Download Modal Logic and Model Theory

Modal Logic and Model Theory
Author(s): Giangiacomo Gerla and Virginia Vaccaro
Source: Studia Logica: An International Journal for Symbolic Logic, Vol. 43, No. 3 (1984), pp.
203-216
Published by: Springer
Stable URL: http://www.jstor.org/stable/20015163
Accessed: 14/06/2009 14:15
Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at
http://dv1litvip.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless
you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you
may use content in the JSTOR archive only for your personal, non-commercial use.
Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at
http://www.jstor.org/action/showPublisher?publisherCode=springer.
Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed
page of such transmission.
JSTOR is a not-for-profit organization founded in 1995 to build trusted digital archives for scholarship. We work with the
scholarly community to preserve their work and the materials they rely upon, and to build a common research platform that
promotes the discovery and use of these resources. For more information about JSTOR, please contact [email protected].
Springer is collaborating with JSTOR to digitize, preserve and extend access to Studia Logica: An
International Journal for Symbolic Logic.
http://dv1litvip.jstor.org
giangiacomoModal
and Model
Logic
Gerla
Theory
and
Virginia
Taccaro
obtained by
Abstract. We propose a first order modal
logic, the QS?E-logic,
first order modal logic QS4 a rigidity axiom sch?mas :A ->O A,
adding to the well-known
entails the possibility
where A denotes a basic formula. In this logic, the possibility
of extending a given classical first order model. This allows us to express some impor?
tant concepts of classical model theory, such as existential completeness and the state
of being infinitely generic, that are not expressibile in classical first order logic. Since
we are also induced to compare the expressive
they can be expressed in I/^-logic,
Some
of
and
?,.
Lm
powers
questions concerning the power of rigidity axiom
QS4?E
are
also
1.
Introduction
examined.
work
This
order
represents
logic and
modal
a "bridge"
an attempt
to establish
first
between
we define
To this purpose,
classical model
theory.
a new modal system, the QS&E-logic, obtained by adding to the first
order modal logic QS4* a "rigidity axiom" schema (see also [3] and [4])
A->?A
(E)
A
where
theorems
mula
model
some
is any basic
formula. We
prove
completeness
we propose
In the semantics
for QS&E.
which
of the type
to another
O A is interpreted
in which
model
recognized
concepts
3.1) and
complete
expressible
3.17^ and
(Prop.
in classical
as the possibility
A holds.
This
of model
first
theory,
infinitely
order
such
and
compactness
a for?
for QS4E
a given
of extending
us to express
enables
as
being
existentially
are not
4.4) which
generic
(Prop.
see [6] Corollary
logic
(for example,
can derive
we
the well-known
results
Then
14.13).
of the class
inductiveness
Theorem
of infinitely
and of
generic models
a
of existentially
of
models
complete
given
theory
(Prop.
5.4)
from a more
about QS?E
(Prop. 5.3). Other
examples
general
proposition
of the possibility
of classical
of proving
results
model
via modal
theory
are
3.1 (equivalence
in Prop.
between
logic,
expressed
(iii)) and
(ii) and
about
the
the
class
in Prop. 4.4 (equivalence among
completeness
also
tizable
and
in ?
being
(see
infinitely
[6] pag.
(ii), (iii), (iv) and (v)). Since existential
generic
we
Ill),
are
are axioma
which
concepts
the
compare
powers
expressive
of ?S4Eand
Lm?m,
of view, we utilize
the above point
reversing
between
modal
and
classical
model
tionship
theory
logic
some results
the
axiom.
concerning
rigidity
Finally,
the
stated
in order
rela?
to prove
204
G.
Y.
Gerla,
Yaecaro
2. The 0S4E-Logic
a classical
sequel L denotes
its modal
extension.
Namely,
of ML
and
connectives
be primitive
In
the
first
we
and ML
->,
order
?-?, V,
with
language
assume
0
equality
d to
a,
H,
v,
abbreviations.
3,
QS?E
is used for the logic obtained by adding to the quantified modal
with equality QSi the axiom schemata
A~> DA
(E)
A
where
for a basic
stands
of an
tion
atomic
is an atomic
that
formula,
of L.
formula
Recall that the following formulas are axioms
and hence of QS?E.
(1)
n(A-*>B)->( oA->
(2)
nA-+A
(3)
?A~> D ?A
(4)
lx1
(5)
UA v DJ5-> D(A vB)
D(A AB) +* ?Aa?B
(6)
Observe
also
that
in QS?
x ^y
<->n(x
(8)
x ##
<->n(x ^y).
For
any
of ML,
J3X,...,
and r
then
Bn
e F.
r\-QmmA
Sometimes
r-Q8?EA, respectively.
entails
that, even
in
hold,
general.
An adequate
and
(or theorems)
of QS?
xnA)
formulas
following
are
of
consequences
(E).
=y)
theses"
nitions.
...
the
A we
Jfi-formula
or the nega?
formula
DJB)
...oan D-?->D(3a?i
(7)
not
logic
as usual.
If JTis a set of formulas
\-q$&-eA
...
for suitable
a
that
ABn->A
YQS B1
we shall write
of
YA and F hA instead
YQSiEA
from hypo?
of "deduction
Such a definition
A \- nA does
if the necessity
rule is assumed,
define
means
is obtained by the following
semantics for QS?E
defi?
A QSi-frame
is a pair W,
<) where W is a class and < is a reflexive
(
on W. A QS
structure
8 = (~#, W, <)
-modal
transitive
relation
for ML
consists of a QS4-frame
=
and a family M =
(W, <)
(Mw)weWof
and Jw
is the domain
of L, where
Mw
Bw
(DWJJw)
is
??
?
w'
of
that
such
<
(Jf^
Jfw,
implies Mw
interpretation
MWJ
is the
of =
a submodel
for every w, w' e W. The interpretation
of Mw)
in Wy
a single element
selected
in every Mw. Once
wQ has been
identity
classical
models
the
the pair (8fwQ) is called a QS?E-model
semantics
(
<),
W,
of L.
is obtained
as it is usual
imposing
by
for other modal
for J?I. Hbte that the proposed
conditions
logics,
but
not
also
only
on the
on
the
frame
interpretations
logic and model
Modal
a QS?E-mo?&l
for L defines
class S of classical model
=
use
and
g. We
M
is
the
map
<is
identity
S,
structure.
In particular,
this QSiE-mo?&l
to denote
a QS4U-modal
defines
of a given
first order theory
Every
for which
Symbol
of models
In
W
?
to
order
have
205
theory
of
account
take
the usual
extended
a set.
necessarily
If w e W, %,...,
structure
the
same
the
class
structure.
a type
of 0S4U-modal
structures
a
that
is
W
class and
assuming
such
definition,
an J??-formula,
ap eDw and A is
at w by ax,
in
8
is
{A
satisfied
then
the
we
not
relation
..., ap) is defined
by
A[ax,...,
8,w?
ap]
as
on
usual
for
of
the
recursion
A,
(cf,
example
[2]). The
complexity
is when A has the form
clause
DJB. In this case we set
only
interesting
eW
for
w'
w
if
such that w < w%
and
N
every
only
if,
B[a19...,
8,
ap]
A
if
write
We
we have
f=
N
8,w?
87w
B[ax,
A[at,
8,w'
...,ap].
...,a,p]
of
i=
A
of
elements
8
is valid in 8,
for every
sequence
%,...,
Mw,
A9
ap
if 8, w f=
A for every w eW and ^qs?bA (or simply \=A),A is QS?E-v?lid,
A
\=
it 8
for
every
structure
?S4U-modal
#.
If Tis a set of formulas of ML, a QS?E-mo?el
(#>w0) is a QS?E-rnodel
e
T1=
o? T it 8?wQ? A for every Jl T. We write
A),
??4JBA (or simply JP1=
of J1 is a QS?E
and J. ?? a logical consequence of F, if every 0$4E-model
-model
of A.
on
to prove,
is easy
by induction
forms
of substitution
weak
following
of A,
complexity
of equivalents
do hold.
It
is an J?L-formula
If A'
(51)
8 ?B ?-*B'
implies
If B does not
of A
formula
(52)
fall
in the
of the
N (JB <-*JB')~>(^
3.
and
Completeness
Now,
theorem
we
the
occur?
S?A++?.
of
range
form
DC,
D,
then:
i.e. does
not
occur
in a sub
<r?A').
compactness
shall prove
a strong
theorem
completeness
and a compactness
for QS4E.
3.1
Proposition
(Strong
with
consistent
ML-formulas,
-model of F.
Proof.
out
B
that
some
from A by replacing
a formula
o? A with
2?', we get:
obtained
of a subformula
rences
the
The
the strong
Lei F be a set of
Theorem).
to QS4?E. Then
there exists a QS?E
Completeness
respect
strong
completeness
result
completeness
can be proved
for QS?E
by pointing
for QS? already known
(see, for example
[5]). To this purpose, observe that if Fin consistent with respect to QS4?ET
then
FuE'9
where
JE/'=
is consistent
with
{O (Va?!... xn(A~> uA))?A
respect
to QS4s.
Indeed,
any basic formula}
if we
can derive
a contradiction
206
G.
Y.
Gerla,
Yacearo
...
where
aAp aBx a
\-QS4tAxA...
ABq->C
and BX9
formulas
of E' and F, respectively.
suitable
A19...,
Ap
...9Bq
is an over-logic
of QS?9 we have
also
A ...
Since QS4?E
a
VQsmAx
AJ.^
...
a
Rule
and
we
Axiom
Schema
a!?!
by Necessity
Now,
(E),
aBq-+C.
for i = 1,...,
we get
Then by Modus
obtain
Ponens
a
p.
YQS?EAi
VqswBi
a contradiction.
and therefore
a ...
ABa->0
\-q8hEC9
of FuJET and the strong
From
the consistency
theorem
completeness
in 0S4,
FuE'
from
C
then
are
=
(89 wQ)9with 8
for QS?9 it follows that there exists a 0$4-model
(J(9
is a set, w0 e TT, <
is a reflexive
and transitive
relation
W9 <), where W
=
=
a
on W,
of
is
e#
that
such
models,
family
Jf^
(Dw9 Jw)
(Jlf?,)??^
=
??
as identity
and Dw s Dw> for every w9w'
is interpreted
< w'.
eW9
w
as
we
assume
to
not
it
is
restrictive
that
for
w0 <
Moreover,
know,
w e W. We
every
is a submodel
Jf^
pothesis,
89w0?
w
g4
Since
prove
w ?
~|JL-> q( 1^.)}.
..?xn(A->
aA)and$,
Vxx...xn(
e
are
of
such
that
elements
an
9an) JW(A)9 whence
Dw
(%,...
A
?
and
? A \aX9...,
therefore
then
wf
aw]
(%,...,
?j
89
[%,...,
an]9
if
then
$, w' t=
an].
(JL). Conversely,
aj gJw\A)
A[%,...,
(a19...,
? Vxx
^-ary,
then$,
if
%,...,
So,
89w
for every w9 w' e W9
if w ^.w'
then
that,
of Mw,. Now,
let jB be a basic
formula,
then, by hy?
if J. is atomic
and
DJB). In particular,
...xn(B->
uNxx
shall
89w
(%,...,
This
-model
an) eJw(A).
that
in such a way
is a submodel
the 0^4-E-model
that
Proposition
3.2.
of Mw,
given
W
type
1=
JL[%,...,
=
an~\ and
and hence
(#, w0)
IJDw)uW]
=
weW
is a set of ML-formulas9
card{L). Then,
can work
we
and A
is a QS?E
is constructed
3.1
a proper
is sometimes
class,
for the 0S4E-logic.
theorems
If F
therefore
^(JL)nD^.
by Proposition
is a set and card[(
that W
the fact
despite
L?wenheim-Skolem
$, w
have
In conclusion, ^(A)
that Mw
proves
of r.
Observe
we
? "Ti.-* D( T?),
out
is any ML-for
mula9 then r ?qS*eA if and only if F YQS?EA.
Proof.
If r
assume
is obvious.
the proof
is inconsistent,
Otherwise,
is consistent.
then Tu 1-4.
3.1, there
By Proposition
Fi= JL.
the hypothesis
contradicts
This
of Fu~l?.
a
axiom
of
that every
to prove
of routine
it is
matter
? A and F non hA:
a 0?4.E-model
exists
r
Conversely,
QS?E is QS4JS-valid. Moreover, the rules of inference
This proves
necessity rule), preserve QS?E-validity.
(in particular the
that VA implies
?A.
... a
for suitable Ai e F
hAxa
if r VA9 then
An-+A
Now,
...
if (89 wQ) is any Q?4jE-model
reason
Now,
?Axa
AAn-+A.
a
s F and, therefore,
of
is
{89 w0)
?>S4E-model
{AX9 ...9An}
Thus F1= A.
Proposition
3.3
(Compactness
Theorem).
Let F
and
for that
of F,
then
of A.
be a set of ML-for
Modal
logic and model
mulas
such that every finite
a
207
theory
subset of F has a QS?E-model.
Then F has
QS?E-mo?e?.
It
Proof.
4.
we
Now
not
shall
definible
and model
that
some
in classical
first
show
3.1.
Proposition
models
complete
Existentially
are
from
follows
completeness
known
well
order
of models,
classes
in a
are,
logic
which
definable
sense,
in QS4?E. The following definitions generalize a notion of classical model
theory. A QS?E-model
(89wQ) of ML is existentially complete if, for every
w eW
that
such
If E
is a class
w0 < w9 MWQ is existentially
complete
for L and M e E
of classical
structures
in Mw
[6].
it is easy to prove
that the 0$4U-model
(?7,M) is existentially
complete with regard to
the above definition if and only ifM is existentially complete in E with
the
to
respect
classical
Proposition
definition.
The
4.1.
{S9Wq) is existentially
(i)
<->A
<->A
89 Wq ? 0-4.
89 wQ ? 0-4.
(ii)
(iii)
are
following
equivalent:
complete ;
for
every
for
every
existential
L-formula
universal-existential
JL;
A.
L-formula
of Mw9
then
Proof.
Let
be elements
(i)=>(ii).
89w0
aX9...ap
? 0-4 [a19...,
if and only if there exists w e W
that wQ < w and
such
ap]
a formula
to say
of L9 this is equivalent
89 w ? A[aX9...,
ap]. Since A is
so
is existentially
that Mw ? A [%,...,
in Mw9
complete
ap~\. But MWq
?
and
therefore
A
conclusion
?
In
...,
w0
Mw
89
[ax,
A[a19...9
ap1
ap~?.
and thus 89 w0 ? 0-4. <-> .4..
89 wQ ? ?A->A9
A
the
.L-formula
Let
be
universal-existential
(ii) =>(iii).
Va^...
...
.B quantifier-free.
Then
with
#n 3 #!...
by (4) f=oV^j....
...ymB9
xn3yx
so
...ymB<->3yx...ymB9
...
1?jiyl... ymB.
^JS^oV^i.oon3y1
The
the
above
S2
by
of
...ymB9
and
yields
89
(iii) is proved.
(2)
it follows,
Proposition,
S9w0?
oHyx...
wQ ?Va?x...
#n3yi...
hypothesis,
89w0?<>\fxx...xn3iyl...ymB-*$x1...
converse
(iii) =>(i). Immediate.
From
by
Now,
...ymB-^fxx..:xn03yx...ymB.
for example,
class of all fields then a field F is algebraically
if E
that
is the
closed if and only if (E9F)
is a QS?E-model
L
of F = {J.?-?<>J./JL
where
existential
X-formula},
is the language
of the theory of fields. In general,
if ? is a class of structures
of the same type,
of axioms
in QS4E
F as a system
then we can assume
for the existentially
of
E.
elements
complete
Now
we
want
-modal
structure
w < w'9 Mw,
is an
of model-completeness
tial
completeness
to express
8 is model
the notion
complete
extension
elementary
is relative
of model-completeness.
if, for every w9w'
of Mw.
to QS4?E-mod&\
A class E
to ^S4JB-models.
Observe
structures
of models
A QS&E
such that
eW
that
the
while
concept
existen?
of L
is model
208
G* Gerla,
Y.
Yaeearo
in the sense of the classical
definition
[6] if and only if the asso?
complete
structure
to the above
ciated QS?E-modal
is model
complete
according
definition.
are
Let
4.2.
Proposition
be a QS?E-modal
8
structure,
the following
then
equivalent:
S is model complete]
(i)
? A*-* D-4 for
8
(ii)
every L-formula
A*9
8 ?A?-* uA for every ML-formula
Proof,
(i) =>(ii). Obvious.
A.
(iii)
(ii) => (iii). We
of d in A. If n =
If n >
hypothesis.
on
by induction
proceed
0 then A is a formula
the number
of L
1, let DJS be a subformula
and
of A
the
such
n of occurrences
assertion
holds
by
.L-for?
is an
that B
mula. From (ii) it follows that 8 ? JB?-?dJB.Then, if A' is that formula
obtained from A by substituting B for dJBwe have, by inductive hypothe?
sis, S ?A'<-> D.4/ and, by SI, S ?A++A'. Still, by SI, it follows 8?A^nA.
(iii) =>(i). Immediate.
In
other
only
equivalent
of D.
5.
Infinite
Let
"theory"
{AIS
to the L-?ormula
?bea
modal
is model
8
i.e.
among
A(xX9...,
if and
complete
Jfi-formula
is
occurrence
xX9...,
a formula
free
of L whose
xp)
w e W and %,...,
elements
ap
xp9
So, the relation 8, w \?A [%,...,
A QS4JE-model
of L and
...9xp)
only
($, wQ) is infinitely
al9 ..., ap elements
or $, Wq\?
..., ap].
A\aX9
The notions
of infinite
ones
if 8
ap] ((89w) infinitely forces A
as the
satisfiability
inductively
concerns
for what
the negation,
if and
ap]
..., ap].
8jW\?
"UB[a19...,
8, w non\? B[a19
of L
that
every
"collapses",
from it by deleting
obtained
every
structure,
are
variables
in al9 ...,
ap) is defined
? A [%,...,
ap1 except
have:
classical
says
?A}
forcing
or bound
of Mw.
4.2
Proposition
words,
if the
if for all w' e W9
relation
we
for which
such
w < w',
that
generic if for every formula A
of
either
89w0\?A[aX9...9
MWq9
coincide
and infinitely
forcing
generic
a
to
class
is the modal
structure
associated
w
89
(x19...
with
ap]
the
of models
[6].
Now,
introduce
in order
Definition
a
suitable
5.1.
to
the
express
translation
The
map
infinite
r from
r from
forcing
i-formulas
L-formulas
we
in the QS4?E-logie,
jMX-formulas.
into
into ML-formulas
is
logic and model
Modal
defined
by
209
theory
on
recursion
the
of
complexity
JD-formulas
by
setting:
if A is atomic, then xA = A ;
if A = JBvO, then x{BvG) = rJSvrO;
if A =BaC9
then x{BaG) = tJBatC;
if A = lxhB(xh), then r(3%JB(%)) = lxhxB(xh);
(i)
(ii)
(iii)
/iv)
if A
(v)
=
~1B9 then
x is not
Obviously
from
that
A<-+B
infer
to the
choice
of a,
=
t(~1B)
DHrJB.
cannot
i.e. we
equivalence,
x is strictly
translation
related
of L. We made
connectives
this
with
compatible
the
The
v,H,
x(A)?-*x(B).
3, as primitive
the infinite forcing as defined,
a choice in order to be able to describe
in [6]. Observe
for example,
in literature,
known
already
The
following
The
Proof.
The
following
Proposition
(i)
[7].
shows the
8, wQ \?A
is by
proof
proposition
relation
[a19...,
ap]
if
induction
on
the
gives
The
5.3.
a modal
into
translations
language
are
see
proposition
5.2.
Proposition
that
following
us
two
between
and
only
complexity
useful
properties
t=and
if
IN.
8,w0?
(rA)
of A.
of
r.
hold:
hT("1-4)->nT-4;
\~xA-^DrA.
(ii)
(i). It follows from (v) of Definition 5.1. and (2).
(ii). The proof is by induction on the complexity of A. If A is atomic,
Proof,
then
xA = A
and
(ii) follows
from
axiom
schema
(E).
h t(13)~>d(t(JB))
Let A = jBvO, then by inductive hypothesis,
and
From
(5)
Thus,
hx{G)->n(x{G)).
\-r(B)Vt(C)-^n(r(B))va[r{G)).
we also have
and
therefore
J-t(JBvO)
bx(B)vx(G)~>n(x(B)vx(G))
->a(r(J5vO)).
If A = BaG, by inductive hypothesis, we get \rx{B) ax{C) ->
a
n(x{B))
i.e.
\-x(BaG)
and, by
(6), Ix{B)ax{G)->?(x(B)ax{G))9
ad(t(0))
^Dx(BaG).
If A =lxhB(xh)9
then, by inductive hypothesis,
bx(B)->?(x(B))
and hence
From (4), it follows that bBxhx{B)
V3iXhx(B)->3xhn(x{B)).
and therefore \-x(3xhB)->DxQxhB).
->n(3xhx(BJ)
=
if
A
that'h
~\B, we must'prove
Finally,
?(l(xB))^n{?(l{rB))),
but this follows from (3), so (ii) is proved.
Proposition
(i)
(ii)
(iii)
5.4.
The
following
are
equivalent:
{S9wQ) is infinitely generic,
89 w0 ? r(iv"li)
for every L-formula A;
89w0? Q(tA)<-+tA for every L-formula A-,
210
G.
~1(xA)*->xC~]A)
for every L-formula
?
A.
w0
every
A*-*x(A)
for
89
L-formula
(iv)
89w0?
(v)
Proof,
condition
QS?E-mod?l,
(ii) of Definition
5.1
(ii)=>(iii). From 89 wQ ? r(iv"1i)
thus,
8,w0?
therefore,
=> (iv).
(iii)
This
xAvnixA.
89w0?
Yaccaro
A;
It follows from the definition
(i)o(ii).
Y.
Gerla,
and
of infinitely generic
5.2.
Proposition
it follows 89 wQ ? xAvx{~]A)
that
proves
0{xA)-^xA.
It suffices
to observe
that
and,
? ~in{~~]xA)-+xA
89 w0
and
(iii) 89 wQ? nC~]xA)*-*~~]xA.
by
(iv) =>(v). We proceed by induction on the complexity of A.
If A is atomic, (v) is obvious.
If A = B vO, A = BaG or A = lxhB(xh), the inductive step follows
the
from
then
IB,
r.
of
definition
=
If A
"IB if and
8,w0?
by inductive hypothesis,
only
if S, w0 non? B
if and
only
if,
8, wQnon ? x{B). Then, (iv), yields 8, wQ ? IB
if 89w??
t("~]JB).
only
A or 89wQ?
Since
(v)=>(i).
89w0?
?i<->ri
from
and 89 w0 ? "lA^x(lA)9
if
and
? xAvx(lA).
Condition
and
QS4?E-mod?l
of
any
axiom
in a class E of models
In
this
we
sense,
generic QSiE-mod&h
example,
structure M
F
as an
axiom
{x(Av~]A)?A
is infinitely
generic
is a QS4?E-mod?l of F.
of L if and only if {E9M)
regard
for
system
are
F =
for
systems,
a classical
In particular,
.L-formula}.
89w0
89 w0
5.2.
Proposition
suitable
hypothesis,
it follows
(i) follows from the definition of infinitely generic
5.4 shows that the infinitely
Proposition
the models
~~\A and, by
89 w0 ? ivli
the
infinitely
generic
structures.
6.
classes
Inductive
theory,
result
a well-known
we
and
infinitely
generic
classes.
inductive
complete
constitute
To
an example
to classi?
of 0S415-logic
of an application
on QS>4?Mogic
from a general
derive
result
will
the existentially
model
in classical
theory.
Namely,
to give
In order
cal model
this
some
aim
given
of models
of L
the union,
(J Mn9
ascending
that if n < m
of an ascending
of
structures
those
generalize
a sequence
(-3fn)n6jy
of
For the definition
that
introduced
of models
chain
class
ofus
then Mn ? Mm.
see [2].
chain
neN
^wfe
Definition
inductive
such
are
definitions
An
in literature.
of a given
structures
in E
A
6.1.
if for
subclass
every
neN
of L
then we
of a class
chain
ascending
from \J Mn e E it follows that
of all models
E'
say
E
of models
(Mn)neN
of
of L
elements
\J Mne E'. If E' is inductive
neN
that
E'
is inductive.
is called
of Ef,
in the class
logic and model
Modal
Definition
An
6.2.
of L if E' =
Proposition
Besides,
is
A
in a class E of models
is inductive
is inductive
in E. A
is inductive if it is
E.
class
6.3.
if A
JfX-formula
{Ml E9M ?A}
in every
inductive
211
theory
Let
be an ML-formula,
is inductive.
A
then
nA
is
inductive.
0A*-?A
inductive9
is inductive,
of simple
conside?
nA
by means
an
E
of
A
for?
of
rations. Moreover,
let
be any class
models
inductive
L,
=
E1
of
? <>A*->A} and
chain
mula,
(Mn)neN an ascending
{M eE/ E,M
The
Proof.
elements
formula
of Ef such that M =
{J Mn e E. We
__ neN
shall prove
that M e E\
__
? 0-4. Since every Mn
that E9M
is a sub?
for
neN.
every
E9Mn
As, by hypothesis^
? 0-4.?-?.4.? we also get E,Mn
?A for every n e N. A being
E,Mn
inductive,
it follows
? A.
In
that E9M
M
?
and
therefore
<>A-->A
conclusion,
E,
that
? OA*->A.
is E,M
of M9
model
we
Assume
? 0-4
have
Z9B\rOA<->A.
Proposition
the classes
in E,
Let E
6.4.
of models
respectively.
are
which
${E)
Then9
be a class
of models
existentially
and *f{E) are
of L
complete
inductive
and ?{E)
and <f{E)
and infinitely
generic
in E.
Proof. S(E) is inductive by Proposition 4.1. (ii) and Proposition 6.3.
From (2) and Proposition 5.3 (ii), it follows that VxA^UxA for every
Jfi-formula
A.
from
Then,
Proposition
5.4
(ii) and
Proposition
6.3,
it
follows that <f{E) is inductive.
7. QS4E-logic
L
and
m-Logic
of QS?E
to compare
section we want
the expressive
powers
some difficulties,
the models
since
raises
a. This
comparison
can ask
of QS4E
In spite of this, we
and of Lm m are different
"objects".
into
the
from
Jfi-f
there exists a translation
the
ormulas
whether
/
JL^?,
if M
for any class E of
? f(A)
-formulas
such that E9 M
?A if and only
we can
o? L, any M e E and any formula
A of ML.
models
Conversely,
In
and
this
La
ask whether
Jii-formulas.
prove
Then
this, we
a model
there
exists
The
a
answers
similar
to both
in the sequel
suppose
of L is any
set, and
translation
into
a-iormulas
In
to
order
negative.
from
are
Lm
questions
that L is the pure identity
a QS?E-mod&l
structure
any family (Xw)weWof sets, with W,
( <) a QS4E-?mme9
language.
for ML
is
such that Xw ? Xw>
for every w, w' e W such that w < w'. In particular
class 0 of sets
every
a QS?E-mod&l
defines
shows
The following
structure.
proposition
that,
if E is the class of all sets, then, relatively
structure
to the QS?E-mod?l
the modal
operators.
E, we can eliminate
Proposition
7.1.
Let
of all models
of L. Then for
A* such that E? A<-?A*.
L
be the pure
identity
A
every ML-formula
and E the class
language
there exists an L-formula
212
G.
Proof.
we
First,
assume
A
that
uB
B
with
of Quantifiers
of Elimination
from the theorem
=
an
Gerla,
Yaccaro
Y.
.L-formula.
for the pure
Now,
identity
of the type
to a formula
(see [2], Theorem
1.5.7), B is equivalent
...
are
without
the
A
where
L~?ormulas
quanti?
AJs
(AxvBx)
a(ApvBp}9
a ...
h dB?-* D
of L. Then,
fiers and the BJs are sentences
A{Apv
((AxvBx)
language
vBp)) and by (6)
(9)
\-nB<r*n(AxvBx)A...
from
Moreover,
for i =
(5) it follows,
p,
1,.,.,
h QA{ v DBi ~>D {AtvB4)
(10)
and
AD{ApvBp).
(1)
by
(11)
As
b D (A- VJ5<) -> 0-4*
each
bAi*-* ?A{
(12)
h d("U<->-B<)->(D"TA?->dB<),
A{
and
i.e.
V DJB,-.
is quantifier-free,
from
axiom
VAi<r^ 0-4^.
from
(10) and
Then,
schema
(11)
(E) it follows
it follows
that
that
ha(?,vJB<)?->?<VDB<.
of the pure identity
sentences
the BJs are sentences
These
Now,
language.
are equivalent
to the assertion
of the model
that the cardinality
belongs
or to a finite
or to the comple?
to an empty
either
set ? of finite cardinals,
case dB* is always
of such a set. In the former
false in the QS4E
ment
modal
(13)
structure
E?
E,
and
uB^BiAlB^
In the latter, E9M?
if and
only if card(M)
that there
expressing
(14)
uBi
if and only if for all JF 2 M9 card(M') $1,
> maxi.
are more
If m = maxi
and
m
than
elements,
Gm denotes
then
an
.L-formula
E??Bi*+Cm.
In conclusion, from (9), (12), (13) and (14) the desired result, for the for?
mula
dB,
follows.
let A be any J?L-formula.
In this case one proceeds
by induction
Now,
=
on the number
n o? occurrences
of d in A.Jin
is obvious.
the
assertion
0,
a subformula
an
If n > 0, there exists
of
A
JD-formula.
B
with
dB
Now,
let G be an i-formula
obtained
such that E ? dB<-*0
and A' the formula
from A by substituting G for dB. Then, from SI, it follows that E ?A*-*A\
Since
the modal
an X-formula
Observe
E and
that
<r^BiA^]Bi
to
alleged
degree
such
A*
that
of A'
that
there
is n~-l,
by inductive
hypothesis,
27 NA'*-* A* and hence
E? A*-*A*.
(13) and (14) hold only for the $$4JE-modal
can not
exists
structure
h dB?
assertions
be substituted
they
by the stronger
7.1 we are not
that in Proposition
and h DJB*?->(7m. It follows
E? A** A* by VA*-*A*. On the contrary,
substitute
Proposi
logic and model
Modal
213
theory
tion 6.3 shows that such a translation
of QS?E
in the classical
logic is
impossible.
The following proposition proves that, in general, there is no reduction
of
J&?it<D-logic
to QS?E-logic.
7.2.
Proposition
no translation
exists
We
Proof.
7.1.
in Proposition
into the ML-formulas
Then
such
there
that
if and only if E9M ?f(A)
A
every Lm ^-formula
be as
the Lm ^-formulas
f from
M?A
for
E
and
L
Let
Let
absurd.
by
proceed
e E.
every M
and
A
be
a formula
of
Lmvm
such
that Jf ?A if and only if M is finite. Then, by hypothesis, E9 M ?f{A)
if and only ifM is finite. Now, from Proposition 7.1 it follows that there
ii and only ifM K(/(-?))*
exists an i-f ormula (f(A))* such that E9M?f(A)
wh?e it is well-known that finiteness is not definable in first order logic,
a
contradiction.
7.1, it follows that E, M? Aii
From Proposition
eE
M
where
is an L-? ormula.
and A*
relatively
that,
In spite of that,
a QSeE-modsl
which
L0
have
logics
is also
from
translation
QS?E
the
this
*-! ormula,
into
does
Lmit(0
that there exists
shows
proposition
following
from
does not allow
translation
any
no
of QS?E
there is
in L
translation
in general,
m. Then,
a
to E9
proves
exist.
A
Since
and only ifM ?A*9
an L
incomparable
QS?E
m and
into
these
powers.
expressive
where X and Y are sets
Proposition
7.3. Suppose & = {X,Y,Z}
with n elements, X $ Z and Y c Z. Then there exists no translation f of
into
ML-formulas
if and only if M
09M ?A
for
Proof.
and M
A
every ML-formula
the
e&.
proceed
by absurd. Let A be an L-? ormula which
of just n elements.
Then 0, X ? ?A9 and 09 Y non
Y non
N/(a(-4.)) and
?f(?(A)).
hence, by hypothesis, X
models
?f(A)
We
the existence
Indeed,
that
such
Lmv0>-formulas
X
and
Y have
same
the
of the pure identity
same
and
cardinality
therefore
expresses
? nA and
This is absurd.
are
isomorphic
language. This entails that X and Y verify
J&? ?-formulas.
8. Overiogics of QS?E
An
can
interesting
such
obtain
new
natural
conditions
to extend
question
overiogics
on modal
the
is to
by
structures.
"rigidity"
the
examine
either
adding
From
axiom
new
the
schema
of QS?E.
We
or by imposing
of view,
it is
first point
class of
(E) to a larger
overiogics
axioms
214
G.
be
QS?Ex
can
We
formulas.
the
A
(E) to every i-formula
so obtained.
The
following
extend
overlogic
Gerla,
Y.
Yaccaro?
Let
quantifierfree.
shows
proposition
coincides with QS?E.
that QS?Ex
8.1.
Proposition
It
coincides
QS?Ex
QS?E.
for every A quantifier-free
we have::
on the
We
induction
(a)
proceed
by
"0S4e-4-<->CL4.> (b)
\-qS4EA++qA.
an
of A. Let A be
atomic
then by (E)
complexity
formula,
bQ84lEA<r+ nA.
we
In order to obtain
consider
(b),
~}A, then, by (E), bg8?E~~\A<r+n~lA?.
From
that
this it follows
i.e. (b). Let A = BvG,
then by
bgS4E0A*->A,
so
and
inductive
hypothesis,
bQmEG^uG9
bgS?EB<-?nB
bgSmB\/G\
then
but
+*nBvnC9
bQ8?E?BvnG->n(BvC),
bQSUSBv?<->n{BvC).
If A = BaG,
In the same way we prove
that
we
bQmEBvG<-><>(BvG).
=
as
If
A
then
above.
inductive
by
proceed
HB9
hypothesis,
dB,
bg8mB^
so
from
that
is
the
inductive
<->~lB,
while,
bg8eEA*-*0A,
bg8mO"iJB
it follows
that
hypothesis
bg8iEB<r?<>B
bgS4EA<-*nA.
Proof.
Now,
we
to prove
with
suffices
that
(E) to every
extend
existential
formula
of L9 and
A
denote
this system with QS?E2
8.2.
Proposition
QS?E2
coincides with QS?E.
=
a quantifier-free
From
formula.
8.1
dB and, therefore,
Proposition
bQsmB*-*
bg^^w^Bixj,}
we have
Since
+->]i.xhnB(xh).
bQS?ElxhnB(xh)->n3xhB(xh)9
bQSmlxhB
that
This proves
o A.
(%)-> d3%J5(%).
bg8?EA*->
Proof.
A
Let
Now we use QS?Ez
every
Proposition
8.3.
From
is an existentially
riot existentially
B
(E) to
for the overlogic obtained by extending
-4.
L-? ormula
universal
Proof.
with
3xhB(xh)9
it follows
that
The
overlogic
4.1
Proposition
it follows
complete QS?E-mod?l.
This
complete.
is a proper
QS?E3
proves
that
extension
every
model
of QS?E.
of QS?Ez,
that are
But QS?E has models
that
QS?Ez
is a proper
extension
of QS?E.
with
It is possible
the case
to consider
in which
of such a type, but we conclude
extensions
many
denote
We
to
is
extended
every ?-formula.
(E)
this overlogic with QS?E*
Proposition
8.4.
The
logic
QS?E*
is a
collapsing
proper
extension
of QS?E.
Proof.
Proposition
It suffices to use the equivalence
4.2.
between
(ii) and
(iii) of
logic and model
Modal
to consider the overiogics
It is also interesting
obtained
some
by adding
For example,
215
theory
Proposition
as
is added
The
8.5.
are
of QS?E which
as new
formulas
axioms.
be the system in which the Barcan formula,
let QS?E+BF
JBJF, $3xA->lxOA9
modal
of the well-known
a new
axiom.
logic QS?E+BF
is a collapsing
extension
of QS?E.
Proof.
and
If A
We
prove
bgS?E+BFA*+oA.
or A
is atomic,
in the Proposition
for every
?-formula
that,
A,
hQS?B+BFA+*nA
on
We
induction
of A.
the complexity
proceed
by
=
=
=
or
or
A
A
the
is as
proof
BaG,
"IB,
BvG,
If A = 3%B(i%),
8.1.
then
and
fiis^asM+j^-B-^aZ?
we
^QS$?hnB{xh)->n3xhB(xh),
is
that
by
inductive
hypothe
Since
bQ8?E+BF?xhB(xk)-+lxhnB{xh).
have
bQ8?E+BF3xhB(%)->D3%B(xh),
bQ8m+BFA->nA.
On the other hand, from BF and the inductive hypothesis
->B
it follows
that
-%#(%)->1%-B(%)>
\"gsm+BF$B
Then
bQ8?E+BFlxh<>B{xk)->lxhB{xh).
that
is
h??4??+^0^.<->A.
bg84E+BF0l
Since the Barcan formula is a theorem of QS5, the Proposition 8.5
also proves that the QS%E system is a collapsing extension of QS?E.
Finally,
we
observe
that,
from
a
semantical
point
of view,
obtain overiogics of QS?E by defining
? referring to particular
of QS?E-mo??ls.
investigate
For
example,
we
can
the
overlogic
we
can
subclasses
obtained
by considering only these QS4U-modal structures determined by classes
of classical models. It is an open interesting question to give a suitable
set of axioms for this overlogic.
can also consider
In
other words, we
plete.
of Mw"
by the stronger
We
for
and
every w9wfeYY
defined
that
in
way
logic
structures
only the QS?E-modal
can substitute
: "M&
the condition
one:
aMw,
w < w\ As
is an
S modal
com?
is an extension
extension
elementary
it is proved
in Proposition
of Jfw",
3.2., the
collapses.
References
For Modal
D. Reidel
P. Company,
II] K. A. Bowen, Model
Theory
Logic,
1979.
London
Model
Amsterdam
[2] C. C. Chano and H. J. Keisler,
1973.
Theory, North-Holland,
?
[3] K. Fine, First-order modal theories I
Sets, Nous,
15, (1981), pp. 117-206.
[4] K. Fine, First-order modal theories III ? Facts, Synthese,
53, (1982), pp. 43-122.
in Modal
and Tense
[5] D. M. Gabbay, Investigations
Logics with Applica?
tions to Problems
in Philosophy
and Linguistics,
D. Eeidel
P. Company,
Dordrecnt-Holland/Boston-U.
S. A.,
1976.
210 G. Gerla, Y. Vaccaro
and W. H. Wheeler,
Division
[6] J. Hirschfeld
Foreitig,
Arithmetic,
Ringsf
Lecture Notes in Mathematics,
vol. 454, Springer Verlag,
1975.
and A. Tarski,
Some theorems about the sentential calculi of
[7] J. C. C. McKinsey
Lewis and Eeyting, The Journal
vol. 13 (1948), pp, 1-15.
of Symbolic
Logic,
Institute
University
of
Mathematics
of
Napoli
Italy
Meceived November
?tudia
15, 1982
L?gica XLIII,
3