DAVID

MAKI~SON

The Giirdenfors Impossibility Theorem in Non-Monotonic Contexts

Abstract. Gfirdenfors' impossibility theorem draws attention to certain formal difficulties in defining a conditional connective from a notion of theory revision, via the Ramsey test. We show that these difficulties are not avoided by taking the background inference operation to be non-monotonic.

Background G/irdenfors' impossibility theorem, presented in I-3] and 1,5], shows that it is not possible to define a conditional connective > between propositions, via.the Ramsey test, from the notion of theory revision, if certain natural properties are assumed of the latter, and the language itself is assumed to be in a certain sense non-trivial. Attempts to "escape" this theorem have taken three main directions: (1) weaken the background assumptions on the operation q- of theory revision, (2) weaken the Ramsey test, (3) weaken the expressive range of the theories to which the test is applied. (1) G/irdenfors 1,5] Sections 7.2 and 7.6 (see especially page 162) tentatively suggests a quite substantial weakening of the usual postulates on the operation q- of revision. As far as is presently known, his suggestion may indeed suffice to avoid the impossibility theorem - provided, as pointed out by Arl6 Costa 1,2], the postulate (K*5) of "success" for q- is understood as not applying to the limiting case of the inconsistent theory A = S. As will be seen in what follows, the limiting case A = S also plays a special role in non-monotonic contexts. (2) Rott 19] suggested certain weakenings of the body "x > y e A iff y ~ A q-x" of the Ramsey test. However, it has been shown by G/irdenfors 1,4] and 1,5-1 Section 7.5 that the particular proposals of Rott in this direction do not provide "an escape route. (3) Levi 1,6] has suggested that the theories to which the Ramsey test is applied should not contain in their language the conditional connective > that the test seeks to characterize (nor any related modal notions). The body of the Ramsey test would then take the weakened form "x > y is 'associated with' A iffy~Aq- x", where A is a theory whose language does not contain >, and where x, y are propositions that do not contain >. This strategy, which is recommended in [6-1 on philosophical grounds, appears to succeed in avoiding the impossibility theorem. .

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It should be noted however, that another less radical "linguistic" strategy fails. It might be thought that the impossibility theorem could be avoided by merely disallowing iterations of the connective >, i.e. restricting the application of > to propositions not already containing >. On this approach, the body of the Ramsey test would once again be expressed with elementhood on bothsides, as "x > y e A iff y~A4-x", but with the proviso that A is a theory whose language does not contain iterated > and neither of x, y contains >. Notwithstanding the attraction of being "principled" in a philosophically interesting way, yet less draconian than that of Levi, this approach fails. An examination of the very economical proof of G/irdenfors' theorem given by Rott [10] Section 2, for example, reveals that it continues to go through under such a restricted applicability of >. Essentially this point was also mode, rather discursively, by Levi [6]. It will also be reflected in our own treatment of the non-monotonic situation. Recently it has been suggested, in the informal oral network, that one source of the problem may lie in using the wrong notion of logical closure or inference in the various conditions governing theory revision and in the very concept of a theory itself. Theories, in the context of the Ramsey test, have always been taken to .be sets of propositions closed under Cn, where Cn is some consequence operation in the sense of Tarski that includes at least classical propositional consequence Cno. Being a consequence operation it is monotonic. The suggestion under consideration is that we should work instead with some appropriate non-monotonic inference operation C >t Cno. Purpose

Our purpose in this paper is to show that non-monotonicity of C is not a viable escape route. Even non-monotonic C, provided it satisfies some weaker and independently motivated conditions, gives rise to the impossibility theorem. As in the monotonic context, it continues to do so even in the case of non-iterable >, and it does so in a particularly acute way in the limiting case

A=S. Conditions on C

Let Cno be classical propositional consequence on a language S. Let C: 2 s ~ 2 s be any "inference" operation, taking sets of propositions to sets of PrOPositions, that satisfies the following conditions - which are known not to imply monotony: (C1) (C2) (C3)

Cno(A)~_ C(A) for all A C(A)= CC(A) for all A If xEC(Au{y}) and xeC(Aw{-qy}) then xeC(A).

Conditions (C1) to (C3) hold, for example, of all "classical preferential entailments" in the sense of the author's [7], building on ideas of Shoham [12];

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3

for verifications see [7]. Of course, given the negative character of the notion of "non-monotonicity", as something that is not monotonic, there will be plenty of ill-behaved operators that don't satisfy the conditions. More to the point is the fact that the inference operations C associated with Reiter's version [8] of default logic do not usually satisfy condition (C3) - a failure that is however generally regarded as unfortunate. Note for future use that conditions (C1), (C2) together immediately imply the following conditions: (C4) (C5)

If x~C(A) and yeC(A) then x ^ y ~ C ( A ) , If x~C(A) and yzCno(X) then yr

Conditions on Theory Change Let - and 4- be operations of theory contraction and revision respectively. We do not require - to satisfy aU of the "basic postulates" ( - 1) t o ( '-- 6) of [1], but only the following, formulated in terms of C rather than as usual with Cn. For all A with A = C(A), (=a) ( - b)

If x e A then A - x = A (vacuity) If x E C(A - x) then x 6 C(0) (success).

N o r do we require that theory revision ~- be exactly definable from theory contraction via the Levi identity, but only the much milder constraint that it is bounded below and above as follows. F o r every A with A = C(A),

(4)

A - - - I x ~_ A4-x ~ C ( ( A - 7x)~3{x}).

The Ramsey Test: A Minimal Version We also formulate the Ramsey test in a very m o d e s t way. This test has as body the equivalence (Ramsey)

x>y~A

iff y ~ A 4 - x ,

but behind this simple biconditional there lurks subtle variation, according to what A, x, y are allowed to range over. In the simplest (and strongest) version, A is any set of propositions, closed under Cno, of a language S that allows unlimited iteration of the connective > , and x, y are any propositions i n S. For our purposes we take the Ramsey test as enunciated for a language S allowing application of the connective > to propositions not already containing > ; for all sets A _~ S with A = C(A); and for all x, y e S that do not contain any occurrence of >. Note that this is considerably weaker than t h e simple version of the Ramsey test, in all respects, In respect of S, x, y the differences are evident. In respect of A, we note that whenever A = C(A) then A = Cno(A), although examples can easily be constructed against the converse. For we have C(A)~_ CnoC(A)~_ CC(A)= C(A) by conditions (C1) and (C2), s o C(A) = Cno C(A); so if A = C(A) we may substitute A for C(A)on both left and right to obtain A = Cno(A ) as required.

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Impossibility Theorem with Non-monotonic C In our version for a non-monotonic inference operation of the G/irdenfors impossibility theorem, we consider separately the limiting case that A = S and the principal case that A # S, for the former gives us a particularly acute and immediate form of the impossibility result. For the proof of the principal case we abstract to the non-monotonic context the elegant strategy of Segerberg [11] and the economy of means of Rott [10]. The limiting case was pointed out to the author by Hans Rott and Horacio Aft6 Costa independently in correspondence~

Theorem Let S be a language admitting truth-functional connectives. Let C be an inference operation satisfying the conditions (C1) to (C3) above, and - , -~ operations taking sets A _ S closed under C, paired with propositions in S, to subsets of S. Let > be any two-place propositional connective such that x > y s S for all.x, y ~ S that do not themselves contain any occurrences of > . Then: (1) LIMITING CASE. T h e Ramsey test taken together with the conditions ( - b ) and (4-), all applied only to the limiting case that A = S, is incompatible with the non-triviality condition that C ( 0 ) v ~ S;

(2) PRINCIPAL CASE. The Ramsey test, together with the conditions ( - a), ( - b) and (4-), all applied only to the principal case that A v~ S, is incompatible with the non-triviality condition that there are propositions x, y ~ S , not containing any occurrences of > , such that x ~ C(y), y~C(x), C(C(x)wC(y)) ~ S and x A y ~ C(0). PROOF OF THE LIMITING CASE. First note that conditions ( - b) and (4-), in the limiting case that A = S , imply that if 7 x ~ S q - x then -i x ~ COD). For if 7 x ~ S-i-x then by the limiting case of(4-), -i x ~ C ( ( S - 7 x)w {x}), so by (C1) and (C3), 7 x ~ C ( S ' - - - l x ) , so by the limiting case of ( - b ) , - i x ~ C ( O ) . Then note that by the limiting case of the Ramsey test, we have x > y ~ S i f t y ~ S 4- x for all x, y ~ S not containing > . Hence in particular, -1 x E S 4- x for all such x. Combining these two gives us -i x 6 C(O) for all x ~S not containing > , so by (C5), C ( O ) = S contrary to the non-triviality condition. PROOF OF THE PRINCIPAL CASE. In what follows, it is assumed that the Ramsey test and conditions ( - a ) , ( - b ) , (4-) are all formulated under the condition A ~ S. Let S o be the set of all propositions in S that do not contain any occurrences of > , and let S 1 be the set of those that do not contain > except possibly as principal connective. From the Ramsey test we have the following

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consequence, which might be described as a "low level mon9tony for 4-": for all A, D ~ S with A = C(A), D = C(D), and for all z e S o, If A r t s ! ~_ D n S 1 then ( A 4 - z ) n S o ~ ( D 4 - z ) n S o. For suppose, A, D :~ S, A = C(A), D = C(D) and z e S o, whilst A n S 1 ~_ D n S 1 and w e (A 4- z) n S O; we want to show that w e (D 4- z) n S o. .Clearly w e S O so we need only show w e D 4 - z . Since w, z e S o and A = C ( A ) ~ S we apply the Ramsey test to conclude z > w e A , so z > w e A n S ~ , so since A n S ~ ~ D n S 1 we have z > w e D n Sx __%_D = C(D) ~ S so by the Ramsey test again, w e D 4-z as desired. Now, suppose for reductio ad~absurdum that x, y ~ S o are such that xq~C(y), y(EC(x), C(C(x)wC(y)) ~ S and x ^ y(~C(O). P u t A = C(x), B = C(y), D = C(C(x)wC(y)). Note that -](x ^ y) does not contain > , and that D, B, A are all dosed under C and are distinct from S, so we may apply the principal case ot; the Ramsey test and conditions ( - a), ( "--b), (4-) to them. Note also that since A, B c D .we have A r t S 1, B n S 1 ~ D n S ~ . Hence by "low level monotony" as established, we have (A4- "-l(X ^ y ) ) n S o _ (04- -7(x ^ y))cTSo and (B4- "-](x ^ y ) ) n S o _ (D4- -q(x ^ y ) ) n S o. But we also have A ~_A.-[--q(x^y) so A n S o ~ _ ( A 4 - - 7 ( x ^ y ) ) n S o and

B ___B4- -](x ^ y) so B n S o -= (B4- 7 ( x ^ y))nSo. The ~erifications of these two are similar; we consider the former. Since yr = C C ( x ) = C(A) using condition (C2) we have by condition (C5) that - 7 7 ( x ^ y ) r so by condition ( - a ) , A = A - q - l ( x ^ y ) _ A 4- -7 (x ^ y) by condition (4-). Putting these inclusions together gives us A n So, B n S O _ (D 4- -7 (x ^ y)) n So, so by the definition of A and B, x, y e D 4- -7 (x ^ y) ~ C(D -- -1 -7 (x ^ y) u { 7 ( x ^y)}) by condition(-[-), so by condition (C4), x ^ y

But clearly also

C(D- -7 -qtx ^ y)u{-7 (x ^ y)}).

by (C1), x ^ yeC(D--]

7 ( x ^ y ) u { x ^ y}),

so by condition (C3) followed by (C5),

7 7 (x ^ y) c ( o "- -7 7 (x ^ y)) and thus by condition ("-b) followed by (C5), x ^ yeC(O)

giving us a contradiction.

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Acknowledgements The author would like to thank Charles Cross who helped pose the problem of this paper, Isaac Levi who helped clarify the background, Hans Rott and Horacio Arl6 Costa for the limiting case part of the theorem, and Wlodek Rabinowicz for helPing redefine the conditions on C.

References [1] C. E. ALCHOURRt)N,P. GARDENFORSand D. MAKINSON,On the logic of theory change: partial meet contraction and revision functions, The Journal of Symbolic Logic 50 (1985), pp. 510-530. [2] H. ARLOCOSTA,Conditionals and monotonic belief revision: the success postulat, to appear in Stadia Logica. [3] P. GARDENFOR$,Belief revisions and the Ramsey test for conditionals, The Philosophical Review 95 (1986), pp. 81-93. [4] P. GARDENFORS,Variations on the Ramsey test: more triviality results, Stadia Logiea 46 (1987), pp. 321-327. [5] P. G~.RDENFORS,Knowledge in Flux, Bradford Books, The MIT Press, Cambridge, 1988. [6] I. LEVI, Iteration of conditionals and the Ramsey test, Synthese 76 (1988), pp. 49-81. [7] D. MAKINSON, General theory of cumulative inference, in M. Reinfrank and others eds., Non-Monotonic Reasoning, Springer-Verlag, Series LNCS, Subseries LNAI n~ Berlin, 1989, pp. 1-18. [8] R. REtTER, A logic for default reasoning, Artificial Intelligence 13 (1980), pp. 81-132. [9] H. Rorr, Ifs, though and because, Erkenntnis 25 (1986), pp. 345-370. [10] H. ROTT, Conditionals and theory change: revisions, expansions and additions, Synthese, to appear. [11] K. SEGERBERG,A note on an impossibility theorem of Giirdenfors, Nofts, to appear. [12] Y. SHOHAM,Reasoning about Change, The MIT Press, Cambridge, 1988.

LES ETANGS B2, LA RONCE 92410 MILLE D'AVRAY, FRANCE

Received January 6, 1989 Revised January 22, 1989

Studia Logica XLIX, 1