TWO N E W INTERPRETATIONS O F MO DA L I TY
James W. GARSON
1. Topological logic (
1
) It is possible to develop n e w semantics f o r modal lo g ic b y
taking o u r cue fro m semantics designed fo r topological logics.
In topological logic, a n indexed operator 'T' i s added t o th e
notation o f p ro p o sitio n a l lo g ic, a lo n g w i t h a n in f in it e s e t
X = x ' , x" , ...1 o f index variables. The fo rmu la 'Txp ' is read
'p is the case as o f x', wh e re the in d e x ' x mi g h t range o ve r
dates, positions, coordinates o f space-time, o r t h e denotata
of indefinite expressions such as ' f o rt y years ago' a n d ' f o rt y
feet f ro m h e re '. Qu a n tifie rs a re in tro d u ce d o v e r t h e in d e x
variables, b u t there a re n o in d ivid u a l va ria b le s o r predicate
letters.
Throughout t h is paper, t h e le t t e r 'p ' ranges o v e r p ro p o sitional variables, t h e le t t e r 'x', o ve r in d e x variables, a n d th e
letters 'A ', 'B', and 'C', o ve r formulas o f topological logic. The
formulas o f topological lo g ic have the forms p, A , (A ---> B),
TxA, and V xik. The other logical constants are introduced b y
definition in the usual way.
To p ro vid e semantics f o r topological logic, le t a D-model U
be a p a ir < D, u > consisting of a non-empty set D o f contexts
(dates, places, o r wh a t have you), and an interpretation func;lion u wh ich assigns a t ru t h va lu e t o each proposition i n a
context. So u is a fu n ctio n f ro m D X P in t o th e se t {O,1} o f
truth values, wh e re P i s t h e se t o f p ro p o sitio n a l va ria b le s.
(u: D X P 2 1 A va lu a tio n y is a fu n ctio n f ro m X in t o D.
(v: X —> D.)
We use letters 'b', 'c', 'd', and 'e' as metavariables o ve r D,
and letters ' v ' and ' w' as metavariables o ve r valuations. Th e
symbols ' 3 ', '(x)', ' D ' , '&', ' ' i f f ' , and ' a r e used in th e
metalanguage w i t h t h e ir u su a l interpretations. A q u a n tifie r
which b in d s a va ria b le i n t h e metalanguage i s ta ke n t o b e
444
J
.
W. CA RS O N
restricted to its corresponding set; so, for example, ' ((I)
. a '(d
ates
b b) erDe v i The predicate t j
U tai t co n te xt d o n va lu a tio n y ' ) i s d e fin e d re cu rsive ly a s
follows:
v
A
( r eu a d
' t d
eU
u vh
v
B
f o v r— Bm i f f
u l th,
p a(
A
i 1
Uw
i
d
x B i f f (w)(w = v D B ) , wh e re w = xV i f f w
f 3
s
agrees with y for all values other than at x.
f —
t
u r>
A uf in a l (cla
e u se i s n e e d g o ve rn in g t h e t ru t h co n d itio n s f o r
C
formulas
o
o
d )f the fo rm 'TxB'. I f 'x i s thought o f as ranging o ve r
dates
coordinates, then we wo u ld like TxB to be true
n o r, spatial
i
in m
context
d
ju
st
in case B is true in the context denoted b y x:
p f
o
) fd
uv
T xl B i f f
eD-v. =
u
1 v
A fo rmu laBA is D-va lid if f (v)(d ) A o n e ve ry D-mo d e l U.
However,
D x ma y range o ve r the denotata o f in d e fin ite e x pressions 1
(lik e ' f o rt y fe e t f ro m h e re
of
x
is
not
.
.taken to be a context but rather a function from con), wtohcontexts.
eJ r e Then
t anh F-model
e
texts
should be a trip le < D, u, F >
d e n vo f at dao ma
t ini Do o f ncontexts, a n d a n in te rp re ta tio n
consisting
function u Cas before, and a non e mp ty set F of unary functions
from D in t o D. Th e n a va lu a tio n assigns t o each va ria b le a
member o f F. We use letters 'f', 'g ', and 'h ' as metayariables
over F. The sentence 'B is the case as of fo rty years ago' is true
in the context 1970 just in case B is true in the context obtained
by applying the function denoted by 'fo rty years ago' (the function f(d) d — 40) to 1970. So in this case the tru th clause f o r
formulas of the fo rm TxB should read:
uv
u
v
TxB i f f v(1)(a)13,
TW O N E W I NTERPRETATI O NS O F MO DA L I TY
4
4
5
where v(x)(d ) is the re su lt o f applying the function v(x) t o d.
A fo rmu la A is S p - v a l i d (S p -sa tisfia b le ) i f f u
((some)
i
v and d, on e ve ry (some) F-model U such that S p . Let
v
C(F)
Ab e th ea condition
t
e th
v a t eF is
r th
y e set o f constant functions
on D. I t is n o t d if f icu lt t o sh o w t h a t A i s D-va lid i f f A i s
2. Semantics for modal logic
No w le t us tu rn to modal logic. Suppose that D is the set of
all possible worlds, i.e. the set of a ll positions in lo g ica l space.
Then ' T x A ' re ce ive s t h e in te rp re ta tio n ' A i s t h e case a s o f
possible wo rld x'. The box ma y be introduced by definition into
the notation of topological logic through the standard definition
of necessity:
Del Di: D B = in V xTx13.
So the formulas o f modal lo g ic are taken to be certain closed
formulas o f topological lo g ic. Semantics f o r mo d a l lo g ic ma y
be obtained simp ly b y defining a s v a lid th e mo d a l fo rmu la s
of topological lo g ic wh ich are either D-valid, o r S p -v a lid .
If we choose th e f irs t a lte rn a tive , t h e t ru t h co n d itio n f o r
modal formulas of the form DB is
D-vi.
Uv
D B
U
i f f (e)
v
U
w
for D B i f f d V
x T x B
i f UwU
f
w
U
v
((W)w= ) ( w
modal
I x
formula,
T
x
Band hence contains no free variables. We ma y
design
v
a
semantics
sp e cifica lly f o r modal fo rmu la s using th e
)
ii
notion
o ff D-va lid ity, b y choosing t ru t h clauses i, i i , i i i , a n d
D-vi,
deleting mention o f valuations. The result is the f a milia r
ff
semantics
f
fo r S5, hence the D-va lid modal formulas are e xa ctly
the
( theorems of S5.
e Weaker modal systems ma y be captured semantically using
the
) notion of S p -v a lid it y . Then the tru th condition fo r O B is:
B
Uv
U
v
i F-vi. D B i f f (f) f(d )B,
f
f
(
e
)
B
,
s
i
446
J
.
W. G A R SON
uw
since u v DB i f f d
v
uw
U
w
U
v
(W)x(W w ( x ) ( d ) B ) i f f (f )f (d )B i f f r ( d ) 1 3 . Semantics f o r
V
modal
formulas alone is obtained by deleting mention of valuations in truth clauses i, ii, iii, and F-vi.
N B
The re su ltin g semantics i s a denotational ve rsio n o f t h e
i
f
f
transformation
semantics developed b y v a n Fraassen (
(
w
)
2
though
it
was
discovered
independently. He re we introduce a
(
w
),
set
F
a
of
l
unary
functions
(transformations)
wh e re Krip ke wo u ld
=
introduce
a binary relation R. Krip ke semantics ma y be defined
x
in
a
simp
lifie
d ve rsio n (
V
3
<
D,
u
,
R
>
,
wh
e re D and u a re as before and R is a b in a ry
D
)T
relation
a s on
D.
The
tru th clauses are i, ii, iii, (wit h mention o f
x
valuations
deleted)
and
fB o l l o) w s :
A
K
i
m
o
d
e
l
K-vi.
u
f
if d
s
a In
i fK rip ke semantics each a xio m (A ) o f modal lo g ic co rre stponds
rto a condition
i
p A(R) on R so that when KS is the system
f
lK (p luee
s a xio ms ( A
A
l )
( (R ifAf A„ is) S(R)-valid.
l) , KsA
,
(The system K consists o f the p rin d
e
(ciples
o
f
p
ro
p
o
sitio
n
a
l
lo
g ic, t h e necessitation ru le : 1— A D
a
n
d
D
R
I— DA , and the a xio m ( A —> B) —* (DA — O B ). )
S
)( TheRsame) is true of the modal semantics ju st presented. Each
&
modal
a xio
i
s m (A ) corresponds to a condition A(F) o n F so that
.t
when
DShis D plus axioms ( A
.A
1
e
).(D):
n
c, ( oDA A n
j
—
. ) c
&
n
(u A ncorresponding
their
conditions o n R and F fo llo ws: (
t
O
A
.
)
)A
F
4
a
n
d
i
o
n
„A
S
)(D) L I(A —>F• O A ) D ( R ) e R d e D ( F ) F 0 (
l, iD As s—*A T ( RA) Rd d
i((T)
5
T ( F )
3 f(f(d) = d)
lR
)t D A —> O D A 4(R) Rb c&Rcd DRb d 4 (F) 3 f(g(h(d)) f ( d ) )
(4)
o
()(B)
h FA )D O
& A . B.( R.) Rd
& e DRe d
B p
3 f(f(g(d)) = d)
fo(E)
e
— > D O A E(R) Rb c&Rb d DRcd E(F) 3 f(f(g(d)) = h(d)).
fsn
to
1
hm
—
ee
n
ics
ro
A
cm
i
om
l
ro
l
rn
A
TWO N E W I NTERPRETATI O NS O F M O DA L I TY
4
4
7
In general w e ma y produce A (F) f ro m A (R) b y t h e e q u ivalence:
RF Rde i f f 3 f(f(d) e ) .
The reader ma y e a sily ve rif y that the conditions o n F in the
above list are equivalent to the result of replacing a f(f(d) = e)
for Rde throughout in the corresponding condition o n R.
We now intend to prove the above assertions. We will prove
Theorem 1. A is S(F)-satisfiable if f A is S(R)-satisfiable, wh e re
S(F) is th e translation o f S(R) b y RF, a n d wh e re S(R) e n ta ils
D(R).
Proof. (le ft to right) We are given that A is S(F)-satisfiable, so
there is a model Up = < D , u, F > a n d a context d such th a t
u
S(F) a n d d FA. We ma y produce a K rip ke model UK f ro m UF
such th a t S(R) a n d ' t - T
, I is defined b y RF: Rde if f 3 f(f(d) = e). Cle a rly S(R) holds in
R
K
u
u ,
A
, model.
b y We prove that
this
b y
sh o win g i K A i f f d 1 A
lfore a llt dtGD,
i nusing
g an induction on the fo rm of A. The proof is
U
K
straightforward,
and the o n ly case wo rth re vie win g is the one
=
u
where
AD has the
<
, fo rm DB . But ,
Iu
,
(e)(RdeD
u i l l
K
E
I
1
3
( e )
R
>
(,, , FR1 3d) e D
,
e ) .1j
((e)(b 3 f(f(d)
yB
K
) e
w
h
e
F
ttheory)
h
e
ir
efi f f F O fB .
B
i )n d( b y
(right
R c F t t o) le ft) W e a re g ive n t h a t A i s S(R)-satisfiable, s o
u
there is a model UK = < D , u, R > a n d a context d such th a t
ii v fe
f
(S(R)
f
h ya n d N) o w N
o
wwe define Up fro m UK so that S(F) and
ip j o t
fh c(i e
fs F i sf e F iff f: D —› D & (d)Rdf(d).
d
) •)
B
i A
(f ,
b
yf b
i y d
e
n l
t
i
e
t
y
t
t
448
J
.
W. G A RS O N
No w we show that U
r S(R)
if
i s holds
s u and
c hRF holds, then so does S(F), since S(F) results
t h replacing
from
a t
3 f(f(d) e ) f o r Rde in S(R).) Th e p ro o f o f RF
S
follows:
( Suppose
F
)
Rd e . L e t t h e f u n ct io n g b e d e fin e d b y
b
g(d)
= e yand g(c) = b, f o r c 4z1, wh e re b , is a me mb e r o f D
chosen
p
r soo th a vt Rcb
ci EDn i s ag consequence o f t h e assumption t h a t S(R) e n t a ils
.R T hhence
a Ft 3 eRce. No w g e F, fo r Rcg(c) f o r a ll cE D, because
D(R),
s. u cc = hd, g (c ) = e, a n d w e h a v e t h a t Rd e , a n d wh e n
when
a
(c d , C g (c ) l = b
b
c i t af,o llo ws
eSo
r t h a t f ( f ( d ) e ) , n a me l y g . N o w suppose
cl,3 f(f(d)
a yna= de). Then
n
Rdf(d). But f(d) = e, so Rde. Wit h RF in hand
bb
,
e
u,
we
lh ma
oa y pcsro ve e x a c t l y as i n th e p re vio u s case.
Theorem
ab
te
e1 amounts to a consistency and completeness proof
with
respect
t o th e semantics o f th is section f o r a n y mo d a l
de
n
system
DS
(a
s stro n g as D) f o r wh ich w e k n o w th e co rre sfc
h
ponding
co
n
d
itio n S(R) o n R i n K rip ke semantics. W e ma y
oo
s
simp ly use RF to produce the appropriate conditions o n F f o r
re
n
this system, and then the completeness p ro o f is a co ro lla ry o f
es
a
Theorem I a s fo llo ws: W e w i l l kn o w th a t 1 —
co
h
valid.
I , i A sis S(R)-valid
if f A
Ds i lBl y Theorem
A
S ( R
) is S(F)-valid, hence
t
- ns
h The conditions o n F resemble, b u t a re we a ke r t h a n t h e
A
a
following:
if
t
f
R T'(F) c3 f(d)(f(d) = d), (F contains the id e n tity function.)
A
b 4•(F) 3 f(d)(g(h(d)) = f(d)), ( F i s clo se d u n d e r co mp o sitio n
i
e
of functions.)
s
. B'(F) 3 f(d)(f(g(d)) — d), ( F contains in ve rse functions.)
S
(E'(F) 3 f(d)(f(g(d)) = h(d)).
It F
tu rn s o u t th a t these stro n g e r and mo re in te re stin g co n d i) m a y b e substituted f o r t h e ir we a ke r counterparts, f o r
tions
when
th is is done, the set o f modal formulas defined as va lid
v
remains unchanged. (')
aThis is imme d ia te ly apparent in the cases o f T'(F) and LI'M.
l p ro o f th a t A i s T(R)-satisfiable i f f A i s T ( F )
The
may
- si a be
t i scarried
f i a b lout
e as in Theorem I, wit h the additional re ma rk
d
.
TWO N E W I NTERPRETATI O NS O F M O DA L I TY
4
4
9
that since R is re fle xive , i t fo llo ws b y De f
t
function
f
i, it shit afotllo wst that
Rdd,
h Rdf(d).
e
In the case of 4'(F), the proof is the
same
i dn save
o
e n tht a ti n to wy we mu st sh o w th a t g ive n De f
te n sitivity o f R, i t fo llo ws that F is closed under composition
tra
,oof functions.
a fn d
Let
t hg and
e h be a rb it ra ry members o f F. B y DefF,
Rdh(d)
t
h and Rh(d)g(h(d)). R is tra n sitive so Rdg(h(d)); hence the
function
goh such th a t goh(d) g ( h ( d ) ) i s a member o f F.
e
m Thee case o f B (F) in vo lve s complications because i n o rd e r
for
m F bt o co n ta in in ve rse functions, F mu st co n ta in one-one
functions
o n ly. I n th is case w e ma y define F i n UF so t h a t
e
r
fse F if f f is a one-one function f ro m D t o D and (d)Rdf(d). (
7
This
added re strictio n leads to d ifficu lty, f o r we can no longer
o
)f
show
that RF, since the function g used to establish this mig h t
not
be
one-one, a n d so n o t a member o f F. (We have n o asF
surance
that b
.
e
that
b
R
i
s
re f le xive a s w e l l a s symme tric, (wh e n t h e mo d a l
F
d
system
a t issue is B o r stronger) then we ma y le t g be defined
r
fo that
so
o rg(d) = e and g(c) = c fo r c d . This function is cle a rly
one-one,
a
y since Rcc, it is in F. Given a proof of RF we ma y
m n and
cf
show th a t a s before. We must also show that S(F), wh ich
d
.
iwill require
(at least) a demonstration that F contains inverses.
)(
But let g be any member of F, and let g be its inverse. Then b y
H
dthe doe fin itio n o f F, Rg(d)g(g(d)), a n d b y th e symme t ry o f R,
w
e
v
)Rg(g(d))g(d).
Sin ce g(g(d)) — d, w e h a ve Rdg(d), hence g c F.
e
r
, wh e re we do not kn o w that R is reflexive, b u t that
— In cases
w
d
it is symmetric, we must resort to a more complicated strategy
h
a vo lvin g t wo stages. First , f o r each K -mo d e l UK, w e sh o w
in
e
n
how
to define a new K-model UK = < D ' , u', R ' > wh ich leaves
n
d set o f va lid formulas unchanged. Th is n e w model w i l l be
the
w
constructed so that there are enough members o f D' t o insure
e(in effect) t h a t b
kone-one
e b
function g with which to demonstrate RF. UK is defined
nfrom
d
UK as fo llo ws. D' i s defined re cu rsive ly f ro m D b y st io
pulating
;
th a t D g D' a n d d ED' D { d } D ' . Th e n w e p ro vid e
w
an
fu n ctio n a, wh ic h maps D' in t o D as fo llo ws:
t association
h e n
i
w
l
b
e
p
s
l
t
o
t
i
o
i
e
l
s
b
450
J
.
W. G A RS O N
aid) = d, for d D , and
a ttd i) = b
a
So,( df o) ,r instance M I D (f o r d e D ) is associated wit h b
b df o b rb y
a ,
where
d
G
D
bb
d
R
' . n
dt uwr we
No
may define u and R':
s
b
u p) = u(a(d), p) fo r d
bo du'(d,
.t R'de iff Ra(d)a(e) fo r d, eGD'.
t
A
o thoughtful inspection o f this definition should convince one
u'
b
that i t i s possible t o p ro ve K A i f f K a(d) A, f o r d D ' b y
e
induction
on the fo rm o f A. A s a special case o f this we have
t
u
h d KA i f f ,
'
e
is 1a ll we need fo r the remainder o f this proof. No w we define
m
UF1fro
( m U
e
. Ath• e ir domains and interpretation functions, and we le t F be
on
m
K f se
the
o t or f one-one functions su ch t h a t (d)Rdf(d). RF is th e n
b
proven
e
dx a cas before save th a t g is defined so th a t g(d) = e a n d
g(c)
te lDy = I c f o r c d . Th is fu n ctio n is cle a rly one-one ( { 0
ra{ (l( ss
cw
that
f o r a ll c W h e n c = d, the n g(c) = e, and we
}, i R'cg(c)
n
h
have
and wh e n c d , th e n g(c) = c 1, hence we mu st
ef c R'de;
e
oo a
prove
that R'ct c 1 f o r a ll c B u t R'cl cl if f Ra(c)a({ c }), and
d
sir ( c}) = b
a({
e
Ra
a
dc cd(c )b
n
a
)bd.() e u'
that d KA if f ( I
,F
ff.F -Iut o. r t a s
11
o
)h
msu lt yie ld s , 3 / c A i f f h e n c e i j
r)b =eee rre
vious
f o r e
rmust
,o
c.,, IdrF eA
show
. Fthat
i nUr'ahas
l lthe
y property
w S(F),e wh ich entails showing
b
h
s ,e F contains inverses. But this proof may be supplied exactly
W
that
h
i
dw
n
a
o fcbefore, once we note th a t th e symme t ry o f R entails t h e
as
e
slct o h try of R'.
e
w
symme
t
o
g
e
u
R
h
In' the case o f E'(F) simila r d ifficu ltie s arise, and th e y ma y
eo
tc r h
e
r
a
s ddisposed of using the same strategies.
tbe
w
i
hh 1
{sU
t
h
tK
cb
o:
a
h
}e
aw
p)
.a
e
ntr .
th )
N
n
d
e
o
cU
a,
w
h
t w
F
to
ai h
h
sgt i
TWO N E W I NTERPRETATI O NS O F M O DA L I TY
4
5
1
So i t is possible t o p ro vid e consistency a n d completeness
proofs o n F-semantics f o r the strong conditions o n F f o r a n y
modal syste m DS, wh e re S is a selection o f the a xio ms T, 4,
B, and E. Th e proof ma y be ca rrie d o u t f o r a n y such system
using th e two-stage stra te g y ju s t outlined. Wh e n conditions
on R other than B are present, these conditions w i l l remain in
force f o r R' and the corresponding conditions on F w i l l f o llo w
without d ifficu lty. A s w e h a ve seen, f o r mo st selections o f
axioms, th e re a re less complicated methods available.
3. A group theoretic semantics for modal logic (8)
Another semantical approach fo r topological lo g ic has been
investigated. (
it
9 yie ld s interesting results f o r modal lo g ic. A n F-model co n tains
) I t a set F of unary functions on D. Instead, le t us p ro vid e a
binary
operation o o n D, s o t h a t a o -mo d e l U i s a t rip le
i s
<D,
u
,
s o mo e, wh e re D and u a re as before. Th e n a va lu a tio n
assigns
w h ato t each variable a member of D. (v: X ---> D.) Th e tru th
lclauses
e re ma in a s i n F-semantics sa ve t h a t t h e cla u se f o r
formulas
s
s of the form TxB reads:
s a t UvU
v
i s f TxB i f f c
l
y TxB
i nis taken
So
to be true in the context 1970 ju st in case B
o wcontext one gets b y applying o to 1970 and the
g tru e in the
is
i
denotation
n
t (o)f Bx.. Wh e n x denotes — 40 (40 ye a rs ago), th e n
1970o-40
u
i
tmight yie ld 1930.
i If vwe define
e
th a box in topological lo g ic b y Def 0, we obtain
the
fo
l
y llo ,win g tru th condition f o r modal formulas o f the f o rm
bDB:
u
t 0-Vi. E I B i f f (c) ( 1
, „ predicates
13.
The
S(o )-va lid and S(o)-satisfiable a re defined o n
analogy w i t h S (F)-va lid a n d S(F)-satisfiable. A g a i n mo d a l
axioms correspond to conditions on o as follows:
D(o) e c ( d o c = e), B u t t h is i s guaranteed, a s o i s a n
operation.
452
T(o)
4(o)
B(o)
E(o)
J
3
3
3
I
.
W. G A RS O N
e(doe d ) ,
e((boc)od b o e ) ,
e((cod)oe = c),
e((cod)oe = cob).
The condition A (o ) ma y be obtained f ro m A (R) b y th e e q u ivalence
Ro: Rde iff I c(doc = e),
which we will demonstrate in
Theorem 2. A is S(o)-satisfiable iff A is S(R)-satisfiable, where
S(o) is the translation of S(R) b y Ro, and S(R) entails D(R).
Proof. (le ft to rig h t) L e t U
o =
< D ,
satisfies
u
, S(o), such th a t 1 j
(o1
>
R
is .defined
°b A
Le e by
t Ro. Cle a rly S(R). We prove that (
1U induction
by
on
the fo rm of A, the interesting case being when
K
o
1a
j
1
(
b
e t h ed f o rem Dl B : (
A
h
a
s
m
o
.<A
i Df f ,
1w
h
i
( 3 c(doc = e) e
1u
c.jK h, B
lYf f >
iR
( e, )
d
E
t
a
(
'
1
3
)
(w R dh e e
f f e t o le f t ) eL e t UK = <
ri (right
D
D
R
>
( e,K u) ,
U
satisfies
S(R), such th a t d
b
e
c)
)
B
o( isA defined
of R by
K
L in
e terms
a d.
Kt
(
o
i1
f
f
m . o4 d
e- l
(=c doe = c,e when Rdc, and
w
h
i
c
= b ,
)<= doe D
h
e
u d
,
,(We
,
remember
that b
o
>
3
No
d
w
w
i
s
h
we
e
mu
st
p
w
h
e ro ve th a t S(o) b y demonstrating Ro; Suppose
1
1
c nh toheesn ed one = e a n d h e n ce 3 c(doc — e). N o w suppose
rRde,
°s —
1 o 3 = e).
) Then either Rdc and e = c, o r —Rdc and e c(doc
iIn
bthe
dh d.first
t R
a case
t Rde, and in the second Rd b
dfR c d. b
,fd h e n c e
R d e .
(. ) o
N
w
c
)
i
f
f
TW O N E W I NTERPRETATI O NS O F M O DA L I TY
4
5
3
we ma y p ro ve ( 1
K -The
A conditions
i f f
presented f o r o re min d one o f the stronger
c i
conditions
from group theory which fo llo w:
( ) A .
a T(o ) s3 i ( d o l = d & iod = d), (D contains an id e n tity elei ment.)
n
t 4'(o) h(boc)od b o ( c o d ) , (o is associative.)
e B(o) R e(doe = i), ( D contains in ve rse elements.)
p E(o)
r 3 ee(doe
v = c).
i
o
u
s
c
aselections o f these conditions ma y be substituted f o r
Certain
s corresponding
e
the
se le ctio n o f we a k e r co n d itio n s wit h o u t
.affecting th e se t o f fo rmu la s defined a s va lid . Th e re i s n o
problem wit h replacing T(o) wit h T(o ) as we show in
Theorem 3. A is T(o)-satisfiable i f f A is T'(o)-satisfiable.
The proof fro m le ft to rig h t is t rivia l. N o w suppose th a t A is
T(o)-satisfiable, and le t U = < D , u, o > b e a model such that
T(o), a n d le t d be a context such th a t
N o w
we define
a model U' such that T'(o) and (
) , f o r some i not in D, wh e re u'
1<D', u', 6 > , where D D
1agrees
j'
wit h u on arguments f ro m D, and u'(i, p) = I f o r a ll p,
and
6 agrees with o fo r arguments in D, and i6d = d and
A
, bwhere
y
d6i
s t =i dp four al ll ad D
t 'i . nNogw U' satisfies T'(o), fo r i is an id e n tity
u'
t
h
a
t
element. W e mu s t n o w s h o w t h a t A , b y p ro vin g th a t
U
'
=
by in d u ct io n o n t h e f o rm o f A . Th e
(d )E D(
u
interesting case is when A has the fo rm DB : d DB i f f (e)
dA
E
doeB i f f (e)D
E
( s i n c e
D
6 a g re e s w i t h o o v e r D ) i f f
(e )
needed
t o complete t h is case i s a demonstration t h a t (e )
eD
e D
.1U1• ,
„
,
U
d6eD, since (e ) d 6 e D
d
OB. Th e
.( d6eD ( e ) ' e D
D e
'
U
16
'
U
•
'0
(
b
y
t
h
e
454
J
.
W. G A RS O N
proof f ro m rig h t t o le f t is t rivia l, a n d since D' — D U {i } w e
u' u '
need o n ly sh o w (e) ( 1 6 e B D (iz,i13. But dOi = d, and by T(o),
ED
u' u '
3 e dDo e = d;
hence
(e
)
d
6
e
B D d B. A s a n imme d ia te
E
D
co rro lla ry of Theorem 3 we have that T A if f A is T ' ( o ) v aWe
l i d tu
. rn n o w to the proof that the a xio m (4) corresponds to
the associativity of o on this semantics.
Theorem 4. A is 4(o)-satisfiable i f f A is 4'(o)-satisfiable.
The proof from rig h t to left is again t rivia l. Let U < D , u, o
be a mo d e l such th a t 4(o) a n d le t d b e a co n te xt such th a t
ti•Pi. Let U' = < D' , u', 6 > , wh e re D i s the set o f a ll non-null
sequences c l
association
fu n ctio n a f ro m D' t o D defined as fo llo ws:
i
. . a(d)
. d = d, fo r d E D; and
„ a(de)
o f = a(d)oe, fo r d E D' and e ED.
So,
f
m eo r minstance, a(bcd) a ( b c ) o d ( b o c ) o d , f o r b , c, d E D .
No
b w
e wr e sl e t u '(d , p ) = u(a(d), p ) a n d w e d e fin e 6 s o t h a t
doe
= de. Cle a rly o is associative, so we need o n ly show that
o
f u'
d A, wh ich is a consequence of
D
u
u '
.
T L: c i
o (d)
for
L
is
d Asinece d E D , a n d a (d ) = d, w e h a ve A if f
i
f
proven
b
y
in
d
u
ctio
n
o
n
th
e
f
o
rm
o
f
A
,
t
h
e
in
te
re
stin
g
case
f
i
f
u ,
n
e
being
d wh e n A has t h e f o rm O B : ,
ED
u
i ( Ad )
'1 = 1 B
i e f)B f(b y the definition o f a). No w if we can show
iff , (e)e t ) a (d
,(
e
)
a
(
d
)
w
that
o
e( e ) D
e D'
e
eD
U'
plete
since
(e)
o
(
d
e
)
p
i
f
B
i f f
( e )
rf
u'
u
of
the einduction)
if f (e) dn 6 e b b i l l '
d
B
o
(
O
(vemissing
bi equivalence
y
is t,riv ia lf r o m rig h t to left. No w suppose
B
.
T h e
td
h
e
)
p
ry o p o o f t
h
e
'
o
f
h
i
s
a
- 1 e3 s
tn
h
e
6
( c i
e ) B
,
t
TW O N E W I NTERPRETATI O NS O F M O DA L I TY
4
5
5
(e) EoD ( d e ) B . Let c be a n y sequence d
i
Then
ae
. . . d w e mu s t s h o w t h a t u
(d
.u. .)o doB .f
Nm o e wm b a e r s
(n
=
o. . .d( a (cdf ) o) d
D
.
(). o. f .D .such
1
b
( ath a( t da(d)ob
) o ( d. . . ( a ( d ) o c 1
u
)iB. . . o d
1
from
(e)
a
)ny. . . o d E D
E
D
(n
)a d n
a (d)ob = a(dc), hence 1 - 1
e
)tp. )i mN e o w
i
t
, immediate co ro lla ry o f Theorem 4, we have a consistency
an
B
fsp
,
l
o
l
l
o
w
s
proof fo r the system D4, f o r b y Theorem 2 we
iand
( d hccompletess
)
tw
yhave
i thaa t A is D(R)&4(R)-satisfiable i f f A i s D(o)&4(o)-satisB
,
w
h
i
c
h
ten
Bu
t
D(o)
is
guaranteed b y the definition o f an o-model,
i(fiable.
s
e
m
g
so
A
is
D(o)&4(o)-satisfiable
i f f A i s 4(o)-satisfiable i f f A i s
h , e
)att
y
4'(o)-satisfiable.
So
it
follows
that
I— D4A iff A is 4' (o)-satisfiable.
i
r
e
rpd
( edr )s
h We
(od
) vw i l l n o w p ro vid e wh a t amounts t o a consistency a n d
e
r,completess
e
s proof
u for lthe system S4 by proving
ec B ,
t
.
a
to Theorem 5 . A i s T(o)&4(o)-satisfiable i f f A i s T'(o )&4 '(o )A
n
hn satisfiable.
s
d
ad
proof is identical to that fo r Theorem 4, save that D' is the
stThe
i et o f a l l sequences o f me mb e rs o f D, in clu d in g t h e n u l l
o
ttsequence - such th a t d - — d f o r d c D ' : a n d wh e n i t comes
,hi
x
to
p
ro
(eo
dvin g t h e missin g equivalence, w e mu st l e t c b e a n y
o f me mb e rs o f D , in clu d in g W h e n c - , w e
)rsequence
n
procede
a s before, a n d wh e n c = -, w e h a ve t o sh o w t h a t
o
e4 u ,
u
b
i( „ ,
B
.
so (
U t o
3 e(a(d)oe
= a(d)). S o f ro m (e ) E L
_ a(dern., w e o b ta in a ( c l )
B
D
a) d
the
B , desired result.
u
m
: e
simila r results are not available fo r the condite3 Unfortunately,
)
tion
m
e D B'(o), as the a xio m (B) is n o t B'(o)-valid. (") Ho we ve r, in
the presence of both T'(o) and 4' (o), matters improve, so that we
b( o
will be able to p ro vid e a consistency and completeness p ro o f
e( r
for S5 by demonstrating
rb ,
o 3
c (
) d
o )
d B
= .
b B
o u
c t
) w
456
J
.
W. G A RS O N
Theorem 6. A is T'(o)84(o)&13'(o)-valid D A is D-va lid .
Proof. Suppose A i s T'(o)&4'(o)&B'(o)-valid, i. e . A i s n o t
T'(o)&4'(o)8LB*(o)-satisfiable. Suppose f o r re d u ct io t h a t A i s
not D-valid, so there is a model U = < D , u > and a context d
such th a t d
u
where o i s defined a s f o llo ws: First , w e p ro vid e a one-one
—
mapping
A .a fro m Re (the set of reals) to D. We ma y assume that
N is oo f th
D
w e appropriate ca rd in a lit y wit h o u t cost, f o r i f D i s
lsmaller
e th a n Re, standard techniques w i l l p ro vid e a mo d e l
with
a la rg e r domain on wh ich A is tru e a t d, a n d fu rth e rt
more
D need n e ve r be o f higher ca rd in a lity than Re, as those
u
who
a re f a milia r w i t h He n kin -style completeness p ro o fs f o r
s
modal
d
e lo g ics w i l l see, f o r the set o f a ll ma xima lly consistent
sets
wit h Re. No w le t us in d e x each member
f
iis equinumerous
n
of
D
b
y
le
ttin
g
d
i
b
e
th e unique d such th a t: a (d ) = j. Th e
e
operation
o
is
defined
by:
a
m
o diodk = k •
d
This
operation
has th e three properties T'(o), 4'(o), and B'(o),
e
l
U
since + does. A ll that remains is a demonstration that A ,
„
for once this is shown, we w i l l kn o w that A is T'(o )8 4 (o )&
<
D
B'(o)-satisfiable,
wh ic h completes t h e reductio.
A
ma y
,
be
u p ro ve n b y sh o win g u
d
,teresting case being when A has the fo rm DB . Then E I B i f f
A
i f f
u
o
d
>(e) 1 3 i f f (e) e
,ct j '' . A .
e
show
B
w e w i l l have completed the
fB
(othba t yr(e ) u B i f f (e)
'
at h l e
l
proof o f Theorem 6, f o r (e)
i
ff (1 3 ' DB. Th e p ro o f f ro m
di n d Du c ,
1
tt i v he
left t o rig h t i s t riv ia l. N o w suppose (e ) d
eh y p o
U'
u
it d h 'n' e n s
„
-i j s ) .
'
has
t h e p ro p e rt y 3 1 e Re
„OBo,+ t 1h e =
(j
n
( k )
en d R) e
m
f
o
r
c
k
a
n
y
je t
,
ww
m
eh
G
R
e
e
.
Sr
o
e
TWO N E W I NTERPRETATI O NS O F M O DA L I TY
(k) ERe B h
entails
e
n B j+1
c,
e
4
5
7
d rnB, f o r a n y m e Re.
Since e ve ry member o f D is indexed, i t fo llo ws th a t (e) B .
No w th a t we h a ve sh o wn Theorem 6, w e ma y sh o w t h a t
s5
right
A ma y be ca rrie d o u t in standard fashion b y induction o n
thei form of the proof of A. No w suppose A is T' (0)&4' ( o )
, f
valid.
Then b y Theorem 6, A is D-va lid ; but we remember that
A
is
D-va
S zf W ( o ) lid
- iff
We w i l l n o w re vie w the results concerning the strong conA
ditions
when using o-semantics:
i
s
TModal System C o n d i t i o n s on o
o has an identity element ( T h m . 3)
'
D4
o
is associative
( T h m .
4)
(
S4
o
is
a
semi-group
(
T
h
m
.
5)
o
S5
o
is a group
(
T
h
m
.
6).
)
&
The
4 a u th o r h a s ve rif ie d t h a t mo d a l syste ms wh ic h co n ta in
axiom
(B) o r a xio m (E), a n d wh ich a re we a ke r than S5, a re
•
not( co n siste n t wh e n t h e st ro n g co n d itio n s a r e u se d w i t h
0-semantics.
o
)
4. &
Mo d a lit y with in topological logic
1Once we have defined the box in topological lo g ic via De ll:
it 3
becomes possible t o capture mo d a l systems a xio ma t ica lly
by' choosing th e appropriate syste m o f topological lo g ic. W e
( p ro ve n i n [3 ] t h a t I - T
have
D(o)-valid),
(")A wh e re TQ
is
Ao i f f
i
s the system fo rme d f ro m the p rin )
ciples
of quantificational
plus the ru le R: A D 1 " x A and
D ( F
) - v a l i logic
d
the
( - axioms:
i
f
f
Av
i a(— A) Tx s— A —› —TxA
l(A —) — TxA -* Tx — A
i(A —>) T x (A —> B) ( T x A T x B )
d(AQ) x T y A ---> Ty V xA, fo r x
.
T
h
e
p
r
o
o
f
f
458
J
.
W. G A RS O N
It fo llo ws b y Theorem I (Theorem 2) that 1- TQ
A isi faf formula
1 - of modal logic.
1 Modal systems stro n g e r th a n D ca n b e captured i n to p o logical
logic
, A
w hsimpely by
n adding the modal axioms not in D to the
system TQ. Let S be a selection of modal axioms and TQS and
DS, th e systems th a t re su lt f ro m adding S t o T Q and D re spectively. Then
Theorem 7. 1 - Ds
The
A proof
i f f is t riv ia l f ro m le ft to rig h t, f o r a ll the p rin cip le s o f
D are
present
T Q
s A in, TQ, and axioms in S appear in TQS. The proof
from
rig
h
f
o t to
r le f t is n o t t rivia l, f o r i t mig h t t u rn o u t th a t b y
using
and formulas o f TQS not available in DS, one
a principles
n
might
y p ro ve a mo d a l f o rmu la n o t p ro va b le i n DS. Suppose
-.1't)sA.
Io
t is easy to show the consistency o f TQS wit h respect
m
to dF-semantics
a
l (o r o-semantics), hence I-- To A D A is S R-va lid .
Butf b y o
Theorem
I , A is S R -v a l i d D A is S(R)-valid, and this,
r
wemknow,u entails
l that tDSA.
a
UnAive rsity o f Pennsylvania
J a m e s
W . GARSON
.
(
BIBLIOGRAPHY
"
)
J. GARSON, A n e w in t e rp re t a t io n o f m o d a lit y (abs trac t), J o u r n a l o f
Sy mbolic Logic, v ol. 34 (1969), p. 535.
121 J . GARSON, The completeness of an intens ional logic : definite t opologic al
logic, f ort hc oming i n No t r e Da me J o u rn a l o f Fo rma l Logic .
131 J . CARSON, I ndef init e t opologic al logic , f ort hc oming in J ournal o f Philosophical Logic.
141 J . GARSON, Th e Logic s o f Space and Time, doc t oral dis s ertation, Un iv e rsity of Pittsburgh, 1969.
[5] N . RESCHER, Topologic al logic , J ournal of Sy mbolic Logic , v ol. 33 (1968),
pp. 537-548, wit h J. Garson.
[61 B. VAN FRAASEN, Fo rma l Semantic s a n d Logic , Ma c Milla n , 1971.
NOTES
(I) To p o lo g ic a l logic s and t h e ir semantics are discussed more t horoughly
in 12], 131, [4], and [5].
TWO N E W I NTERPRETATI O NS O F M O DA L I TY
4
5
9
(9 s ee [6], p. 151 ff.
(
world).
3
I t is eas y t o s how t h a t t h e t wo v ers ions a re equiv alent .
()
and
T (B) i n [61, p. 152. O u r res ults w i l l b e a b i t mo re general t h a n his .
4
()h
an 5
veF
.F )a
s 0 i n t he semantics f o r modal logic s . W h e n t his is done, w e obt ain a
semantical
c harac t eriz at ion o f t h e t heorems o f K . I t i s n o t d if f ic u lt t o
- n
T
e
mo
h
m
F
show
t hat 1— KA if f (d) "
d ee
ra l
(
U
. ca
n
results
5
are a bit more general.
d
T o
t
a
() o n
A
h n
si
was
v vsuggested
7
e
e r y to me
F by- Bas v an Fraassen.
e d
sc
()a o d e l ,
m
r ie
s
early
n research
T
8
w
h
e on the
r results of this section.
e tn
h
(
F
h
)
e
a ire
r(39 I n [ I ) , w e re p o rt e d mis t ak enly t h a t t h e s y s t em B K T B i s s u c h
e
M
F
r o
r B
that
)e
ie ysa
s
n
e
p
A
I
(")
Theorem
ta See
p
o
s
s 2 of [3].
n (D
p
o
ii f h
ist f b
l
y
o 1(r
A
s
p
a
e
m
p
d 9F
it n
pte
e
u
y
.
i )s
s T
p
kln
f hive
T sa
re' (
f isin
o te
a)
i a
& so
rrtB '
ip
c cte
( otsN
o )
u o
d
u
- h
iu
r
n
l en
e
a
v e
tna l
t osl
i dlh
s[ .
i re
li
4
s
a
B
e eq
ym
]e
ti
s m
u
ti
,lfm
i se
n
h
C
i
b
n
n un
a
e
h
l
e
a
r pcst
a
p
e pe
a
r
,o
t oo
s
m
tw
r
n
r rfe
K
e
h
a ttcr
ro
-s
c sh
i
o
5
u
o
w
t ce
p
n
.a
l
n
i lw
k
d
t
se
n aa
ie
fs
e
g iyt'
i
xu
t m
w
is
tn
h se
G
o
rc[
e m
h
(
n
t6
e
s aa
st
i1
m
t dvfh
,
o
e
i ee
o
p
ln
p od
ra
yip
u ne
(c
.
n
h
l pfD
t