FINITE MATRICES FOR QUASI-CLASSICAL MODAL LOGICS
S. K. THOMASON
ABSTRACT
It is shown that every finite matrix which determines a quasi-classical modal logic is equivalent to one based on a finite Boolean algebra,
in which the designated elements comprise a filter.
This note adresses a lacuna in recent work on tabularity in modal
logic.
A logic is a set of propositional formulas, in the connectives
0 , containing all classical tautologies and closed under modus ponens
and substitution ; it is classical if it is also closed under RE (from
A — B infer El A — D B). The smallest classical logic is called E, and a
logic is q u a s i
- A matrix is a structure 9)Z = (M, 0, D, * , D) such that (M, 0, D, * ) is
an
c l aalgebra
s s i c aand
l D c M ; D is a l liter if whenever a Db e D and a e D
then
i f b e D; '..))1 is Boolean if (M, 0, D) is a Boolean algebra and D is a
filter.
i
t A formula A is valid on 9M, A , i f h(A) E D fo r every
h from the algebra o f formulas into (M, 0, D, -x-).
chomomorphism
o n t
Two
matrices
are
equivalent if the same formulas are valid on each. A
a i n s
matrix
is
characteristic
for, o r determines, a logic if the formulas in
E
the
. logic are exactly the formulas valid on the matrix. A logic is
tabular if it has a finite characteristic matrix.
Every Boolean ma trix determines a quasi-classical logic, since
{A I h(A) I f o r all h : .
Boolean
i . - * 9 ) Ithen
}
i{A 193Z
s
A } may be any set of formulas closed under
substitution-let
(M, 0, D , * ) be the algebra of formulas and let D be
a
the
c ldesired
a s sset.i Eve
c ary lquasi-classical logic has (and every ma trix
determining
l
o
g such
i
ca lo g ic i s equivalent t o ) a countable Boolean
characteristic
matrix-let
(M, 0, D , -») be the algebra o f formulas,
.
B
u
t
i
f
s
)
)
1
i
s
n
o
t
342
S
.
K. THOMASON
modulo equivalence provable in E, and let D be the filter of equivalence classes of formulas in the logic.
The problem is this. Recent discussions of tabularity (for example
j I, 2, 6, 7j), in order to utilize methods derived from universal algebra,
have limited themselves to (at most) the class o f Boolean matrices,
without considering th e possibility that a logic having n o fin ite
Boolean characteristic matrix might yet be tabular in the sense defined
above. I shall prove that that cannot happen :
Theorem. Any finite matrix which determines a quasi-classical logic
is equivalent to a finite Boolean matrix.
But first, three comments:
I. Th e lacuna is recent. Dugundji 131 proved that S5 is not tabular,
in the strict sense. Scroggs 181 proved that every proper extension of
S5 is tabular— the matrices he constructed happened to be Boolean.
2. I n the cases of 11, 6, 7j the lacuna is easily resolved. Fo r those
works are limited to classical (indeed, normal) logics. By 14, Lemma
3.21, fo r any finite matrix V there is an equivalent finite matrix 9.3r
such that every rule "weakly valid" on V is "strongly valid" on 9.W.
if WZ determines a classical logic then modus ponens and RE are
weakly valid on s.).)Z, and hence strongly valid on V ' , and this means
that D i s a filter in TZ' and equivalence modulo D' is a congruence
relation. Reducing V ' modulo this congruence relation yields an
equivalent Boolean matrix 9)1". (And if V determines a normal logic
then V " is isomorphic to the power-set matrix based on a finite
Kripke frame (W, R).)
3. The lacuna affects only tabularity, not the finite model property.
Indeed the finite model property is vacuous absent some restriction on
matrices—if L is any logic and A L then there is a finite matrix on
which every formula in L is valid but A is not [5, Theorem 2.41.
(Originally, o f course, the finite model property was intended as a
property not o f "logics" but o f "systems" comprising axioms and
rules.)
Proof of Theorem. Let V = (M, 0 , D , rn X
= card(M), determining a quasi-classical logic. Let ;1(rn) ( F ( m ) ,
I, - E, l )Dbe) the algebra
all of whose variables are included
b e of formulas
a
among
p
f i n i t e
i m satisfying
F(m)
a t r i x ,
, w
p
i
t
h
2
,
,
p
m
.
L
e
FINITE MATRICES FOR QUASI-CLASSICAL MODAL LOGICS 3 4 3
I. i f E H A — B then A
—ifBh(A)
, = h(B) for all h :
.
Thus A —B if and only if there is a finite chain A = A , — A2 '*-• A3
1(m)-0)Z
'"
t h e n
E H-A3 A 4 , etc.
A
—
Then
•
— is a congruence relation. Fo r suppose A —A' and B —B',
B
.
say AE H A —A' and h(B) = h (B') for all h (the more complex cases
follow
= e a sily); t h e n h (A —, B ) h ( A ) h ( B ) = h ( A ) h ( B ' ) =
h(A—)B')
fo r all h, and EH(A—)13') ( A ' —>B'), so (A—)B) ( A '
B
—*B').
Similarly,
if A —A' then E A — E A . Now let (M', O. D,-x-) be
s
( mu) /
let—D'
c = 11A, 1 V A I . Then (M', 0, D) is a Boolean algebra, since
IA,
; h= I
then
B fi to n A ' (use the definition of —); hence [A, E D' — A , and it
follows
wr hh that D' is a filter and V ' = (M', 0, D, -x-, D' ) is a Boolean
matrix.
e Ana
e cAnd
vt
n ' i s a fin ite : i f [ A , [ B , th e n h (A ) h ( B ) f o r some
h:;i(m)---0V;
e FrE
since there a re ju st mm su ch homomorphisms h ,
card(M')
E( A
mm
Let
m
Hm
l M = (a
.are
A )1Ih.qt. . . .
i—leah( (mq. )aj(I i= I , , n ) and let A' = A lp ,
ra,
(n,B
i Ae,s[A1 3
1
t,,. taq2, o n p o f each q
occurrences
., )n
. ,h A
i,sh
i„F.'(p
[nid
1
i/su A
3qs=„y,/w,u a nB, d qI A ' „ ç t ID'. FT h e n 931' 1
oVI;ZA',
b
,)u
t(i
iq
r( l,=p,h
t I ,hp ), m. ) then k (A ' ) = IA ') ( D ' . Since A ' i s a substitution
m
thati n ' fA .
•.ainstance
ch eAo
=
4
e
) (-' , ts sof i A,n it cfollows
Conversely,
suppose
A
is a formula in q
h
eA
( f .p
I•9ka
eM
iAc3' (N' pand h(A)çt D'. Let [ A
,3h
i,tr '—)9
h
A[A,,
q =
h
i)=
Z
i1
mb,i )n. . ,. , A
n
[A',
n
=
h
a
'(A'),
( I = I, , n ) ,
')a[h
1
a
,
s
=
p
h
(n qa ndd s in c e Vh ' (A , )' h ( q )
oe f
h'(A')
=
h(A)
OD'.
Since
[A'
,
9
N
I
;
L
A',
and
since
A' is a subs/=
q
I
,
,
q
#
A
,
ini(tr tao
titution
instance
of
A,
931
i#
A.
y =
sdr),q
n
a
e hsf r ( o aj
F
,
n
itn,I)h
, erI a , m
.(')=
h
E
u
,
l
i el
i
e
IA
,n .
d
a(a
=
fSimon
qam
l )Fraser
University
S
.
K. THOMASON
e
t
-rIl,;A uwh
,s- i h,
'
IefA
'
(
s,eh
a
n
=
o
-v q
)yA as i
,up
lvia
m
' le a
)tl)n
a ev e
o
d
tn=
n na t
h
f.[h
e
d cr
n
sp
'(n ei i
344
S
.
K. THOMASON
REFERENCES
11 jBlok, W. J. The lattice of modal logics: an algebraic investigation,./. Symbolic Logic
45 (1980), 221-236.
12j Blok , W. J., and Kohler, P. Algebraic semantics f or modal logics, J. Symbolic
Logic, to appear.
13j Dugundii, James. Note on a property of matrices for Lewis and Langford's calculi
of propositions, J. Symbolic Logic 5 (1940), 150-151.
141 Harrop, Ronald. On the existence of finite models and decision procedures f or
propositional calculi, Proc. Carob. Phil. Soc. 54 (1958), 1-13.
15j Horror). Ronald. On simple, weak and strong models of propositional calculi, Proc.
Carob. Phil. Soc. 74 (1973), 1-9.
16, Maksimova, L. L. Pretabular extensions of Lewis S4, Algebra i Logika 14 (1974),
28-55.
171 Rautenberg, Wolfgang. De r Verband der normalen verzweigten Modallogiken,
Mathenzatische Zeitschrift 156 (1977), 123-140.
181 Scroggs, Schiller Joe. Extensions of the Lewis system S5, J. Symbolic Logic 16
(1951), 112-120.