Reliable Belief Revision

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KEVIN KELLY, OLIVER SCHULTE, VINCENT HENDRICKS

RELIABLE BELIEF REVISION

ABSTRACT. Philosophical logicians prop osing theories of rational b elief revision havehad

little to say ab out whether their prop osals assist or imp ede the agent's ability to reliably

arrive at the truth as his b eliefs change through time. On the other hand, reliabilityisthe

central concern of formal learning theory. In this pap er weinvestigate the b elief revision

theory of Alchourron, Gardenfors and Makinson from a learning theoretic p ointofview.

1. CONSERVATISM AND RELIABILITY

There are two fundamentally di erent p ersp ectives on the study of b elief re-

vision. A conservative metho dologist seeks to minimize the damage done to

his current b eliefs by new information. A reliabilist, on the other hand, seeks

to nd the truth whatever the truth might b e. Both aims supp ort hyp otheti-

cal imp eratives ab out how inquiry ought to pro ceed, imp eratives that might

b e considered principles of inductive rationality.Itwould seem that there is

some tension b etween the two p ersp ectives. The conservativesentimentisto

smo oth over the e ects of new information, whereas reliable inquiry mayre-

quire more radical changes. The inductive leap from a hundred blackravens to

the universal generalization that all ravens are black is not conservative. Nor

was Cop ernicus' revolutionary rejection of conservative tinkering within the

Ptolemaic system. Conservatism and reliabilism are re ected in two equally

distinctive logical p ersp ectives on the problem of b elief revision. Conservatism

is the motivation b ehind the theory of b elief revision prop osed byAlchourron,

Makinson, and Gardenfors (AGM). Reliabilism is the principal concern of for-

mal learning theory.

Although the AGM approach is not motivated by reliability considerations,

its authors seem to hop e for the b est.

Even if the analysis of b elief systems presented in this b o ok do es not dep end on any account

of the relation b etween such systems and reality, I still b elieve that rational b elief systems

to a large extent mirror an actual world. (Gardenfors, 1988, p.19)

But if this happy situation obtains, it must have b een broughtaboutbythe

pro cess of b elief revision. Therefore, an imp ortant question for any prop osed

theory of rational b elief revision is whether its norms assist or interfere with

an agent's ability to arrive at the truth as evidence accumulates. Indeed, when

putative principles of inductive rationality can b e shown to prevent one from

b eing as reliable as one otherwise mighthavebeen,we are strongly inclined

to side with reliability rather than with the principles.

For general intro ductions, cf. (Osherson et al., 1986) and (Kelly, 1995).

Hilary Putnam (Putnam, 1963) resp onded to Rudolf Carnap's theory of con rmation

in just this way. 1

2 KEVIN KELLY, OLIVER SCHULTE, VINCENT HENDRICKS

In this pap er, weprovide a learning theoretic analysis of the e ects of the

AGM axioms on reliabile inductive inquiry. We consider a variety of ways

in which the AGM theory can be interpreted as constraining the course of

inductive inquiry. A representation of the reliable AGM metho ds is established.

2. THEORY DISCOVERY

We will adopt a fairly simple and abstract construal of reliable theory discov-

ery (Kelly,1995).We assume that the scientist is studying some system with

discrete observable states that may b e enco ded by natural numb ers, so that

in the limit the scientist receives an in nite stream " of co de numbers. Let N

denote the set of all such data streams.

An empirical proposition is is a prop osition whose truth or falsity dep ends

only on the data stream. We therefore identify theories and hyp otheses with

sets of data streams. Let b e a countable set of empirical prop ositions. We

may think of as the set of all prop ositions wewant to nd the truth ab out.

We assume only that is closed under complementation. A -theory is de ned

to be any result of intersecting some collection of prop ositions drawn from

. The complete -theory of a data stream " is just the intersection of all

prop ositions in that contain ". Set-theoretic relations among prop ositions

represent logical connectives and relations in the usual way: entailmentisset

inclusion, inconsistency is disjointness, conjunction is nite intersection, and

so forth.

Let e be a nite data sequence. The empirical prop osition expressed by

e is just the set of all in nite data streams extending e, which will be de-

noted by[e]. We refer to such prop ositions as fans. e q denotes the result of

concatenating datum q onto the end of nite sequence e.

A theory discovery method takes nite data sequences as inputs and out-

puts prop ositions. In particular, pro duces an initial conjecture (;)onthe

empty sequence ;.We don't require that the output prop osition b e in . We

will think of the metho d as reading ever longer initial segments of some in nite

data stream ". It is therefore convenienttolet "jn denote the initial segment

of " of length n: i.e., ("(0);:::;"(n 1)). Nowwe de ne two concepts of reliable

theory discovery. Uniform discovery requires that the metho d b e guaranteed

to eventually pro duce only consistent conjectures entailing the complete

truth. Non-uniform or piecemeal discovery requires that each prop osition in

has its truth value eventually settled by the metho d's conjectures, but there

may b e no time bywhich all prop ositions in have their truth values settled.

De nition 1 Reliable Theory Discovery

(Kelly and Glymour, 1990) and (Kelly, 1995) de ne these concepts in a rst-order logical

setting.

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RELIABLE BELIEF REVISION 3

uniformly discovers the complete truth () for each data stream "

there is a time such that for each later time m, ("jm) is consistent and

entails the complete -theory of ".

piecemeal discovers the complete truth () for each data stream "

and for each proposition P in there is a time such that for each later

time m, ("jm) entails P just in case P is true of ".

Clearly, uniform success entails piecemeal success. Also, there is no guarantee

that the successive theories pro duced by a piece-meal discovery metho d get

ever more \verisimilar". A piece-meal metho d is p ermitted to add as much new

falseho o d as it pleases at each stage, so long as more and more prop ositions

have their truth values correctly settled.

A few examples may clarify the di erence b etween the two reliability con-

cepts. Let be the set of all fans and their complements. It is trivial to

nd the complete truth in a piecemeal fashion simply by returning [e]on

input data sequence e. But no metho d can do so uniformly, since for each data

stream ", f"g is the complete -truth, and whereas there are uncountably

many such singleton theories, the range of eachdiscovery metho d is countable,

so most such theories cannot even b e conjectured.

Even when there are only countably many distinct -complete theories,

piecemeal success may b e p ossible when uniform success is not. Let the prop o-

sition H say that 0 o ccurs at stage n and forever after. Let b e the set of

n 1

all such prop ositions and their negations. requires piecemeal solutions to

p erform nontrivial inductive inferences, since eachhyp othesis H is a claim

ab out the unb ounded future. This time, there are only countably manydis-

tinct -complete theories . To succeed piecemeal, let metho d conjecture

H \ H when the last non-0 o ccurring in the data o ccurs at p osition n.

n+1 n

Nonetheless, uniform success is still imp ossible. For supp ose for reductio that

some metho d succeeds uniformly. Then a wily demon can present 0,0,0,... un-

til 's conjecture entails H , whichitmust eventually do on that data stream.

Then a 1 is fed (e.g., at stage m) followed by all 0s until 's conjecture entails

H ,whichitmust eventually do on that data stream, etc. The data stream

m+1

" so presented never stabilizes to 0, so pro duces in nitely many conjectures

inconsistent with the complete -theory of ".

The same argumentshows that no piecemeal solution could p ossibly suc-

ceed by eventually pro ducing only true hyp otheses, since the construction

forces an arbitrary piecemeal solution to pro duce in nitely many false conjec-

tures. Hence, piecemeal success is sometimes p ossible only if in nitely many

false conjectures are pro duced. On the other hand, whenever uniform success

is p ossible, it can b e achieved by a metho d whose conjectures are eventually

true. Given successful metho d , pro duce whatever says if it is not a consis-

Such a theory either says exactly when the data stream stabilizes to 0 or says that the

data stream never stabilizes to 0.

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4 KEVIN KELLY, OLIVER SCHULTE, VINCENT HENDRICKS

tent prop osition entailing some -complete theory and pro duce the (unique)

-complete theory entailed by the current conjecture of otherwise.

3. BELIEF REVISION

A b elief revision op erator + takes a theory T and a prop osition P and returns

a revised theory T . Then wewrite

T + P = T

In our setting of empirical prop ositions, the AGM axioms amount to the fol-

lowing:

(AGM 1) T + PP.

(AGM 2) If P6= ;, then T + P6= ;.

(AGM 3) If P\T 6= ;,then T + P = P\T.

(AGM 4) If P\(T + Q) 6= ;,then T +(P\Q)=P\(T + Q).

We employ an elegant representation of these requirements due to A. Grove

(1988). A Grove system for prop osition K is a collection of subsets of N

such that

1. is nested (i.e., totally ordered by ),

2. For each consistent prop osition P there is a unique, -minimal element

of whose intersection with P is nonempty.

3. N 2 .

4. K is the least memb er of (with resp ect to ).

Let denote the least element of whose intersection with P is nonempty.

This is well-de ned bythe second condition on Grove systems. Let (P )=

\P. Then wehave:

Prop osition 1 (Grove 1988) For each AGM belief revision operator + and

for each proposition K thereisaGrove system such that for each propo-

sition P

(P )=K + P :

For each proposition K ,let beaGrove system for K . Then there exists

a unique AGM belief revision operator + such that for each K ; P

(P )=K + P :

4. INDUCTION AS BELIEF REVISION

There are several ways in whichthe AGM theory of b elief revision might

interact with inductive metho dology.Anambitious approach views induction

as nothing but the revision of one's previous b eliefs in accordance with the

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RELIABLE BELIEF REVISION 5

total evidence available. On this prop osal, one's revision op erator uniquely

determines one's inductive metho d once an initial theory is sp eci ed. There

are twoways in which this might happ en. According to the rst, inquiry b egins

with a xed theory K and this theory is rep eatedly up dated on increasing data.

We will refer to this as repeated revision. Rep eated revision of K bya belief

revision op erator + uniquely determines a discovery metho d according to

the following relation:

1. (;)=K .

_ _

2. (e q )= (;)+[e q ].

Then wesay that is representedrepetitively by K and +. If is rep etitively

represented by some AGM op erator starting with K , then we say that is

repetitively AGM. A second prop osal is that it is always one's current theory

that is revised as the data comes in. We will refer to this as sequential revision.

Sequential revision starting with K uniquely determines a discovery metho d

as follows:

1. (;)=K .

_ _

2. (e q )= (e)+[e q ].

Then wesaythat is representedsequential ly by K and +. If is sequentially

represented by some AGM op erator starting with K , then we say that is

sequential ly AGM. The two concepts are equivalentforagiven op erator + if

the op erator satisifes the following principle:

_ _

K +[e q ]=(K +[e]) + [e q ]:

We refer to this as the irrelevance of earlier subdata (IES) principle. The

IES principle do es not follow from the AGM axioms, however, so rep eated

revision by a given op erator is not the same as sequential revision by that

op erator.

It follows immediately from the AGM axioms that a sequentially AGM

discovery metho d has various prop erties. First, it is data retentive, in the sense

that its current conjecture always entails the data. Second, it is consistent,in

the sense that its current conjecture is always consistent with its current data.

Third, it is stubborn in the sense that its current conjecture always entails its

preceding conjecture unless the preceding conjecture is refuted. Finally,itis

timid in the sense that it only adds the data to its previous conjecture unless

its preceding conjecture is refuted.

In fact, these prop erties characterize b oth the sequentially and the rep eti-

tively AGM discovery metho ds, so although it matters for a particular revision

op erator whether it is interpreted sequentially or rep etitively, the same meth-

o ds are representable either way.Wemaynow refer to discovery metho ds that

are data retentive, consistent, stubb orn and timid simply as AGM metho ds.

Both prop osals are develop ed in (Martin and Osherson, 1995). Sequential revision is

there referred to as `iterated' revision.

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6 KEVIN KELLY, OLIVER SCHULTE, VINCENT HENDRICKS

Prop osition 2 The fol lowing statements areequivalent:

1. is data retentive, consistent, stubborn, and timid.

2. is sequential ly AGM.

3. is repetitively AGM.

The pro of is given in the app endix.

Data retentiveness is a fairly tame requirementforan ideal logic of dis-

covery: one mayaswell rememb er the data. Consistency is as well: while we

pro duce contradictions, we couldn't p ossibly be on the path to the truth.

Stubb ornness and timidity are another matter. Consider a stubb orn scien-

tist whose only b elief is that a given coin will come up heads at least once.

Supp ose it never comes up heads. He can never retract his b elief unless he

adds other \auxilliary hyp otheses" so that his whole b elief set is eventually

refuted, a ording him an opp ortunity to retract the original false b elief. But if

this scientist is also timid, then he never gets an opp ortunitytoaddanysuch

auxilliary hyp otheses, so he is hop elessly frozen for eternity in a false b elief.

Since there is no problem nding the truth ab out this hyp othesis (conjecture

its negation until one head is observed and then conjecture it forever after),

it is clear that there are some initial b elief sets from whichnoAGM op erator

can recover, either in the rep etitive or in the sequential sense.

So for some initial b elief states, sequential and rep etitive AGM up dating

do es stand in the way of nding the truth. The question remaining is whether

some AGM metho d with the right kind of initial b elief set can succeed when-

ever success is p ossible. The preceding discussion already provides a clue: if

one always pro duces refutable conjectures, one never ends up in the awkward

situation of not getting an opp ortunity to add the right kind of auxilliary

hyp otheses. A trivial way to ensure this is to always pro duce an empirically

complete hyp othesis (i.e., a hyp othesis that singles out a unique data stream).

Prop osition 3 Suppose that the complete truth is uniformly [piecemeal]

discoverable. Then the complete truth is uniformly [piecemeal] discoverable

by an AGM method.

Pro of: Let uniformly discover the complete truth. Let (K ;e) denote the

choice of some data stream in K\[e]if K\[e] 6= ; and let (K ;e) denote an

arbitrary data stream in [e] otherwise. Then de ne

(;)=f( (;); ;)g;

_ _

(e q )= (e)if (e) \ [e q ] 6= ;;

_ _ _

(e q )=f( (e q );e q )g otherwise.

By \ideal" we mean \abstracted from all computability considerations".

On the other hand, computable inquiry can b e very severely restricted by imp osing the

consistency requirement (Kelly and Schulte, 1995a; Kelly and Schulte, 1995b).

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RELIABLE BELIEF REVISION 7

By the de nition of , is consistent and data-retentive. By the second

clause of 's de nition, is stubb orn and timid. So by prop osition 2, is

an AGM metho d. Since uniformly discovers the complete truth, wehave

that on each data stream ", there is a stage n after which always pro duces

consistent conjectures entailing the complete truth. Supp ose that stabi-

lizes to some conjecture f g along ". Then = ", so succeeds uniformly.

Supp ose then that never stabilizes to any conjecture along ".Then selects

a new conjecture after stage n. Thereafter, 's conjecture always entails the

complete truth b ecause 's conjecture do es.

Nowsuppose discovers the complete truth in the piecemeal sense. Con-

struct just as b efore. Again, if happ ens to stabilize to some conjecture on

", then succeeds uniformly.So supp ose never stabilizes to a conjecture.

For each H2, there is a stage n bywhich 's conjecture entails H if H is

true of " and entails H otherwise. So there is a stage m after n such that each

successive conjecture of also has this prop erty. 2

An AGM op erator + is said to be maxichoice just in case K + A is the

result of adding A to some minimal sup erset S of K consistent with A,inthe

sense that any prop er subset of S that includes K would not be consistent

with A. The metho d constructed in the preceding pro of can b e represented

by a maxichoice op erator, since (e q ) [ (e) is always a minimal sup erset

of (e) consistent with [e q ]. This is the strongest sense of \minimal change"

considered byGardenfors (and rejected for b eing to o strong), but even it do es

not interfere with the p ossibility of reliable inquiry (although we have seen

that the AGM axioms themselves interfere with reliability for agents starting

out with the wrong kinds of b eliefs).

On the other hand, conservativeintuitions ab out minimal b elief change are

hardly comforted by the piecemeal discovery metho d just constructed.

leaps from one complete theory to another in order to ensure that it can change

its mind when necessary without violating stubb ornness and timidity. This

raises the question whether we can say something in general ab out how strong

the conjectures of a reliable AGM metho d must b e. A few top ological concepts

are helpful. A limit point of an empirical prop osition K is a data stream along

which K is never refuted. The empirical closure of K (denoted cl(K )) is the

set of all limit points of K (i.e. the set of all data streams along which K is

never refuted). H is empirical ly decidable with certainty or empirical ly clopen

given K just in case the truth value of H is eventually decided by the data,

assuming that K is true. That is, on each data stream in K , there is a time

after which the data together with K either entail H or entail the negation of

H . H is empirical ly decidable with certainty by stage n or empirical ly n-clopen

given K just in case the truth value of H is determined by stage n,given that

K is true. Nowwehave:

(Martin and Osherson, 1995, Prop. 23) show that if the notion of logical consequence

is weakened, an initial b elief set can b e found that succeeds with anyAGM op erator in the

rep etitive sense. We do not consider weakened consequence relations here.

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8 KEVIN KELLY, OLIVER SCHULTE, VINCENT HENDRICKS

Prop osition 4 Let be a discovery method.

1. If is data-retentive, stubborn and timid and piecemeal discovers the

complete truth, then for each e, each proposition in is empirical ly

clopen given cl( (e)).

2. If is data retentive, stubborn and timid and uniformly discovers the

complete truth, then for each e,thereisan n such that each proposition

in is empirical ly n-clopen given cl ( (e)).

Pro of: (1) Supp ose otherwise. Then for some e, and for some H 2 ; H is

not clop en in cl( (e)). Hence, () there is a data stream " along which (e)is

never refuted such that at no stage do es the data read along " entail either H

or its complement. Since is data retentive, (e) [e], so for some n; e = "jn.

Hence ("jn) = (e). But since is stubb orn, timid and data retentive, for

each m n; ("jm)= (e) \ ["jm]. So by(), do es not piecemeal discover

the complete truth.

The pro of of (2) is similar. 2

The rst condition is equivalent to saying that the top ological b oundary

of each prop osition in must b e excluded from each conjecture pro duced by

. In a setting with rst-order sentences instead of prop ositions, this would

mean that eachof 's conjectures entails that each sentence in is equivalent

to a quanti er-free sentence (Kelly, 1995, Thm. 12.7). This is a striking re-

quirement, but in the prop ositional setting it can always b e accomplished with

conjectures of arbitrarily high probability if probability is countably additive.

5. RELIABILITY AS A REVISION THEORETIC AXIOM

AGM belief revision theory makes no reference to reliability. What would

happ en if wewere to add as an AGM axiom that each op erator b e a reliable

solution to a sp eci ed, solvable inductive inference problem ? It would be

nice to have an exact representation, analogous to Grove's, of the class of all

AGM op erators that are reliable solutions to . Wehavenotyet found sucha

representation for the notions of reliability de ned ab ove. But it is much easier

to obtain such a result if we mo dify our notions of reliability so as to require in

addition, that on each data stream there is a time after which all the theories

pro duced by are true (recall that uniform and piecemeal discovery are b oth

p ossible even when no conjecture is true). Then we sp eak, resp ectively,of truly

discovering the complete truth, uniformly or in a piecemeal fashion.

Recall from section 2 that the complete truth is truly uniformly dis-

coverable just in case it is uniformly discoverable. On the other hand, true

piecemeal discovery is prop erly easier than uniform discovery and piecemeal

discovery is prop erly easier than true, piecemeal discovery.

Cf. (Kelly, 1995), prop osition 13.15.

The rst noninclusion is witnessed by the example provided at the b eginning of the

pap er. Wehave already seen that this inductive problem is not uniformly solvable. But the

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RELIABLE BELIEF REVISION 9

Two more concepts are necessary in order to represent the \truly" reliable

AGM metho ds. K is empirical ly closed just in case K contains all its limit

points (i.e. just in case K is identical to its own empirical closure). Then K

is guaranteed to b e refuted eventually by the data if it is false, for otherwise

there would b e a data stream making K false along which K is never refuted

(i.e., K is missing one of its limit points). Let be aGrove system and let

S 2 . Let core(S ) denote the union of all R 2 such that R is a prop er

subset of S .Thenwehave:

Prop osition 5 Truly Reliable AGM Learners

1. is a repetitively AGM method that truly, piecemeal discovers the com-

plete truth () thereisa Grove system such that

(a) 8e; ([e]) = (e)

(b) 8S 2 ; core(S ) is empirical ly closed, and

2. is a repetitively AGM method that truly, uniformly discovers the com-

plete truth () thereisa Grove system such that

(a) 8e; ([e]) = (e)

(b) 8S 2 ; core(S ) is empirical ly closed, and

Pro of of (1): (=)) supp ose is a rep etitively AGM metho d. Then bypropo-

sition 1, there is a Grove system such that 8e; ([e]) = (e), so wehave1a.

1c follows from 1a and prop osition 4. Finally, supp ose for reductio that 1b

is false. Then let S 2 be such that some " 2= core(S ) is a limit point of

core(S ). But then for each n, core(S ), so " 62 (["jn]) = (["jn]).

["jn]

Hence, pro duces in nitely many false conjectures along " and so fails to

truly piecemeal identify the complete truth.

((=) Supp ose there is a Grove system satisfying 1a{1c. Let " b e given.

Since is aGrove system, there is a least element S 2 suchthat " 2S.

So " 2= core(S ). There is an n such that ["jn] \ core(S ) = ;, else " is a

missing limit point of core(S ), contrary to 1b. But since " 2 S , for each

m n; (["jm]) = (["jn]) \ ["jm]. By 1c, wehave that for each H2; 9k

n:8j k; (["jn]) \ ["jk ] either entails H if " 2 H or entails H otherwise.

By 1a, truly piecemeal discovers the complete truth.

The pro of of (2) is similar. 2

The requirement that the metho d stabilize to the truth undercuts the piece-

meal strategy of succeeding byalways pro ducing a complete, false theory.We

trivial metho d that always rep eats the current data succeeds truly in the nonuniform sense.

The second noninclusion is witnessed by the example also presented at the b eginning of

the pap er. It was shown ab ove that this problem has a piecemeal solution. But the demonic

argument against uniform solutions forces an arbitrary metho d to pro duce in nitely many

false conjectures, so no true piecemeal solution is p ossible.

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10 KEVIN KELLY, OLIVER SCHULTE, VINCENT HENDRICKS

leave op en the interesting question whether AGM metho ds can solve every

problem solvable in the truly piecemeal sense.

6. ARBITRARY REVISIONS

So far, wehave examined an ambitious interpretation of b elief revision theory,

in which one's b elief revision op erator uniquely determines one's inductive

inferences. It has b een shown that even on this interpretation, the AGM axioms

do not restrict the reliability of inquiry, at last for ideal metho ds that needn't

worry ab out computational limitations. On the other hand, AGM metho ds

have some awkward prop erties such as timidity and stubb ornness, and these

requirements must b e \steered around" to arrive at the truth.

But there is a more temp ered interpretation of belief revision theory,ac-

cording to which a revision op erator is just a way of consistently forcing into

one's b eliefs whatever one decides to force into them for whatever reason. Say

that a metho d is rep etitively AGM with arbitrary revisions just in case it al-

ways up dates its initial b eliefs on the conjunction of the total data and some

arbitrarily selected prop osition. The de nition of sequentially AGM metho ds

with arbitrary revisions is similar, except that it is the initial b elief set that

is revised.

De nition 2 AGM Learners With Arbitrary Revisions

1. is repetitively AGM with arbitrary revisions () there is an AGM

_ _

revision operator + such that for al l e q thereisanA such that (e q )=

(;)+([e q ] \A).

2. is sequential ly AGM with arbitrary revisions () there is an AGM

_ _

revision operator + such that for al l e q thereisanA such that (e q )=

(e)+([e q ] \A).

Neither concept implies stubb ornness or timidity, since A can be chosen to

b e inconsistentwith (;) (or resp ectively, with (e)). Data retentiveness still

follows.Sodoweakened versions of stubb ornness and consistency. Stubb orn-

ness obliges to moveforward until contradicted by the data. An `internal'

version of stubb ornness requires to moveforward until contradicted either

by the data or by 's new b eliefs. Accordingly, is internal ly stubborn in

the repetitive [sequential] sense just in case 's current b eliefs either entail its

initial [current] b eliefs or are inconsistent with them. Finally, say that is

quasi-consistent just in case 's b eliefs are never b oth inconsistent with the

data and consistent. Then wehave:

Prop osition 6 Let be a discovery method. Then

1. is repetitively [sequential ly] AGM with arbitrary revisions () is

data retentive and internal ly stubborn in the repetitive [sequential] sense.

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RELIABLE BELIEF REVISION 11

2. This remains true if we add the condition that is quasi-consistent.

Pro of of 1, sequential case: (=)) Let be sequentially AGM with arbi-

trary revisions. Let + be the AGM op erator that witnesses this fact. Since

is de ned in terms of an AGM op erator revising on at least the data, is

data retentive. Now supp ose that (e q ) is consistent with (e). Since is

sequentially AGM with arbitrary revisions, there is an A suchthat (e q )=

_ _ _

(e)+([e q ] \A). By axiom (AGM 1), (e)+([e q ] \A) ([e q ] \A). Hence,

_ _

([e q ] \A) is consistent with (e). So by axiom (AGM 3), (e)+([e q ] \A)=

_ _

(e) \ ([e q ] \A) (e). Hence, (e q ) (e).

((=) Let b e data retentiveandinternally stubb orn. For each prop osition

A let Grove system = fN ; Ag. Let + b e the op erator represented bythe

collection of all such according to prop osition 1. Then wehave

()if S\R= ; then S + R = R:

_ _

By data retentiveness, (e q ) [e q ], so

_ _ _

() (e q ) \ [e q ]= (e q ):

By internal stubb ornness in the sequential sense, there are two cases. Case

_ _ _

1: (e) \ (e q )=;. Then by() and (), (e)+([e q ] \ (e q )) = (e)+

_ _

(e q )= (e q ).

Case 2: (e q ) (e). Then by(), (AGM 3), and the case hyp othesis, we

_ _ _ _ _

have that (e)+([e q ] \ (e q )) = (e)+ (e q )= (e) \ (e q )= (e q ).

Pro of of 1, rep etitive case: The (=)) side is just as b efore, with (;) in

place of (e). The ((=) side is also just as b efore, except that we employ

Grove system = f (;);Ng, and cho ose + to agree with when (;) is

b eing up dated. The rest of the argument is as b efore, with (;) replacing (e)

everywhere.

Proofof2:(=))Supposethat is sequentially AGM with arbitrary revi-

_ _ _

sions. Supp ose (e q )= (e)+([e q ] \A) 6= ;.By(AGM 1), (e)+([e q ] \

_ _ _ _ _

A) ([e q ] \A), so (e q ) \ ([e q ] \A) 6= ; and hence (e q ) \ [e q ] 6= ;.

The rep etitive case is the same with (;) in place of (e).

((=) This side was already shown without app eal to quasi-consistency. 2

In the pro of, we showed that the extra set A chosen to revise on can always

be (e q ) itself. We also employed a trivial revision op erator + (i.e. the

op erator generated by full meet according to the Levi identity). Strengthening

+ would endanger condition () in the pro of, p ossibly leading to a metho d

whose conjectures are sometimes stronger than the corresp onding conjectures

of .

It is not intuitively clear to us why violations of internal stubb ornness

should be denounced as irrational, but it is at least straightforward (in the

absence of computability considerations) to bring an arbitrary metho d into

compliance with this principle without a ecting its reliabilityordelaying its

garden.tex - Date: October 30, 1995 Time: 19:08

12 KEVIN KELLY, OLIVER SCHULTE, VINCENT HENDRICKS

time of convergence. For supp ose that metho d uniformly identi es the com-

plete truth, p ossibly violating internal stubb ornness. Let agree with on

data ;. On data sequence e q , agrees with if this do es not lead to a vio-

_ _

lation of internal stubb ornness, and (e q )= (e q ) (e) otherwise. Then

is internally stubb orn in the sequential sense and uniformly identi es the

complete truth. The same mo di cation works for a theory learner that

piecemeal identi es the complete truth.

7. MINIMAL REVISIONISM

Thereisaneven weaker way for inductive metho dology to interact with b elief

revision theory. According to this approach, revision theory is nothing but

an explication of what it means to add a prop osition to a theory that might

b e inconsistent with it, and an inductive metho d is allowed to retract on or

to add whatever it pleases for whatever reason. Accordingly, is sequentially

AGM in the minimal sense just in case there is some b elief revision op erator +

_ _

such that for each e q there are A and B such that (e q ) is the result of rst

contracting (e)by A and then revising the result by B . The rep etitiveversion

of this concept is de ned by replacing (e) with (;). On either interpretation,

b elief revision theory is entirely vacuous as a constraintondiscovery metho ds.

Prop osition 7 Every method is a sequential ly [repetitively] AGM method

in the minimal sense.

Pro of: We b egin with the sequential case. Let b e given. Let + b e the op erator

represented by the trivial Grove systems de ned in the pro of of prop osition

6. Let b e de ned according to the Harp er identityas

KA = K[(K + A):

K )=K[K = N (the second to last identityisby Then KK = K[(K +

() in the pro of of prop osition 6.) Moreover, for each A; N + A = A. Hence,

_ _

(e q )=( (e) (e)) + (e q ). The rep etitive case is similar. 2

8. CONSERVATISM, RELIABILITY, AND LOGIC

We b egan with a fundamental distinction b etween conservative and reliabilist

approaches to the problem of inductive inference. According to the former, our

b eliefs should b e repaired in the most elegant p ossible manner when they are

revised. According to the latter, the aim is to nd the right answer whatever

that answer might b e. Belief revision theory b elongs to the former p ersp ective

and formal learning theory b elongs to the latter. In this pap er, wehaveex-

amined various di erentways in which b elief revision theory could b e viewed

In our prop ositional setting, the intersection of the deductive closures of two prop osi-

tions X ; Y is just the union of all prop ositions including b oth X and Y .

garden.tex - Date: October 30, 1995 Time: 19:08

RELIABLE BELIEF REVISION 13

as as constraint on empirical inquiry, ranging from the view that inquiry is

nothing but revision on the data to the prop osal that revision is nothing but

an explication of what it means to force a prop osition into a system of b eliefs,

leaving inquiry to decide what to add or subtract. The result of our inves-

tigation is that the weaker interpretations imp ose few short-run restrictions

on inquiry, whereas the stronger ones imply obstacles that must b e carefully

circumvented if inquiry is to b e reliable.

This study illustrates twoways in which logic can b e broughttobearonthe

problem of understanding inductive inquiry.Onestyle of philosophical logic

prop oses intuitive normative principles and tests and re nes these principles

in a quasi-empirical way, by deducing particular consequences to compare

against one's intuitions, by providing alternative representations of the prin-

ciples, and byproving soundness and completeness theorems for the principles

according to some illuminating semantic interpretation. The extensivework

on b elief revision theory ts into this approach. In this pap er we have at-

tempted to illustrate how the pertinent logical questions change when one

shifts one's fo cus from coherence to reliability. Reliabilist applications of logic

are more akin to the theory of computabilitythantousualwork in philosoph-

ical logic. Whereas philosophical logicians tend to analyze systems re ecting

norms derived from direct intuition or from particular case examples, the logic

of reliability concerns the intrinsic solvability of inductive problems. The log-

ical to ols employed are programming, de nability, and diagonalization rather

than semantic interpretations, completeness and representations.

We hop e to have illustrated howthesetwo approaches can usefully inter-

act. The logical analysis of intuitive axioms of rationality can motivate lo cal

side-constraints on inductive metho ds (e.g., internal stubb ornness and timid-

ity) that can enrich reliability analysis. On the other side, wehaveshown

in a preliminary way how reliability can be imp osed as an extra axiom in

a normative theory of b elief revision, leading to enriched representations of

acceptable inductive practice, as in the representation of truly reliable AGM

metho ds byGrove systems of a sp ecial kind.

We hop e that the range of questions raised by this preliminary study un-

derscores the fruitfulness of reliability considerations in the study of belief

revision. Is there an elegant representation of the reliable AGM metho ds that

may pro duce in nitely many false conjectures? Are all problems solvable with

nitely many false conjectures solvable by AGM metho ds? What happ ens to

our results when more stringent constraints are imp osed on revision than just

the AGM axioms? What happ ens when we allow facts ab out the data stream

to arrive in an arbitrary order? And p erhaps most imp ortantly,how restric-

tivearethe AGM principles for computable inquiry? Our non-restrictiveness

results in this pap er make use of highly idealized constructions in whichun-

It is interesting how naturally the Grove systems could b e \programmed" as inductive

metho ds.

Cf. (Martin and Osherson, 1995).

garden.tex - Date: October 30, 1995 Time: 19:08

14 KEVIN KELLY, OLIVER SCHULTE, VINCENT HENDRICKS

computable logical relations must b e decided. In light of the strong negative

results concerning consistency in similar settings (Kelly and Schulte, 1995a;

Kelly and Schulte, 1995b), one would exp ect the AGM principles to b e restric-

tive for computable inquiry for that reason alone.

APPENDIX: PROOF OF PROPOSITION 2

Prop osition 2 The fol lowing statements areequivalent:

1. is data retentive, consistent, stubborn, and timid.

2. is sequential ly AGM.

3. is repetitively AGM.

Pro of: To see that (3) implies (1), let b e rep etitively AGM. Then for some

AGM op erator +, wehave that for each e; (e)= (;)+[e]: is data retentive

by(AGM 1) and is consistentby(AGM 2). Now supp ose that (e) \ [e q ] 6= ;.

_ _ _

Then by axiom (AGM 4), (e q ) = (;)+[e q ] = (;)+([e] \ [e q ]) =

_ _

( (;)+[e]) \ [e q ] = (e) \ [e q ]. So is stubb orn and timid. An even

simpler argument establishes that (2) implies (1).

To see that (1) implies (3), let b e data retentive, consistent, etc. Toshow

that is also rep etitively AGM,onemust show not only that behaves likean

AGM op erator rep eatedly up dating K , but that 's domain can b e extended

to all empirical prop ositions in a manner that satis es the AGM axioms. It is

useful to employ the Grove representation for this purp ose.

We will construct a single Grove system for (;) that b ehaves just like ,

from whichitfollows by prop osition 1 that is rep etitively AGM. De ne

S = (;);

S = S [ (e);

n+1 n

lh(e)=n+1

S = N ;

=fS : ! g.

It is easy to verify that is a Grove system for (;).

Next, we showby induction on the length of e that ([e]) = (e). It then

follows immediately by prop osition 1 that is rep etitively AGM. In the base

case, wehave: ([;]) = (N )=S = (;).

For the inductive case, supp ose ([e]) = (e). Let q be given.

_ _

Case 1: supp ose that [e q ] \ ([e]) 6= ;. Then = ,so([e q ]) =

[e q ] [e]

_ _

([e]) \ [e q ]= (e) \ [e q ]by the inductivehyp othesis. But since is b oth

_ _

timid and stubb orn, the case hyp othesis yields that (e q ) = (e) \ [e q ],

_ _

and hence (e q ) = ([e q ]).

Case 2: [e q ] \ ([e]) = ;. Let lh(e)=n. First we establish that

() = S :

n+1

[e q ]

garden.tex - Date: October 30, 1995 Time: 19:08

RELIABLE BELIEF REVISION 15

_ _ _

Since is consistent, (e q ) 6= ;. Since is data retentive, (e q ) [e q ].

_ _

But (e q ) S ,by de nition of S . Hence, [e q ] \S 6= ;: Supp ose

n+1 n+1 n+1

for reductio that for some i n there is some " 2 S \ [e q ]. Let k be

the least such i. By choice of k and the de nition of S ; there is an e of

0 0

length k suchthat " 2 (e ). Since is data retentive, " 2 [e ]. So ["jk ]; ["jk +

1];:::["jn]; ["jn +1] are all consistent with ("jk ). Since is b oth stubb orn

and timid, (e) = ("jn) = ("jk ) \ ["jn]. Hence, ["jn +1]\ ("jn) 6= ;.

But "jn = e; "jn +1 = e q and by induction hyp othesis, (e) = ([e]), so

[e q ] \ ([e]) 6= ;,contrary to the case hyp othesis.

_ _

Thus wehave(). Hence, ([e q ]) = (S S ) \ [e q ]. But by de nition,

n+1 n

_ _

(S S )= (e). Since is data retentive, (e q ) [e q ]. Let

n+1 n

lh(e)=n+1

00 _ 00 _ 14

e 6= e q b e of length n +1. Then [e ] \ [e q ]=;: Since is data retentive,

00 _ _ _

(e ) \ [e q ]=;: Hence, S \ [e q ]= (e q ).

n+1

Now we argue from (1) to (2). Let be data retentive, consistent, etc.

Let prop osition K be given. If K is not in the range of , then let be an

arbitrary Grove system. If K is in the range of then we de ne as follows.

First, de ne:

D = f;g;

n+1

D = fe q : lh(e)=n and (e)=Kg.

Now de ne

= K ; S

n+1

[ (e); S = S

e2D

S = N ;

: ! g. = fS

Since the (e)'s added at each stage are mutually disjoint, one can showjust

as in the preceding argument that

(;)

(;)= ([;])

and

_ (e) _

(e q )= ([e q ]):

REFERENCES

Gardenfors, P. (1988). Know ledge In Flux: modeling the dynamics of epistemic states.

Cambridge: MIT Press.

14 0

Our argument turns heavily on the fact that two distinct data sequences e; e of the

same length generate disjoint prop ositions [e]; [e ]. (Martin and Osherson, 1995, Prop. 21)

contains a construction of a reliable rep etitive AGM metho d that do es not require this

assumption, employing a well-ordering of theories.

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16 KEVIN KELLY, OLIVER SCHULTE, VINCENT HENDRICKS

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Kelly, K. (1995). The Logic of Reliable Inquiry. Oxford: Oxford University Press.

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Journal of Philosophical Logic 19: 1{33.

Kelly, K. and Schulte, O. (1995a) \The Computable Testability of Theories Making Un-

computable Predictions,"Erkenntnis. 43:29{66.

Kelly, K. and Schulte, O. (1995b) \Church's Thesis and Hume's Problem". These proceed-

ings.

Martin, E. and Osherson, D. (1995). \Scienti c discovery based on b elief revision". These

proceedings.

Osherson, D., Stob, M. and Weinstein, S (1986). Systems That Learn. Cambridge, Mass:

MIT Press.

Popp er, K. (1968). The Logic Of Scienti c Discovery.NewYork: Harp er.

Putnam, H. (1963). \ `Degree of Con rmation' and Inductive Logic," In The Philosophy of

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Kevin Kel ly and Oliver Schulte

Department of Philosophy

Carnegie Mel lon University

Pittsburgh, PA 15213

USA

E-mail: kevin.kel [email protected], [email protected]

Vincent Hendricks

Department of Philosophy

University of Copenhagen

Email: [email protected]

garden.tex - Date: October 30, 1995 Time: 19:08