Nearly a decade has past since Grove 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

Priest, Graham

Tanaka, Koji

English

PhilPapers

Logique & Analyse 158 (1997), 215-217i.e.:ON AN ERROR IN GROVE'S PROOFGraham PRIEST and Koji TANAKANearly a decade has past since Grove [1988] gave a semantics for the AGMpostulates. The semantics, called sphere semantics, provided a  newperspective of the area of study, and has been widely used in the context oftheory or belief change. However, the soundness proof that Grove gives inhis paper contains an error. In this note, we will point this out and give twoways of repairing it.We follow Grove in matters of notation. To make this paper self-con-tained, we start by rehearsing some. Let L be a propositional language. LetML  be  the  set  of  all  maximally  consistent  sets  of  sentences  of  LI  If  A  is  anysentence, 1,41 is the set of all members of ML  c o n t a i n i n g  A .  I f  X  C  ML,  W O  i sthe theory FIX. I f  i s  a set of sentences, C n ( )  is the set of logical con-sequences of / .  I f  T is a theory in L, c(A) is a certain subset of ML  — i n t u i -tively, the smallest "sphere" containing all extensions of T, some of whichcontain 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, inverifying 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 AB )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, ML  c a n  b e  t a k e n  a s  s e t  o f  a ll  m od e ls  of  the  l a ng ua ge ,  but  we  follow  Grovehere.216 G R A H A M  PRIEST AND KOJI TANAKABut in general It(X)1 X .  Let X be the class of all maximal consistent setsminus any one of them. t(X) is the set of all tautologies, since for any non-tautology there is more than one maximal consistent set which does notcontain it. Hence It(X)I = ML.  A  s i m i l a r  p r o b l e m  b e s e ts  t h e  v e r i f ic a t i o n  of+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 proofthen becomes:31/11 n  IBI n  c(A)IAI n  IBI n  c(A)t(IA A BI n  (( A  A  B ) )T-FA A BC 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 BBy 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 ,  asrequired.Now this new proof requires the function t  to satisfy the condition thatt(S) / A = t(S n  IA1) for any S C ML.  G r o v e  s t a t e s  t h i s  a s  a  p r o p e r ty  o f  t  b u tdoes not prove 4 .4  A  p r o o f  i s  i l l u m in a t i n g,  a nd  g oe s  as  f o ll o ws .5  Note  thatA - )  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 eo v e r,  s in ce  -  Aentails A B ,  A -4 B E x for all x E  I -  AI. So A -) B  E x' for all x' E Sn (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 sen-tence, 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 ECn(t(S) U {A  }).  S o  A  B  E  t( S ).  H en ce  for  all  x  E  S,  A  B  Ex.  Nowfor 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 ri a ldetachment. Thus B E t(S n  IA I). QED.This proof is perfectly correct in classical logic, but it may fail in a non-classical logic, particularly one where material detachment, the disjunctivesyllogism, fails, as it does in standard relevant and paraconsistent logics. In2 As was pointed out by Grove in correspondence.3The  proof  appeals  to  the  following  property  of  t:  For  S,  S'  C  M„  if  S  C  S'  then  t(S')  Ct(S). See Grove [1988], p. 158.4(3),  p.  158.5This  is  the  heart  of  the  proof  of  postulate  +7  that  is  given  in  Priest,  Surendonk  andTanaka [19961. A s imilar proof was given by Grove in correspondence.ON AN ERROR IN GROVE'S PROOF 2 1 7other words, the argument will fail if  one generalises Grove's semantics tonon-classical logics, as is entirely possible.6 Is there any way that the proof may be repaired more generally? The an-swer 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 noth-ing 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 ry  s p h er e  i n  MLbe an lementary class, Grove's original argument goes through.8 This constraint means, in effect, that every sphere represents a theory thatis finitely axiomatisable. This is not an implausible constraint if  one takes itthat spheres represent theories that are accessible to the belief-reviser inquestion, and that the reviser is a finite agent. In this way, Grove's con-struction may be liberated from the vicissitudes of classical logic.BIBLIOGRAPHY[1] J.L. Bell and A. Slomson [1969], Models and Ultraproducts: an Intro-duction, North-Holland.[2] A .  Grove [1988], 'Two  Modeflings for Theory Change', Journal o fPhilosophical Logic 17,157-70.[3] G .  Priest, T.I. Surendonk and K. Tanaka [1996], 'An Error in Grove'sProof', Technical Report: TR-ARP-07-96, Automated Reasoning Proj-ect, Australian National University.[4] K .  Tanaka [1995], 'Paraconsistent Belief Revision', Honours Thesis,Department of Philosophy, University of Queensland.6See  Tanaka  [19951.7Sce,  e.g.,  Bell  and  Slomson  [19691,  p.  141.8This  is  the  approach  taken  in  Tanaka  119951Department of Philosophy, University of QueenslandSt. Lucia QLD 4072, Australiaggp@cltr.uq.edu.aubluesky @cltr.uq.edu.au

On An Error In Grove's Proof

https://philpapers.org/archive/TANOAE-2.pdf

On An Error In Grove's Proof

Abstract

Similar works

Full text

Available Versions

PhilPapers