Logique A n a ly s e 133-134 ( I g9 I ), 23-39
MORE SUBTLE THEORY CHANGE
Fra n k Db RI NG
In this paper, I derive a syntactic procedure for revising theories in propositional logic from considerations of indifference and informational economy
(minimality). The procedure is very flexible. It allows us to make use of
information about the relative epistemic merit (entrenchment) of the sentences in the theories whenever such information is available, and, unlike
other procedures proposed in the literature, yields plausible results even for
very simple entrenchment orderings.()
1. The problem of theory change
The problem of theory change I propose to consider is twofold: how can
old information be deleted from a given theory (a data base, stock of beliefs,
knowledge state, or what have you), and how can surprising news be added?
The first aspect of the problem concerns contraction, the second revision.
The problem arises for every fallible representational system, hence for
every —artificial or natural— agent. A representational system, even one
that never commits a logical mistake, becomes fallible by drawing inferences
that take it beyond the evidence. It therefore needs a method to accommodate unexpected news and give up erroneous information without giving up
too much else. (Error correction is not the same as recovery from contradiction. I will not offer a recipe for resolving contradictions, I will offer
only a recipe to help avoiding them.)
For present purposes, theories are consistent, logically closed sets of
sentences in the propositional language L. L has a finite stock of sentence
letters A, B, etc. and contains as connectives only negation and disjunction.
(The restriction to the finite case is for the sake of simplicity; I shall indicate
where an infinite language would call for a more complex treatment.) Sentences or clauses in L are sets of literals interpreted as disjunctions, and
(
CNRS.
I
Thanks to Mars hall Farrier, Gary Gates, and François Lévy f or comments on an
earlier
draft.
)
R
e
s
e
a
r
c
h
24
F
R
A
N
K
DGR1NG
literals are sentence letters with or without negation. (A, —
-- p l e
4is3the
) disjunction
f o r
e (Ax va - m
calculus
notation
1B) . S
e t scan be translated effectively into into Le) Occasionally,
I will make use of the standard notation.
o f
s Logics,
e n classical
t e n as
c well
e as non-monotonic, are of little help in our problem.
s A logic gives interesting directives about how to expand sets of sentences, namely by closing them under some relation of implication. This is
i
n
relevant
when we want to accommodate new information that is consistent
s the
t theory
a n at hand.
d
with
But our problem is not with expansion, it is with
a
r
d
contraction and revision. It is a problem about giving up information, and
p
r tells
o us
p anthing
o sabout that. Logics, conceived as theories of implicano logic
ition,t furnish
i ostate
n laws
a for objects in theory space, but they do not deterl
mine
trajectories through the space.
We will write T A
traction
- f o of
r T by A, and T f o r the revision of T by A. Revision can be conceived
of
t h e as a composite operation: first, T is made consistent with A, i.e.
contracted
-e x p by
a n
definition
of
revision
in terms of contraction and expansion has come to be
1A
s ; i to h ne n
known
as ;the Levi identity. Alternatively, contraction can be defined in
To
f
terms
of
revision by means of the Harper identity T = T n T :
i t
h
e are equivalent given the Alchourrôn-GArdenfors-Makinson
two
A
i o identities
sr
y
postulates
.e Tx h pefora all three operations.e) I find the Levi identity intuitively
T compelling and will therefore treat contraction for the most part as
more
nb d e d
basic and revision as composite.
by
Theory change should be minimal. This tenet has been labelled variously
yA
principle
o f informational economy, conservatism, o r minimality. The idea
A
is ,that information should be neither adopted, nor given up, gratuitously.
iToTviolate
n the principle is to sabotage the project of finding out about the
t f
o is why. Information is not self-authenticating (except in degenworld.
This
T o cases). Thus, if one were to adopt any of it gratuitously, one would
erate
have
= r no reason to believe it to be true, that is no reason to believe it, that
is
T tno reason to adopt it. Information is not self-refuting either (except in
degenerate
cases). Thus, if one were to give up any of it gratuitously, there
Ah
would
be
no
point in gathering it in the first place. Conservatism is built
' e
into
the
very
foundations
of epistemic rationality.
. c
To
hn
i (s (2
)3
A
)
tS
re
a
e
n
f
so
lr
a
MORE SUBTLE THEORY CHANG E
2
5
The principle determines expansion: T ; should contain the entire content
of T and A, and nothing else. So T = Cn (T U 1/11). Contraction, by
contrast, is only constrained, but not determined. A semantic illustration
shows why:
Figure 1
Think of figure 1 as depicting the set of all "possible worlds" describable
in L. Contents correspond to sets of worlds or points, i.e., to areas in the
diagram. (We say that a clause is valid in a set of worlds i f it is true in
every world in that set; a set of worlds W characterizes a theory T iff all
clauses in T are valid in W and all clauses valid in W are in T, written as
W = I T I .) Contracting theory T by A in a most conservative way means
enlarging IT I by exactly one point from I - 'AI, say W. I T I + co then
characterizes T A
-IAI,
I fwe are
w left
e with co only. The result of a most minimal contraction by
A
followed
by
an expansion by n o w
is
thus
a
maximally
1e A ,x wp h ai c h specific theory. We have created information out of
inothing
t violated
h
e the principle of informational economy. Radically
n sd and thus
conservative
contraction
has the embarrassing consequence that every revisT
a
m
e
sion
creates
information
in this way.
a;
s
w
was
chosen
arbitrarily.
A priori, all points in I - ab
is 1nothing
in
T
that
distinguishes
between
them. So the alternative to select1 1 va r ie s o i n
a
ry , e
ing one arbitrary pointt is toh select
This choice sets T ; to I T I
e allr points.
e
o- p na r ;
+I b
ybut l o g i c a l consequences. This way, information is
I1 / 1 nothing
A
tains
-thrown
,I
away gratuitously. Less radical contraction is therefore as bad as its
Ii
A.
,
,
ea
.
,n
sd
h
rT
i
n:
k
IA
26
F
R
A
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K
D6RI NG
radical alternative. For a better solution, we have to break the symmetry
between the points.
A common move at this point is to deploy an ordering of epistemic entrenchment over the sentences in L that induces a partition of the area
surrounding IT I in the model. The sentences in the representation language
are assumed to be partially or totally ordered under a relation of entrenchment that mirrors their relative epistemic merits. This ordering is then used
to define contraction and revision functions that retract the sentences in the
order of their entrenchment, the least entrenched first and the most entrenched las t.(
4in L that satisfies the following four conditions:C)
) L e t
•
(1
- _•)
r F o r all A, B, and C, if A B and B C , then A C (transib
e tivity).
(2
a F on r all A and B, if A implies B, then A LÇ-T B (dominance).
(3 • ) F o r consistent T, A T iff A B for all B (bottom).
o r d
(4 I f A B for all A, then B is a theorem (top).
e r i n
g
The ordering is relative to T because according to (3 t h e sentences with
rthe lowest
e
l ranking are just those that are not contained in T. T induces
a
t of
i nested spheres around T with entrenchment increasing from the
asystem
o
nto the periphery (see figure 2).
center
d
e
f
i
n
e
d
o
v
e
r
a
l(
thing
l 4 lik e entrenchment may be implic it. F o r instance, Papini's (1992) procedure works
roughly
)
by computing maximal subsets of the original theory that do not imply the sentence
c E is to be retracted. Th e procedure gives automatically more credit to sentences that are
that
l.
explicitly
represented than to those that are merely implied, whic h amounts to treating
explicit
sentences as more entrenched. When more than one maximal subset is found, the
ag
choice
among them is left to the user. I find this move unsatisfactory.
u.
G
(
s89r ff.
5
eo)
s vT
e
h
(e
1
s
9
e
8
a
8
r
MORE SUBTLE THEORY CHANG E
27
Figure 2
Each sphere S, in figure 2 corresponds to a subset of sentences in T. The
bigger the sphere, the smaller the set of sentences it characterizes and the
more entrenched its least entrenched elements. Formally, i f S, and S, are
spheres such that S, C S
that
E txh<e, yr for
e every sentence y valid in S,. The innermost sphere is IT I
itself,
the
outermost
i s
a
t sphere (the box of f i
in
L.
The
minimal
of T by A can now be defined as the union
i s
lg uer e a 2 s ) t contraction
of
h I nand
e the
s
e
ta of I —
ot IT
e intersection
secting
I
—
IA
I
a
n
d
S
l
l
so e f n t e a
going
one
such
smallest
sphere
A
t
h
e
tn I .a, c tou be
t
o
l
o
g
i
e s for contingent /1.(
e
the
contraction
and
revision
function
for any sentence A can be read off the
T hm e a l l e s t
sx6
the spheres characterize.
)r eT p
aby lnoting
ur e which
o sentences
f
vhofiespheres
svsystem
h vl e
es sets of
acharacterized
The
set
by
S_
iis i ond n t
e
r
strictly
more
entrenched
than A. Revision therefore becomes: T
A
c
o
n
s
i
s
t
s
b
-i
Cn
(
o
f
y
nHarper
{ x identity
e
a c tT Al y
A
S
In
this
it is the ordering that does all the work. The
—
=
To
L
A
ti
h approach
s to revision,
e
of
the
derived
revision
and contraction functions depends crucial,plausibility
n
ss) } e nT t e
ly
on
the
details
of
the
sphere
system
induced by the ordering. We therefore
snU c
u s
jneed someeprincipled
account of how to construct that ordering. Reference
ct{
hh
a
u
tsX
I
a
r
t
X
e
a(
limit
>6 assumption in order to assure the existence of a smallest I —
.IA) I - i n t e r s e c t i n g s p h e r e .
T
ST
•
i h
Ae
ni
}
cn
)
ef
,
Li
28
F
R
A
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K
DbRI NG
to a notion of epistemic entrenchment is one attempt to fill this bill. On a
natural construal of entrenchment, however, this is not going to work. In
Glirdenfors's view, from which I take my bearings, it should be "possible
to determine the relative epistemic entrenchment of the sentences 1...1 independently of what happens [...] in contractions and revisions" (Gardenfors
1988, p. 87). This much seems uncontroversial for any substantial notion
of epistemic entrenchment. Gardenfors also claims, somewhat more controversially, that the entrenchment of a sentence is connected with "how useful
it is in inquiry and deliberation" (ibid.). I have two complaints. First, an
ordering of sentences in terms of their usefulness in inquiry and deliberation
will normally not determine a plausible revision function via the sphere
construction just described. Second, more generally, any ordering that has
been determined independently of considerations about revision will yield
a poor revision function via the same construction.
As for the first complaint, suppose there are two sentences A and B in T
that intuitively have nothing to do with each other; say, A is about French
cuisine and B about life an mars. Because of their independence, it is reasonable to demand that A E T ,--- and B E T T h e demand is satisfied just
in case there is a sphere bigger than IT I in which the disjunction of A and
B is valid but neither A nor B is, i.e. i f (A, B) is strictly more entrenched
than A and B individually. (Proof If {
T
A ,„ / 3 }
i s
v a l i d
.i= T An
S ,
at
A
- least one (
A
-a
and
in T Tn h e reasoning for A is the same.) But surely
,- so nott contained
h
e
In
the
A
disjunction
&
of
A
and
B is no more useful in inquiry or deliberation —and
[O n A
,
thus
hardly
more
entrenched—
than A or B individually. It is difficult to
–
d
t h
1
3 other independent
)
imagine
any
specification
of entrenchment on which the
1
,E
e/ 3 ) disjunction
would be strictly more entrenched than its disjuncts. In order
w
bo o r l d ,
to
make
things
come out nicely, the ordering apparently has to be set up
w
h
ywith
t
an eye to its use in revision, which is what we are told to avoid.
ich c
As for the second complaint, there are certain simple entrenchment orderh
lings
e that should not be ruled out a priori. But these orderings yield unrealor
i contraction and revision functions. For example, suppose that the
sonable
e
sh
sentences
in L are arranged in just three tiers, tautologies at the top; continsua sentences in T in the middle, and the remainder of L at the bottom.
gent
irn system of spheres induced by this ordering yields the second of the two
The
n
ed
"bad"
revision functions discussed before where T = Cn –
a
contingent
I, A N
fA oE T.
r The result is less drastic for a richer ordering, but the
.B
i
H
E
f
e
T
{
n
„A
c*,
e
A
/
B
S
3
MORE SUBTLE THEORY CHANG E
2
9
basic flaw is the same: too much information is deleted. Since I agree with
Gdrdenfors that entrenchment should be kept independent of considerations
about revision, I have to reject the proposed construction.
Entrenchment information would be useful nonetheless if we had a way
to select a sensible subset of the worlds in the intersection of I —
that
A I is,
a ifn we
d could
S distinguish sensibly between sentences undistinguished
by
the
entrenchment
relation. This is of course just our original problem,
A
only
now confined to a part of I —
,
lution
problem.
The solution can then be combined with a
1 / 1 1 .to the
I ngeneral
t h
e
sphere
model
s e q
u to
e define
l , plausible revision and contraction functions.
I
w
l theoretic
l
2. Theimodel
clue
d
e
v
e
l
o
p
Our problem is to draw a distinction among the -'A-worlds in figure 1
a
based
on no other information than what is contained in T. It turns out that
s language
o of T provides
the
the constraint we need if we think of the worlds
in the model in terms of their descriptions in L rather than as unstructured
points. "Possible worlds" describable in L are given by maximal, consistent
sets of atomic sentences and their negations. (A set is maximal just in case
it contains every atomic sentence of the language in affirmed or denied
form.) A theory in L is characterized again by a set of possible worlds, as
depicted in Figure 3:
Figure 3
30
F
R
A
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DbRI NG
The items in figure 3 stand for classes of worlds, not for individual worlds
(which can be very complex). For example "ABCD..." stands for the class
of worlds in which A and —
negation).
13 a n dTheory
- - T corresponding to the encircled area contains A and B
together
with
I C
a their
n dlogical consequences. The no
'DAlonger
- w o on
rla
d par:
s there
o u ist a subset
s i dI Te
gion,
members
are
"closer"
to the worlds in T I than any other —
-Ia I whose
o
f
t
h
e
m
,
I
rT e
worlds.
They
are
closer
in
the
sense
that each of IT ' s members differs
Iia
t A -n r sr iu de ee
from
in rI T lI only in T h e structure of the language thus
t( u some
h n world
e
singles out
adset eof -'A-worlds
closest to T:
e
i nl i l ni gp t i c
ri
ed
i -setc S
(Cl) nA non-empty
a- o just
tf -ine-case sfor every member s
o which
f Ss
1 P -with
e atomic,
r e s and s
w o -rofl P.
d t(sIfhP is
s
and
s
i-as gd ris fe f ee r
i
u oss t me
ei
n
c l s-j o
The
information
contained
d
i
f
f
i n e;
P
m
b
ein T and A provides no grounds for finer
s tm
distinctions;
we
are
indifferent
vis à vis possible further subdivisions within
re
ia r lo f l
t
the setsiP
of
closest
r e s
a
led to add the set of closest -'A-worlds as a whole to I T I
'are
A -therefore
w
rld
on
ip o ee
css . f
s
E
smaller
inclusion would impose an arbitrary partition on I —
I. verynS
o
t s
a
t
every
larger
inclusion
violate minimality. The resulting theory T
A
I
,
a
n
d
c o n
rs a j uwould
en ti xthe
d
n by
c the circle and the ellipsis, shows neither
o
comprising
area
enclosed
c t e
itc ni ego n ,
f the earlier defects: the theory T g a i n e d by restricting T A
of
T
p-r e t a is neither
'P
its
s -'A-worlds
ao
t o maximally informative, nor does it consist only of
b
rt o
the
w consequences
r of —
y
m
h
1/1.
l inside
and
d T the
s ellipsis,
A
contains all and only the logical consequences of —
A
and
B,
o
which
is
intuitively
correct.
cIA
S o r r e s p o
i Contraction
n d rsi n g by a disjunction works in just the same way. The selection
n
of
e oworlds falsifying a given disjunction is unique according to (Cl)
t closest
a
because
o
r hthere
e is only one way for a disjunction to be false, i.e. one common
t
m
feature
of
i
the
fl
e selected worlds: all disjuncts must be false there. The contrace
tion
rule
for
arbitrary clauses in L can therefore be stated as follows:
n
P
vi r a
a
m
a
eCon T'n ai s tcharacterized by the union of IT! and the set of —
l
o1 C - sfclosest
u o r lttod T.
w
s
w
d
oi
s
d
a
e Theirrule is stated in model theoretic terms. It would be better to have a
y
st
t
,
jh
h
w
u
e
e
e
n
t i
c
cr c
r
tu e
l
st
MORE SUBTLE THEORY CHANG E
3
1
statement in syntactic terms that tells us how to revise the sentential representations of theories. Our next task is therefore to translate Con into a
syntactic rule. This rule has to effect the removal of all clauses from T that
are not valid in the set of closest —
1 Let
C -C
w =o r{ 4l ,d s4 , } be
Wthe clause in T that is to be retracted. W = I T I ,
and W
all
- the L
i s
removed?
Consider first those clauses whose intersection with C is empty,
that
a
e clauses containing none of the L,. By assumption, these clauses are
t rh is
valid
fe a lin W. They are also valid in W
agrees
inb W on
-cs eb
c asome
u s world
e
y everything except the L,, and this agreel .ewith
ment
is
all
that
matters
for
the
truth of L,-free clauses. The clauses must
(W
o sC l )
therefore
be
included
in
T.
Second,
consider the clauses whose intersece
e
r
y
h
iv
e
s
tion
with
w
o C isr not empty,
l
dand among them those that contain the denial of
ct
at
least
one
of
the
L,.
These
clauses are valid in W b y virtue of their
ih
n
sdisjuncts and
must therefore be retained as well. This leaves us with those
W
ce
l whose intersection with C is not empty and which do not contain
clauses
-a
tthe denial of at least one of the L
u
so seem too sweeping, but we will see presently that it is not. The rule
may
is:
sf. T h e s e
e
sw h a l l
so
RI
b (IL,,
e L
Otherwise remove all clauses from T that contain
sca
r
l
l
of C but not the denial of any of the L
,a
lw) , i t h d r any
a subset
w
s
n
T
d .)
n
Atomic
clauses
are
a
limiting
case of disjunctions. An atomic clause {A} is
.
IrsT f h
i
retracted
simply
by
removing
all clauses from T that contain A as a disjunct.
e
T
si RI leaves only clauses in the remainder of T that are valid in Con (C, W)
m
d
o o
n
m
v
=W U W
. Therefore
all clauses in the closure of RI (C, T) must be
a
e
s
e
valid in Con (C, W). If we can also show the converse, we have proved the
ilfollowing equivalence result for Con and RI plus closure:
n
n
o
l
tio
Equ I f a set of worlds W characterizes a theory T, then Con (C, W) charin
acterizes Cri (RI (C, T)).
f
T
m
w
,Assuming
p
that W characterizes T, we have to show that every clause D that
h
is
la
o valid in Con (C, W) is included in Cri (RI (C, T)). We will first establish
that
yn
s a clause D valid in W is not valid in Con (C, W) if its L
D
.C
d
e - C is not valid in W. Suppose that D - C is not valid in W, i.e. false in
-=
w
ef r e e p a r t
{h
l
ie
L
,cm
„h
e
h
4
n
}a
32
F
R
A
N
K
DbRI NG
some world w E W. There exists another world w' that is just like w except
that it falsifies all the L, E C. In w', D - C must be false as well, and so
must be D because the literals by which it exceeds D - Care all false in w'.
But by (Cl) w' is in W
- Con (C, W). The rest is straightforward. Suppose that D is valid in Con
in
(C,
of what we just proved, D - C is valid in
, sW).
o Then,
t hbyacontraposition
t
W,
hence
contained
in
T,
not
affected
by RI, and consequently included in
D
RI
(C,
T).
D
C
implies
D,
and
therefore
D E Cn (RI (C, T)).
i
s
restores what RI removes without need. Equ allows us
n Logical
o closure
t
to define contraction as follows:
v
a
l
i
d
T =
i
n
c
W
l
-3. Conjunctions
e
a
n
f
d complete account of revision requires an additional rule for sets of clauses
A
C
h
interpreted
e as conjunctions. Conjunctions are not sentences of L, but we
r
need
a rule
n
c for retracting what is expressed by a conjunction in other lani
guages
because
otherwise we could not guarantee consistency for expansion
e
(
by
n a disjunction: before we can expand a theory by ( R
have
1
— sure that it does not imply all of the Ç. Contraction by sets
o C „to make
I as conjunctions raises two specific problems. One is that there is more
read
It C J ,
w
e
( one way for a conjunction to be false, which opens several possibilities
than
v
a
forCinterpreting the notion of minimal change; the other is that contraction
l
i
and
, revision procedures ought to treat logically equivalent sentences alike.
d
Let
T us start with the second problem. We will adopt the following postulate
(read
4 = a s l o g i c a l l y equivalent to...'):
)
)
If
, A 44 B, then T = T Tr()
f
The postulate is met trivially for RI because every non-tautological clause
o
in L is logically equivalent to no other clause than itself. Equivalent sets
r
(conjunctions) of clauses, however, need not be identical, as for example
c
lHA, B ), (A, —
a1 B ) ) 4 4
u1
s{
(
, 1
7e
}
)s
FC
,
oc
l
.
rl
a
4
d
e#
MORE SUBTLE THEORY CHANG E
3
3
(Equivalent theories are identical because they are closed under implication.)
Contraction must be made indifferent to the different ways of expressing
the same proposition. This is best achieved by rewriting the sets of clauses
in a unique, canonical form before submitting them to the syntactically
sensitive contraction prodedure. We define the canonical form of a set of
clauses to be the minimal set of clauses that implies Cll. The minimal set
in the example is f
form
effectively. We will assume that every conjunction is brought into its
{
canonical
form before it is retracted.
A)
} Conjunctions introduce a complication into the the distance measurement
between sets of possible worlds because there is more than one truth-value
.
assignment to the conjuncts that makes a conjunction false. The issue is
T h e r e
whether we should count as closest to a set W verifying a conjunction P all
a
r
e
—
v a we
r should
i
whether
draw finer distinctions among the —
1
o
u
s
the
P's falsification.
1PPnumber
- w o r of
l d conjuncts
s,
b involved
a s ein d
w
a
let T be the closure of (B) , {C} , and {
o- For example,
n
y Dand
sletc Ph =a1(24,
W,
)
,
r a B)
c ,t {eC r i z e d
w
}to}b
y
ABCD...
ABCD...
.or l W
e
a r eABCD...
ABCD...
oe r e s t e
icd n t ABCD...
ABCD...
ABCD...
p i
dm
n
s
th
e e
tut
W,
tc
l
o
s
e
h
sha
t
—
ABCD...
1et
P c
a
w
d o r l d s
n
o
:i
ABCD...
n
i
f
c
a
f
Figure 4
l
e
r
The closeness criterion ( Cl) is not sensitive to the number of conjuncts
f
involved in the falsification of P. It counts as closest any world falsifying
Pr that otherwise agrees with some world in W. According to (CI), the set
o —
of
mP W
1
a o rW than either W, or W2 because each world in W, differs from any
from
2
w
U
lnd s
W
cy l
3
ow s
.eo s
Itr
n
tl
tod
34
F
R
A
N
K
DbRI NG
world in W more than any world in W, or W„. This suggests the following
more discriminating closeness criterion:
(C2) A non-empty set S
- o worlds
f - S in which P is valid just in case every member s
1 P -falsifies
wo of exactly
r lSd sone conjunct C
s
o
f
P ,s a n d
S
with
which
i
-s
t-a h i e rd e
s
According
ia gortose(C2),
e s the set of t
U
W,
U
Ib=PW- sw
o
r
l
d
s m
e
i n eo W
(C2).
The
earlier
proofs
remain intact because for clauses in L the two
2
cc l m
o
t
b
al s loe l s sm
definitions
are
equivalent.
te.
erso e rt s p
There
another alternative, or rather class of alternatives: one could set
M
i
n
i ism
T
c at l s
tT t o seonly
smaller sets W U W, and W U W, and thereby stay
io t y oe s xone of the
f
even closer to the original
theory. It is not clear, however, which of the
c l c e ,ae
W
a
smaller
sets to choose in general because there will be a tradeoff between
l py t
U
W
sr size
the
of the sets and the minimal number of atomic sentences by which
ftheir
a
v
2
t
ht
e worlds
differ
from the worlds in the reference set. If, for example, the
o
r
s
,reference
o set sis (ABCD..., ABCD...) and we want to retract {
o
h nthe closest not-IA, B, GI set is IABCD...), and the closest nota
ID1),
ethen
{ft
s ,a
d
{,i 4 , f B
l
o
p
Tze against
si
of content, and I have no idea how to do this a priori.
D
s
idistance
f
One
might
decide
to go always for the smaller set, which would amount to
t}
iy
oi
retracting
sn
n; only
g the most complex conjunct, because the more complex the
conjunct
the
smaller the closest set of worlds in which it is false. But then
eI
C
one
could
not, as in the example, withdraw {A, B} and {
tt
sh
C
tion,
which
can be desirable because retracting {
ie
.r
l2
e
aves
us
with
}e1 ,a sfB } aaatheory
n c dthat
o ndoes
{j unotncontain
c - IA, B, q
sis weaker than the one characterized by the choice of W U W, U W
,C
i . er .
,
a
t h e o r y
{o
the
relative
of the conjuncts have their place
t}2e Considerations
h s ea p t aabout
r a t e l merits
y
A
in
. the preparation of a contraction, that is in the choice of the object to
a
B
retract.
They should not be taken to constrain the contraction rule itself. By
d
C
choosing
(C2) as the relevant criterion, we do not pretend to have an answer
o
to
D the Duhem-Quine problem. We do not pretend to know what to do in
general
when we find one of our theories in conflict with the evidence. We
.p
t.
only
claim to know what to do —in the simple propositional case— once
we
. have singled out an undesired conjunction and decided not to put the
blame
on any particular conjunct. (C2) does not assist us in reaching this
,decision.
A
B
C
D
.
.
.
MORE SUBTLE THEORY CHANG E
3
5
(C2) gives rise to the following contraction rule, which treats all conjunct
clauses C, alike:
R2 (
ing. Otherwise apply RI recursively to all C,, i.e.,
{
perform (RI ( C
C,,
1
union
of every 2-element subset of P to the resulting
C
set.
...(R1
„
( C „ ,
)
The unions of the 2-element
T ) subsets
) . . of P are the strongest clauses we are
,
allowed to add without. reintroducing
any one of the C, and thus breaking
) ,
T
symmetry; these clauses
imply
the
unions
of all larger subsets of P. The
a
n
)
trace that remains, for dexample, of I
IC), tB, C, DI, 1,4, D)). (The rule that would correspond to the less dis{
t
h
fcriminating closeness definition
(Cl) is R2 without the second step in which
A
e
Tauses deleted in the first stepnare added again.) Contraction can be defined
cl
}
a
d
as
of R2 (IC,, C „ ) , T).
, before
{ B as
, the logical
C } dclosure
,
o R2 preserves what the contractions by the different conjuncts have in
( D
d
e
common;
the effect of adding
the union of the C, is that T Tc/ c n) = Tc; n
}
i
s t
n
T
I
n
our
example,
we
first contract T by l q , which corresponds
s
tto joining
h W and
e W2. Then
h
we contract by IA, B), which corresponds to
n
s
e U W2,t W„ and
joining
W
W
e
o
I
A
toI subtracting
W,
3ponds
t
the sets of
characterizing T f
3B
.of
l l ,worlds
y ,
i F i n a steps
contraction
is
irrelevant.)
A
,m s oe t h a t
w
Thed
etwo rules
a,w
d RI and R2 form a complete account of revision for logically
p
closed
theories
el ) naA n dd, in L Tby sets of clauses in canonical form. Syntactic and
{B
semantic
(u T p approach
e are equivalent in the following sense:
B
,h
y
o
ri ) dt
e,
r
w
C
a
Equ*
I f a set of worlds W characterizes a theory T, and (C2) is the closeo
f
hl
w
h measure
i
c Con, then Con (P, W) characterizes Cn (R2 (P,
ness
for
h
e
htW
T)), where P is a set of clauses in L.
l
cU
o
r
r
t
eW
s
Ihomit the proof for Equ*; it is essentially a combinatorial explosion of the
,e
proof
for Equ.
U
c
W
l
2a
,
tu
hs
ee
us
n
iC
o
n,
i
36
F
R
A
N
K
DbRI NG
4. Closure
It is time now to inject some realism into the discussion and drop the closure
requirement on theories. We want to be able to modify the fragments of
theories that we actually represent on paper, in computer memories, or in
our minds. Contraction and revision should be unaffected by the way the
theory is stated; so we postulate that 0 = (Cn 0) p
set
- , off clauses
o r
aO. nI f we
y simply applied R2 to 0 , this postulate would be
violated
in
two
kinds
c o n s i s t eofncase.
t A clause that is not in 0 might be irretrievably lost if it is a superset of some clause in 0 or if it is implied by some
subset R of 0 without being a superset of any of R's elements. The first
kind of case is illustrated by 0 = { ( An contracted by {A, 13}. RI ((A, B),
Cn (0)) contains (A, - (A,
1 B )—
, b u t
contracted
by A. R I (
1
B
)
.
R I
contains
nor
(AA n( A ,implies B. Mixed case are of course also possible.
remedies are called for. The remedy for the second kind
)iB,lTwo
n,
l C)different
of
case
is
to
bring
0 into canonical form before contracting or revising it.
(u
0 s0 ) ) )
The
first
kind
is
taken
care of by the following reformulation of RI:
ctn r o e n i t a
t ih
na
e t sir
R1' ( { L
B
o
c n o ,n
t
Otherwise replace every clause D that contains a
]
bo
a
i un
s
subset
C' of C by the set of clauses D U (
, L
tfn
o
each
of
the L, in C - C.
„ }
R
tr
11
,
Ih
i
m
pis a clause
Notice
that
if
S
1 containing C, there is no 1 —
T
(1.
e
l case,
i 0S is replaced
e
this
by
) , thef empty
o r set, i.e. deleted. We have RI' ((A, B),
)
1UAW
s1
=, (
Itwo remedies work in general.
0()e t o c o n j o i n .
I
n
f
)A
c, - T
n1
i
o B ) )e ,
d5. Entrenchment revisited
ta
e
n
sh
o
rd e s
eR2 is not without quirks. For example, suppose theory T = Cn ((A, 131)
ik r e
sis first expanded by A and T' = T ; then contracted again by A. The result
i .
d
nis Cn 0 , although intuitively we may want to restore the initial theory T.
In
oEven if we do not think of I
ld weemay take the explicit mention of the disjunction (A, B) as an indica(
tA,
otion that there are reasons other than A for its presence, reasons which count
a
, B ) ,
(
iA
vf
A
)
)
a
s
m
c
e
h
e
pt
ia
rl
e s
u l t
ts
f
yo
te
a
n
C
i
o
x
p
a
n
=e
ts
s
i
o
n
MORE SUBTLE THEORY CHANG E
3
7
against its removal in the course of removing A. Let me flesh out the example a little. Suppose you believed fo r good reasons that there is a ball in
exactly one of two urns, A or B. No w someone tells you that the ball is in
urn A , which you accept. Afterwards you learn that the informant is a
notorious liar. You will then reject what you came to infer from his testimony, but i f in revising your vie w you fo llo w R2, you w i l l no longer
believe that there is a ball either in urn A o r in urn B. Yet o f course you
still want to believe this disjunction. So R2 appears to give bad advice.
Your reasoning in favor of retaining the disjunction is based on a wealth
of tacit assumptions not represented in T'. Most importantly, you think that
your belief in the disjunction has independent —and better— grounds than
your later acquired belief in one o f the disjuncts. This thought can be expressed very naturally in terms of epistemic entrenchment by saying that IA,
{A}. And the entrenchment ordering can be used in combination
with R2 to yield the desired contraction.
It is easiest to approach contraction through revision. Recall that a total
ordering T over the sentences in L induces a system o f nested spheres
around T as shown in figure 2. Call the smallest I Aa n 1
S
d -the
i nsett e
of rsentences
s e c t characterized
i n g
by it P. The desired revision by
s- p
h
e
r e
worlds
that are also in SA. In these worlds, all sentences in P are valid. So
I
A selection amounts to revising T by P U {
the
equivalent
to (
-i
.IsA }disjunction
the
of A and the
of all the P, in P, brought into canoni.
T
( negations
,
)
1
cal
ig form.
s T c a n be computed using R2 in the normal way. (I f L were an
-dinfinite
i e f language,
i n i tthei canonical
o n a form
l l ofy X might not be finite, in which case
R2
uv (X, —)
, would not be an effective procedure.)
Ae In )the, urn example, we wanted to preserve the disjunction A v B in the
contraction
w
n h
e of T' = Cn {{A }) by A. What we want is T'
v /I) v w h i c h
ris
, the
e same as T' ,
indeed
contains {A, B}, as desired. In the simplest case, with which we have
X
an )
i.o concerned
been
A
in the bulk of this paper, all non-tautological clauses in T
sqt u equally
i c entrenched.
k
were
Thus for any non-tautological A implied by T, the
tib n s I p- e
smallest
hcy 2 set
1
t4 iof
I otautologies,
n
the
and contracting T by A while preserving Pis contract-eo
a
ing T by (A v
v
v
...), fo r all tautologies P
i"ifl n tne r e to simply contracting T by A.
tantamount
sg.R
l The
eT cahvirtue
t ii Ist of ithe sapproach
o btaken
v iin othisu paper
s l isy that it allows us to construct
specified contraction and revision functions fro m an
i'- g o syntactically
n
n
s"s7 p h
ooA e w
h
rfs- e
iP
tw
h
sU
ao
t
t{tr
-hl
h
38
F
R
A
N
K
DÔRING
arbitrary entrenchment ordering. The ordering can be as simple or as complex as we please, which is how it should be if epistemic entrenchment is
a substantive notion, not just a new label for the revision problem. The story
is still not complete, though. Entrenchment is theory dependent, and we
have not said anything about how to revise it in the course of theory revision. We therefore cannot yet handle iterated theory revision.
CREA / Ecole Polytechnique
LIPN / Université Paris-Nord
REFERENCES
Alchourrôn, C. E., P. Gdrdenfors, and D. Makinson (1985). "On the Logic
of Theory Change: Partial Meet Functions fo r Contraction and Revi-
sion." Journal of Symbolic Logic 50: 510-530.
Gdrdenfors, Peter (1988). Knowledge in Flux. Modeling the Dynamics of
Epistemic States (Cambridge, Mass. & London: MI T Press).
Gdrdenfors, Peter, and David Makinson (1988). "Revisions of Knowledge
Systems Using Epistemic Entrenchment," in Theoretical Aspects o f
Reasoning about Knowledge. E d . Moshe Y . Va rd i (Los Altos, CA :
Morgan Kaufmann): 83-95.
Genesereth, Michael R., and Nils J. Nilsson (1987). Logical Foundations
o f Artificial Intelligence (Los Altos: Morgan Kaufmann).
Grove, Adam (1988). " Two Modellings for Theory Change." Journal o f
Philosophical Logic 17: 157-70.
Harman, Gilbert (1986). Change in View: Principles of Reasoning (Cambridge, Mass.: M I T Press).
Levi, Isaac (1984). Decisions and Revisions. Philosophical Essays on Knowledge and Value (Cambridge etc.: Cambridge University Press).
Levi, Isaac (1989). "Review: Knowledge in Flux. Modeling the Dynamics
of Epistemic States." The Journal of Philosophy LXXXVII. 8: 437-444.
Lévy, François (1993). "Revision in Classical Logic." Unpublished paper.
Lewis, Da vid K . (1973). Countetfactuals (Cambridge, Mass.: Harvard
University Press).
Mongin, Philippe (1993). " Th e Logic o f Belief Change and Nonadditive
Probability," in Proceedings of the 1991 LMPS Conference. Ed. Dag
Prawitz and Dag Westersahl (Dordrecht: Kluwer, Synthese Library).
Papini, Od ile (1992). " A Complete Revision Function in Propositional
Calculus," in ECA1 92. 10th European Conference on Artificial Intern-
MORE SUBTLE THEORY CHANG E
3
9
gence. Ed. B. Neumann (Chichester etc.: John Wiley & Sons).
Rott, H. (1991). "Two Methods of Constructing Contractions and Revisions
of Knowledge Systems." Journal of Philosophical Logic 20: 149-173.