MODEL THEORY AND MO DAL LOGIC
John L. POLLOCK
The interpretation of first-order logic in terms of models is well
known. Recently attempts have been made to extend the modeltheoretic results obtained for first-order logic to modal logic (9.
The best known of such attempts are those of Kripke H i n t i k k a
[3] and Pi, Carnap [5], and Montague [6]. These involve modifications o f the concept of a model. Although the modifications
lead to interpretations which appear superficially very different,
they can be reformulated in a way which shows that they are really
very similar to one another. It is the contention of this paper that these
attempts to formulate the semantics of modal logic all suffer from
an apparently irreparable difficulty.
Let us suppose that our object language L is an ordinary firstorder language, without predicate constants or individual constants,
and augmented with the new logical constant E (
semi-model
on D to be an ordered pair <D,G > where D is a non2
empty
set,
and
Gua function
). L e t
s
dmapping
e f iindividual
n e variables onto D, and
for
each
n
>
0,
mapping
R
a
onto
S(Dn), the set of all sets of n-tuples of elements of D. I f M
n
is
, the
t hsemi-model
e
s e<D,G
t >, let M * = G , a n d M
model
to
be
an
ordered
pair <G,K> where K is a set of semi-models,
*o
f
n G
eK,
and
for
every
H
=
w e sK, cH a n
a D .r N oyw
prime
model
if
only if K is the
*d
e fl Gia n.and
a set of all semi-models H such
r =
t ei o
that
L
eH, —
t G
n Now
we
can define truth in a model as follows, where val is a
*u
sfunction
ys mapping
m
bthe set
o of sentences onto {T,F) :
.s
y
li) I fs fa 8Rn
, and
. xt
t
h
a
T if and only if < G * ( x
/< = G
,
K
,>. . . l, x
(I) By 'modal logic' I mean the logic of logical necessity, as opposed to
) , . . . , G * ( epistemic
x
n
necessity, deontic necessity, etc.
iphysical necessity,
s
( en
a
r
a
guage
)drather
>i dthe symbols themselves, so that, e.g., 'pig' is not a sentence,
ibut2nthe
i v than
) c name of a sentence.
G *
u a l
T ( f ) ;
v a r i a
313
h
bel e s
t s
h
y
e
n
v ma l
( b0
o
, l,
Ks )
f wx
,
ii) v a /
i(iG
i)va l
(iv)val
c
, , K )(a,1)(x)fx T if and only if for every individual variable y,
=
R )v a l
y)
v a, l
T
—w
(i GK, K
)
p Now
)f f y let us say that a sentence is KH-valid if and only if it is true
=
in
every
model, and CM-valid if and only if it is true in every prime
E
a p
T
T
model.
It
is not difficult to see that the definitions of validity given
T ;
n
iid Kripke
by
and Hintikka (
f
fa
definitions
given by Carnap and Montague correspond to CM3
o
validity
)a c o r( r e s p o n d
n
constants
4
n
d
lt
o and identity,
K H but
- that need not concern us here with our
somewhat
impoverished
d. a l i d i t y language L. These authors all give more
o
y)v
or less the same intuitive explanation of their concepts of validity.
n
o
iT
, h e
lThe
fs
n e idea isnthat a model <G,K> can be interpreted as follows. Kis
a
the
set of "possible worlds", and G is the actual world. Truth in a
yva
ld u
possible
world H eK for a non-modal sentence is defined just as in
ia
tty h
h
ordinary
first-order
logic. Then to say that O p is true in G is just
flo
r
ito
e say that p is true in all possible worlds, i.e., true in every H eK.
f(s
f The difference between KH-validity and CM-validity is that in
o
G
d
i
vthe former
case, any set of semi-models on G* is said to generate a
r,
facounterpart
f
of the set of all possible worlds, whereas in the latter
e
K
e
lv)case,ronly the set of all semi-models on G * is taken as generating
a counterpart of the set of all possible worlds. An immediate ini(e
p
n
G
rtuitive difficulty arises in connection with both of these methods of
T
interpretation.
Intuitively, there is only one set of all possible worlds.
,t
ya
Why
then
are
we
allowed to assume that all of these (prime) models
h
H
K
n
mirror the notion of the set of all possible worlds? A sort of an
e
)cd
answer can be given by saying that in logic we are only concerned
K
v
ip
with the formal structure of the set of all possible worlds, and that
,a
r=
this formal structure is obtained by requiring truth in all (prime)
vltF
a
(r; (
lG
eto 3be an interpretation of logical necessity (in which alternativeness is a n
(requivalence
relation), and which does not employ his restrictions f or non)
areferring
individual
constants. Those latter restrictions are irrelevant for our
H
l H
tpresent purposes.
,o e
m
K (r
g
4
e
e
)=314
)
n
p
T IS
t=
; a
e
om
T
e
.f o
M
n
i ol
n ny
d tt
a
i a
g
v l
models. I find that answer somewhat obscure, and I think that that
obscurity is the source of the purely formal difficulties that I will
point out shortly. But let us suppose for the moment that this answer is unobjectionable. This answer does not clearly determine
whether we should consider all models, or just all prime models. I
think, however, that it does make any intermediate alternative,
which would consider more than just the prime models, but not
all models, highly implausible. I f this type of approach is to work
at all, either KH-validity or CM-validity must succeed in formalizing
our intuitive concept of a truth of the logic of L. I t will now be
shown that that is not the case for either KH-validity or CMvalidity.
The defects in KH-validity and CM-validity are illustrated by
noting that — D[(3x)fx & ( T
is
) CM-valid),
— f x ] and
i that
s — E(x)fx is CM-valid (although it is not
KH-valid).
To
see
that
n o t
K H these
- are in fact defects, let us reflect on what
vit means
a l toi say
d that a sentence is a truth of the logic of L. I would
propose
the
following
( a l t h o : Ausentence p is a truth of the logic of L
if and only if every interpretation of p is necessarily true (
g h
would
seem to be just what we mean by a truth of logic. Given this
5
idefinition, it tis clear that — 111(x)fx is not a truth of logic. There
). T h i s
will be interpretations o f the predicate letter f which will make
(x)Pc necessarily true, and so — 0( x ) fx will be false under some
interpretations o f f Thus any system of interpretation which would
make — El(x)pc valid must fail to formalize our intuitive concept
of a truth of logic.
Given our definition of a truth of logic, it also seems that the
sentence — 0[(ax )fx & ( ix ) , , ,
be
f 1 1valid.
i s Thisais because, given any sentence p, if in order for p to
be
true
t r u there
t h must be
o at least
f n things in the universe (n > 1), then p
lcannot
o gbe necessarily
i c , true. This is presupposed by our ordinary
interpretation
of
a
n
d first-order logic. But now, for any interpretation
off, the sentence (gx)fx & (gx) —fx requires for its truth that there
s
o
be at least two things in the universe. Thus no interpretation o f
s
h
o
u
l
it is necessarily true. That is, every interpretation of the sentence
d—E[(gx )fx & (gx) ,
f x( ] i s
t 5r u e .
315
B) u
t
I
af s s u m
i t n g
th
h
a
i
t
s
i
s
n
o
EIP D E I P )
is a truth of logic (and this is accepted by all of the above authors),
it follows that every interpretation of
—E [(gx)fx & (3x) — fx]
is necessarily true, and hence that
— 0 [(3x)fx & (3x) — fx ]
is a truth of logic.
Now we must prove the following :
(I) — 0[(3x )fx & (3x)—fx] is not KH-valid.
(11) — 11(x)fx is CM-valid.
First let us prove (I). Choose any semi-model G such that G* = >
2. For some f ER1, G*, and G*(f) O .
Then <G,{G}> is a model, and val ( G
Then
for every H E{G}, val ( H
,{
But
then
vat ( 0
,0{
Therefore,
— 0[(gx )fx & (3x) —fx] is not KH-valid.
G
},{
Now
let
us prove (II). Consider an arbitrary prime model <G,K>.
})G
For every f sR1, there is an H EK such that H*(f) G * (as G* 0 ) .
)[} ( g x ) f x
&
Then
val
(H.
19
(X)fX
>
0
[( ( g 3x ) f xx ) & = F.
B u t t h e n W I (G K W
[. ( 3g x x) ) f
(—
x
] <G,K>, it follows that — 0( x ) fx is
As this is true for arbitrary
/—
5xC
=
fCM-vaiid.
F
,
&
fT
a Thus
n x neither
d . ] KH-validity nor CM-validity succeeds in formaliv
a
(zing
a
T our intuitive
.x
concept of a truth of the logic of L I am in(
G
)
clined
to ascribe the defects o f these interpretations to the fac t
K
E
I x relativize the concept of a possible world to a certain set
that
they
f
(
X
)
K,
and
f]
X then vary K for their definition of validity. There can really
T
only
be one set of all possible worlds, and we must find a semantical
T
.
interpretation
which conforms to this fact.
.
State University of New York at Buffalo
J o h n
L. POLLOCK.
(
equivalent
to KH- validity. However, anything valid in that sense is also KH6
valid,
and
so
the above difficulty with KH-validity is also a difficulty with that
)
concept of validity.
I
n
316
[
2
]
,
K
r
i
p
k
e
REFERENCES
[I] Saul A.KRIPKE, „ A Completeness Theorem in Modal Logic", The Journal
of Symbolic Logic, vol. 24 (1959), pp. 1-15.
[2] Saul A . KRIPKE, „Semantical Considerations o n Mo d a l Logic ", Acta
Philosophica Fennica, Fasc. XVI (1963), pp. 83-94.
[3] Jaakko HINTIKKA, „Modalit y and Quantification", Theoria, (Lund), vol.27
(1961),pp. 119-128.
[4] Jaakko HINTIKKA, "The Modes of Modalit y " Acta Philosophica Fennica,
Faso. XVI (1963), pp. 65-81.
[5] Rudolph CARNAP, “Quantification and Modalities", The Journal of Symbolic Logic, vol. 11 (1946), pp. 33-64.
[6] Richard MONTAGUE, "Logic al Necessity, Physical Necessity, Ethics, and
Quantifiers", Inquiry, vol. 3 (1960), pp. 259-269.
[7] John L. POLLOCK, "Logical Validity in Modal Logic", The Monist, vol. 51
N
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