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- 1 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 2 to side with reliability rather than with the principles. 1 For general intro ductions, cf. (Osherson et al., 1986) and (Kelly, 1995). 2 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 3 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 3 (Kelly and Glymour, 1990) and (Kelly, 1995) de ne these concepts in a rst-order logical setting. garden.tex - Date: October 30, 1995 Time: 19:08 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 0 nd the complete truth in a piecemeal fashion simply by returning [e]on 0 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 0 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 1 p erform nontrivial inductive inferences, since eachhyp othesis H is a claim n ab out the unb ounded future. This time, there are only countably manydis- 4 tinct -complete theories . To succeed piecemeal, let metho d conjecture 1 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. 0 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 ". 1 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- 4 Such a theory either says exactly when the data stream stabilizes to 0 or says that the data stream never stabilizes to 0. garden.tex - Date: October 30, 1995 Time: 19:08 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 0 a revised theory T . Then wewrite 0 T + P = T In our setting of empirical prop ositions, the AGM axioms amount to the fol- lowing: (AGM 1) T + PP.