lemma 2 . I n addition vp(P"x, x „ ) = T i f f
r ( Pn x
R
finition
o f f(T). F inally w e prov e t hat v
la x
vin
'
m
that
y, which is possibly not
.)o, ( (such
F
x ) Aa) v ' must
= assign a variable
T
x,
x, so that we need only prove that R,4x)A)EF
(iE
f, variable
f
d to
F the
iff
R
A
ie f
and
t) (R,-R,((x )A)) ( R , - R , ( ( y
fl
betically
the k " sentence to begin with a universal quantifier,
k(T ( y
w
<
v
/fby
x the
)) A )definition
) w h oferegular
r e valuations. This ends the proof.
(i x
QED.
(A
)
A
) x
o
,t )
iE
s
r,h F
a
l
p 6, hthis implies
a
- that 1_,,
faBy Theorem
v
addition
thef following
LOwenheim-Skolem
theorem is a corolq
h
a
s
i
n
i
t
a
r
y
o
l(d x
lary
to
theorem
7.
rle n t a i l m e n t ;
e
n
n
a
v)-iTheorem 8,
A set of sentences of L,,, is satisfiable only if it is
l'
> E
satisfiable by a valuation over a Tq-ms w ith dell
numerable domain.
f
vi
( T
a
k)It is easy t o see that this theorem can be generalized to the
recardinality of the set of variables, whic h need not to kept deF
ivnumerable. The extension to identity theory can also be carried
( P
out in the usual manner.
a
e
n )
b
xb
lcYale University/Indiana University
B .
C. VAN FRAASSEN
y
e
e
t
sp
h
yt
e
.p
d
B
e
u
rth
178
B
A
S
C. V A N FRAASSEN
BI BLI OGRAPHY
[1] EVERT W. BETH, The Foundations of Mathematics, Amsterdam, 1965
( 2 " ed.).
[2] STEVEN A. GAAL, Point Set Topology, New York , 1964.
[3] SAUL KRIPKE, Semantical analysis o f modal logic 1: normal propositional calculi, Zeitschrift f i r mathematische Logik und Grundlagen
der Mathematik, 9 (1963), pp. 67-96.
[4] SAUL KRIPKE, Semantical considerations on modal and intuitionistic
logic, Proceedings of a colloquium on modal and many-valued logics,
Helsinki, 1963, pp. 83-94.
[5] RICHMOND H. THOMASON, S o m e Completeness Results f o r Modal
Predicate Calculi, (f ort hc oming).
[6] BAS C. VAN FRAASSEN, A Topological Proof of the Ldwenheim-Skolem,
Compactness, a n d St rong Completeness Theorems f o r Free Logic ,
Zeitschrift f ar mathematische Logik und Grundlagen der Mathematik,
14 (1968), pp.245-254.