Logique & Analyse 158 (1997), 215-217 i.e.: ON AN ERROR IN GROVE'S PROOF Graham PRIEST and Koji TANAKA Nearly a decade has past since Grove [1988] gave a semantics for the AGM postulates. The semantics, called sphere semantics, provided a new perspective of the area of study, and has been widely used in the context of theory or belief change. However, the soundness proof that Grove gives in his paper contains an error. In this note, we will point this out and give two ways of repairing it. We follow Grove in matters of notation. To make this paper self-contained, we start by rehearsing some. Let L be a propositional language. Let M L b e t h e s e t o f a l l m a x i m a l l y c o n s i s t e n t s e t s o f s e n t e n c e s o f L I I f A i s a n y sentence, 1,41 is the set of all members of M L c o n t a i n i n g A . I f X C M L , W O i s the theory FIX. I f i s a set of sentences, C n ( ) is the set of logical consequences of / . I f T is a theory in L, c(A) is a certain subset of M L - i n t u i -tively, the smallest "sphere" containing all extensions of T, some of which contain A- and T + A is defined as t ( I A M c(A)). For future reference, T I A, the expansion of T by A, is Cn(T { A } ).We can now state the problematic step in Grove's proof. On p. 161, in verifying the postulate +7, two successive lines of the proof are: 1,41 I B 1 n c(A) C 1 ,4 1 I B I c ( A A B) IBI n IT + Al C ICn ((A ,B ))1 n c(A A B ) The left-hand side of this step relies on the fact that: IT+ Al C I A I n c(A) It(1,41 n c(A))I C I A I n c(A) I Alternatively, M L c a n b e t a k e n a s s e t o f a l l m o d e l s o f t h e l a n g u a g e , b u t w e f o l l o w G r o v e here. 216 G R A H A M PRIEST AND KOJI TANAKA But in general It(X)1 X . Let X be the class of all maximal consistent sets minus any one of them. t(X) is the set of all tautologies, since for any nontautology there is more than one maximal consistent set which does not contain it. Hence It(X)I = M L . A s i m i l a r p r o b l e m b e s e t s t h e v e r i f i c a t i o n o f +8. The proble can be repaired as fo llows. 2 W e d o n o t r e p l a c e I A I n c ( A ) by IT + AI at this stage of the proof, but simply carry it along. The proof then becomes: 3 1/11 n IBI n c(A) IAI n IBI n c(A) t(IA A BI n ( ( A A B ) ) T-FA A B C I A I n IBI n c(A A B) C I A A BI n c(A A B) C t ( I B I n IAI n c(A)) t(IAI n c(A)) I B By definition, t(IAI n c(A)) , T + A . H e n c e t ( l A i n c ( A ) ) 1 B , T F A I B , a s required. Now this new proof requires the function t to satisfy the condition that t(S) / A = t(S n IA1) for any S C M L . G r o v e s t a t e s t h i s a s a p r o p e r t y o f t b u t does not prove 4 . 4 A p r o o f i s i l l u m i n a t i n g , a n d g o e s a s f o l l o w s . 5 N o t e t h a t A - ) B is the material conditional A y B. Proof Right-to-left: Suppose that for some sentence B, B E t(S n IA I). Then for all x E S f l A I , B E x . H e n c e , A - 3 B E x . M o r e o v e r , s i n c e - A entails A B , A -4 B E x for all x E I AI. So A -) B E x' for all x' E S n (IAI U I AI) = S. Hence A -> B E t(S). Thus B E Cn(S = t(S) / (by detachment for the material conditional). Since B is an arbitrary sentence, t(S n IA1) C t(S) / A. Left-to-right: Suppose that for some sentence B, B E t(S) I A. Then B E Cn(t(S) U {A } ). S o A B E t ( S ) . H e n c e f o r a l l x E S , A B E x . N o w for any x' E S MAI , A - )B E . x - ' . A l s o A H e n c e B E x ' b y m a t e r i a l detachment. Thus B E t(S n IA I). QED. This proof is perfectly correct in classical logic, but it may fail in a nonclassical logic, particularly one where material detachment, the disjunctive syllogism, fails, as it does in standard relevant and paraconsistent logics. In 2 As was pointed out by Grove in correspondence. 3 T h e p r o o f a p p e a l s t o t h e f o l l o w i n g p r o p e r t y o f t : F o r S , S ' C M „ i f S C S ' t h e n t ( S ' ) C t(S). See Grove [1988], p. 158. 4 ( 3 ) , p . 1 5 8 . 5 T h i s i s t h e h e a r t o f t h e p r o o f o f p o s t u l a t e + 7 t h a t i s g i v e n i n P r i e s t , S u r e n d o n k a n d Tanaka [19961. A s imilar proof was given by Grove in correspondence. ON AN ERROR IN GROVE'S PROOF 2 1 7 other words, the argument will fail if one generalises Grove's semantics to non-classical logics, as is entirely possible. 6 Is there any way that the proof may be repaired more generally? The answer is yes. An elementary class is one of the form IB1 for some sentence B. It is well known that if X is an elementary class then It(X)1. X. Moreover, the proof of this fact is a quite general, model-theoretic, one, and has nothing to do with classical logic. 7 H e n c e , i f w e r e q u i r e t h a t e v e r y s p h e r e i n M L be an lementary class, Grove's original argument goes through. 8 This constraint means, in effect, that every sphere represents a theory that is finitely axiomatisable. This is not an implausible constraint if one takes it that spheres represent theories that are accessible to the belief-reviser in question, and that the reviser is a finite agent. In this way, Grove's construction may be liberated from the vicissitudes of classical logic. BIBLIOGRAPHY [1] J.L. Bell and A. Slomson [1969], Models and Ultraproducts: an Introduction, North-Holland. [2] A . Grove [1988], 'Two Modeflings for Theory Change', Journal o f Philosophical Logic 17,157-70. [3] G . Priest, T.I. Surendonk and K. Tanaka [1996], 'An Error in Grove's Proof', Technical Report: TR-ARP-07-96, Automated Reasoning Project, Australian National University. [4] K . Tanaka [1995], 'Paraconsistent Belief Revision', Honours Thesis, Department of Philosophy, University of Queensland. 6 S e e T a n a k a [ 1 9 9 5 1 . 7 S c e , e . g . , B e l l a n d S l o m s o n [ 1 9 6 9 1 , p . 1 4 1 . 8 T h i s i s t h e a p p r o a c h t a k e n i n T a n a k a 1 1 9 9 5 1 Department of Philosophy, University of Queensland St. Lucia QLD 4072, Australia ggp@cltr.uq.edu.au bluesky @cltr.uq.edu.au