K G ' ' " " A ND THE EFMP
Brian F. CHELLAS
Introduction
For natural numbers k, 1, in, and n,
modal logic containing
i
s
the smallest normal
Gk.1.tn.n /
)
The Iformula
G
f
l
l
p
'• i "D. 1p
• " <T. D p p
1 >B. p - 4 C l O p
p E l p . -4 1 = 1 E l p
' ' 4.
1 5. O p —> El Op
e m figure
b r prominently in such well-known logics as the Feys-von
These
a c "logique
e s
Wright
t " o r system M. Kripke's "Brouwersche system",
and
S4 and S5.
s Lewis's
u
cLess h
well known, but important in what follows, is the instance
f a m
G. < > p L i Op
i l i
of aG'• "r . This is the characteristic theorem of the modal logic S4.2,
i.e.f KT4G.
o
rA logic
m is said to have the finite model property (fmp) just in case
each
u non-theorem
l
of the logic is false in some finite model of the logic.
Alternatively,
a logic has the fmp exactly when each formula consisa
s
tent
a in the logic is true somewhere in a finite model o f the logic. We
say
s o f a normal modal logic that it has the finite model property
essentially
(efinp) if and only if each of its normal extensions has the
:
fmp.
The point of this paper is to examine the question : For what values
k, I, in , n does K G " ' '
1 h a v e
t h e
e f m p ?
i
t
t
u
r
n
s
o
u
t
t
h
a
t
a
256
B
.
F. CHELLAS
theorem, essentially due to Kit Fine, provides an answer for all but a
handful o f quadruples of natural numbers. This is the main result o f
the paper. Except for one (or, perversely, two) the remaining cases
can also be resolved. So a secondary purpose here is to state a
problem for future research.
Background
A normal modal logic is based on propositional logic and may be
characterized by the presence of the theorems
Df. O p
K.
( p
—10
q ) ( 0 p 0
q)
and rules of modus ponens, substitution, and necessitation
RN.
A
LIA
Models and frames fo r normal modal logics are Structures H =
<W, R, P > and 5
worlds"),
R is a binary relation in W, and P associates truth values
1
with
pairs of atomic formulas and worlds. /11 is a model on 5r- ; i s the
frame
- < ofWM ,o d e l s and frames are finite iff their sets of worlds are.
Truth
conditions
for modalities are given by:
R
>
i
n
w
hO A is
i true at x in ;it iff A holds at every y in /it such that xRy.
c
h O A is true at x in J iff A holds at some y in /it such that xRy.
W
,1
i is a model of a logic iff the logic's theorems are true at every world
in I f . 3
s
7
logic.
A logic is determined by a class o f models (frames) if f every
a
; is
member
o f the class is a model o f (frame fo r) the logic and each
s
non-theorem
of the logic is rejected by some member of the class.
a
e
f For
r each k, 1, frn, and n, KG'
t
in,
and frames – i.e. those in which the relation R
a ,n-incestual
m n
i models
s
(
satisfies
e d e the
t econdition
r m i nthat
e for every x, y, and z in W
o
f d
f
if A
o b
y
"
po
r t
h
e
s
sM y i
a c
l
a
s
b
a
l
e
l s
n
o o
f
d
g k
,
x
i 1
,
l
c
?
i
"
Ke' A N D THE EMP
2
5
7
For KD, K T, K B , K 4 , and K 5 th is boils down t o the familiar
properties o f seriality, reflexivity, symmetry, transitivity, and euclideanness. In the case of KG the condition is called simply incestuality.
S4.2 and the efmp
We begin with a lemma.
LEMMA. K T
4
Proof.
0'
Simply note that every extension o f KT4 (S4) contains the
1
"reduction
laws" D A — D'A and O A O i A , for every I .
'
m
THEOREM
1. S4.2 - and therefore KG " for any k, 1, m, I - has
'
'
a normal extension that lacks the fmp.
=
K
T
4
Proof
G I n his "Logics containing S4 without the finite model property"
( S PI Kit Fine defines an extension KT4X of S4 and shows that it
4
does
.not have the fmp. It turns out that an almost exact copy of Fine's
proof
2
) yields the result that the extension KT4GX o f S4.2 lacks the
fmp.
f
o Fine's formula X is the conditional Y—>Z, where Z is the formula
r
<>(<>p A Oci A —"Or),
a
and
Y is a conjunction of these formulas:
l
l
(I)
k
(2) 0 (s —> 0 ( —Is A s ) )
,
(3) O p
1
(4) O A
,
(5) O r
m
(6) 0 (p ( - 1 0 c l A
r ))
,
n
• (
I17I•L
. )
F
o
r
m
o
r
e
b
a
258
B
.
F. CHELLAS
(7) ( q ( - - c O p A —10r))
(8) E l (r—) (--1 0 p A —10q))
To show that KT4GX lacks the fmp it is enough to establish two
lemmas.
Consistency lemma. Y is consistent in KT4GX.
Infinity lemma. Y holds at a world in a model i t of KT4GX only if it
is infinite.
For the infinity lemma it will do simply to remark that Fine proves
the corresponding proposition for the logic KT4X —i.e. that any model
of this logic is infinite if as much as one of its worlds verifies Y. Thus
since KT4GX extends KT4X the result holds here as well.
To prove the consistency lemma it suffices to construct a model of
KT4GX a t one o f the wo rld s o f which Y holds. Th is ma y b e
accomplished by minutely modifying the model Fine employs for the
analogous purpose in his proof.
Specifically, we consider I t — <W, R, P > in which the worlds are
arranged as in the diagram in figure I . In the diagram the arrows
indicating R are meant to be transitive; the fat arrows mean that
worlds 0 and — I go by R to every world below them; and for the sake
of readability self-directed arrows indicating reflexivity a re everywhere omitted. As the diagram shows, the worlds other than 4. 0,
and —1 are divided by levels, three worlds to a level. Thus worlds 1, 2,
mid 3 are on the lowest level, and so on.
The model I t is exactly like Fine's except for the addition o f the
world 4.
Note two features of I t : First, a world located on a level goes by R
to all but one of the worlds on every level below it. Second. distinct
worlds on the same level between them go by R to every world on
every level below them. It follows that within the levels three distinct
worlds are on the same level if none is an R-alternative of any of the
others. This is important shortly.
The following two propositions will secure the consistency lemma.
(A) Y is true at 0 in A .
(B) I t is a model o f KT4GX.
KG
I
•
"
A
N
D
T
H
E
E
M
P
2
5
9
For (A) we verify by inspection that Y's eight conjuncts hold at 0 in
J . Thus s holds at 0 by construction, and since s holds only at 0 and
OR-1 and -1 R0 the formula D (s 0 ( - - , s A Os)) is true at O. That
takes care of (I) and (2). For (3) we note that OR] and p is true at I ;
similarly for (4) and (5). Finally, since p holds at I alone and none of
I's R-alternatives verifies q or r, ( p ( - 7 0 A A ----101)) holds at O.
This settles (6), and the reasoning for (7) and (8) is similar.
For 035, note first that the relation R is reflexive, transitive, and
incestual. So I I is a model of KT4G, and we need to show only that
the frame ,9; = <W, R> of it is a frame for KT4GX - i.e. that X is true
at every world in every model on
Suppose that X's antecedent Y is true at a world w in some model
on T h e n Y's first two conjuncts hold at w. This means that the
formula <>(—Is A O s) holds at ty, and hence this world begins an
unending R-chain of worlds in 9; at which s and -I s hold alternately.
Thus w is either 0 or - I. since otherwise both s and - , s hold at 4.
Note that w is R-related to every world in
Because conjuncts (3)-(5) hold at w this world has R-alternatives x,
260
B
.
F. CHELLAS
y, and z that verify p, q, and r respectively. We argue now that these
worlds are all on the same level in g
Conjunct (6) holds at Iv. Hence Oct and O r are false at x, and so
neither y nor z is an R-alternative of x. Using (7) and (8) we conclude
in general that none o f x, y, and z is an R-alternative o f any o f the
others. R's reflexivity entails moreover that these worlds are distinct.
World 4 is an R-alternative to every world in .9
R-alternative
to 0 and — I. So none of x, y, and z is 4, 0, or —1; each is
7
on
some
level
at once that x, y, and z are all on the
, a n d
e inv „F.
e I rt follows
y
same
level.
w o r l d
i
s
a Now we
n can show that X ' s consequent Z holds a t w. L e t us
designate by xy the world immediately R-above worlds x and y in .9'.
Clearly Op and q hold at xy. But O r is false at all R-alternatives to
xy. So <>p, K>q, and O r are all true at xy. Therefore, since wRxy,
<>(<>13 A <>q A r ) is true at iv. This ends the proof of (B) and so
too the consistency lemma and the theorem. (2)
The remaining eases
Theorem I thus resolves, negatively, the question of the efmp for all
but at most fifteen of the KG'
, t and n are either 0 or 1 and at least one is O. Eliminating by duality,
m,
and
, l discounting
o g i c salternative axiomatizations, this number reduces to
eight
— :
t
o
w
iKGo, o,
t o, o K
t
hKG o
s
e
i
K
o ,Gn
w
i
c
o
Kh,G
h
o
°, ,G
K
k
,o
,
I1 ,G
,
K
1
1
, ,G
o
,
o
K
,
1
K
1o G " 0, 1' 1 = K5
=
o
,'1
=
,K T, and D, are the converses p D p and O p -3 El p of T and
—where1
K
D.
K
T
1
,D
B
,
T
1
,=
(
K
could
2 be thus adapted to show that S4.2 lacks the efmp [9].
D
)
I
a
m
v
e
r
y
i
n
KG
k
•
Four of these logics -K
, KT, K D, and K G
m
S4.2.
all lack the efmp.
( 1 So by theorem I these
•
The
case
of
K5
has
been
resolved
affirmatively.
'"
a r e
A
s u b l o g i Nc s
THEOREM
2.f K5 has theD efmp.
o
T
H
This was proved by Michael
Nagle in his doctoral dissertation 161
E
and reported in the Journal
o
f
symbolic
logic[51.
E
The question of the efmp
M and the logic KD„ was open until recently.
P
THEOREM 3. KD, has the
2 efmp.
6
1
Krister Segerberg presented
a proof of this in a paper read at the
1983 Conference of the Australasian Association for Logic181.
The logic KT, is an extension of both K5 and K D
r
Segerberg's
results entail that KT, has the efmp. But the result is easy
to .seeSdirectly,
o
N since
a g generated
l e ' s models for extensions of KT„ have at
most
a onen world
d (0, 0, I, 0-incestuality is the condition that xRy only if
x = Y ). (
3 There remains thus the question o f K G
)logic
° containing the "Brouwersche axiom", B, that fails to have the
' 1model
4
1property
, 1
finite
? 1 do not know, and so I conclude the paper
with
: this qI u e ry.
s (
t h e r e
4 a
)University
r
i
a
n
F. CHELLAS
n
o o fr Calgary
m
a Bl
(
normal:
3
for if A is a theorem so is El A by modus ponens and T
generated
c )
frames for l a
are
in number.
DI , alsot known
c. Tbut
B ytwo e
x a m One
i n is
i nK g
h as ABS
e or VERUM. It is determined
, othe
by
m oone-element
r e o v e frame
r ,
in which the relation is empty. The other is KTT,, also known
as
o eTRW
n (since
e necessity is equated with truth). This logic is determined by the other
one-element
frame,
in which the relation is total.
c e a
n
s (x
e
e
4t
tLeave
hFellowsa hip f rom the Social Sciences and Humanities Research Counc il o f
Canada
and by a Sabbatical Leave Research Grant from the University of Calgary, both
)
t e
of
which
T
i n
t are gratefully acknowledged. Furt her subventions from these institutions
me twice to travel to New Zealand during my sabbatical leave to work at the
hs
senabled
I wis h especially to thank my host in Auckland, Professor
ei o
cUniversity
n of Auckland.
s
Krister
Segerberg,
f
o
r
our
many profitable discussions, and in partic ular f or his
r
i os
t
e
resolution
of
the
efmp
question
for Klic.
e
n n t
p ss
r
o uo
p
e lf
r
e tK x
t
e sT n
s
i r, o
n
s ea
pr
oe
ri
262
B
.
F
.
CHELLAS
REFERENCES
[11 Chellas, Brian F. Mo d a l logic : a n introduction. Cambridge and Ne w Y ork :
Cambridge University Press, 1980.
[2] Chellas, Brian F. " e . l• -• n and the finite model property". Paper presented at the
1983 Conference o f the Australasian Association f o r Logic . Univ ers ity o f
Western Australia, Perth, 18-21 May 1983.
[3] Fine, Kit. "Logic s containing S4 without the finite model property". In Conference
in mathematical logic—London '70, edited by W. Hodges, pp. 98-102. Lecture
notes in mathematics, no. 255. Berlin: Springer-Verlag, 1972.
[4] Lemmon, E. J. (in collaboration wit h Dana Scott). The " L e n n o n not es ": an
introduction to modal logic, edited by Krister Segerberg. American Philosophical Quarterly Monograph Series, edited by Nicholas Rescher, no. 11. Oxford:
Basil Blackwell, 1977.
[5] Nagle. Michael C. "Th e decidability of normal K5 logic s ". Journal o f symbolic
logic 46 (1981): 319-28.
[6] Nagle, Michael C. "Normal K5 logics". Ph.D. diss., University of Calgary, 1981.
[7] Segerberg, Krister. An essay in classical modal logic. 3 vols. Uppsala: University
of Uppsala, 1971.
[8] Segerberg, Kris ter. " A small c orner o f the big lattice o f modal logic ". Paper
presented at the 1983 Conference of the Australasian Association f or Logic,
University of Western Australia, Perth, 18-21 May 1983.
191 Urquhart, Alasdair. Letter to author. 28 September 1981.