Beyond Rational Monotony: Some Strong Non-Horn Rules for Nonmonotonic Inference Relations

Beyond Rational Monotony: SomeStrong Non-Horn Rules forNonmonotonic Inference Relations

HASSAN BEZZAZI, UFL UA. 369 du CNRS, Citt Scientifique, 59655Villeneuve d'Ascq Cedex, France and University de Lille II, Faculte" deDroit, 59000 Lille, France.E-mail: [email protected]

DAVID MAKINSON, Les Etangs B2, La Ronce, 92410 Ville d'Avray,France.E-mail: [email protected]

RAM6N PINO P£REZ, UFL UA. 369 du CNRS, Citi Scientifique 59655Villeneuve d'Ascq Cedex, France and University de Lille I, Eudil, 59655Villeneuve d'Ascq, France.E-mail: [email protected]

AbstractLehmann, Magidor and others have investigated the effects of adding the non-Horn rule of rational monotony tothe rules for preferential inference in nonmonotonic reasoning. In particular, they have shown that every inferencerelation satisfying those rules is generated by some ranked preferential model.

We explore the effects of adding a number of other non-Hom rules that are stronger than or incomparable with ra-tional monotony, but which are still weaker than plain monotony. Distinguished among these is a rule of determinacypreservation, equivalent to one of rational transitivity, for which we establish a representation theorem in terms ofquasi-linear preferential models. An important tool in the proof of the representation theorem is the following purelysemantic result, implicit in work of Freund, but here established by a more direct argument: every ranked preferentialmodel generates the same inference relation as some ranked preferential model that is collapsed, in the sense of beingboth injective and such that each of its states is minimal for some formula.

We also consider certain other non-Hom rules which are incomparable with monotony but are implied by condi-tional excluded middle, and establish a representation result for a central one among them, which we call fragmenteddisjunction, equivalent to fragmentedconjunction, in terms of almost linear preferential models.

Finally, we consider briefly some curious Horn rules beyond the preferential ones but weaker than monotony,notably those which we call conjunctive insistence and n-monotony.

Keywords: Nonmonotonic reasoning, rational monotony, preferential models.

1 Introduction and overview

The postulates for preferential inference, as formulated by Kraus, Lehmann and Magidor [5]are intended to gather together some properties for inference relations that may be regardedas in principle desirable, even when the inference relations are not monotonic. They are allHorn conditions, that is of the form: if such and such pairs are in the relation, so too is such

J. Logic ComputaL, Vol. 7 No. 5, pp. 605-631 1997 © Oxford University Press

606 Beyond Rational Monotony

another pair. Lehmann and Magidor [6] and [7] have also studied the effects of adding tothe preferential postulates a further rule, non-Hom in character, called rational monotony. Asusually formulated with a negative premiss, it is: if a\~fi and not a^y, then OA7|~/?. Equi-valently, with positive premisses but disjunctive conclusion, it is: if af~/? then either aAj\~/3or a|—iy. In the two papers mentioned, it is shown that every inference relation satisfying thepreferential rules is determined by some model of a certain kind, also called preferential, andthat every inference relation satisfying in addition rational monotony is determined by someranked preferential model.

It is known that rational monotony implies certain other non-Horn conditions of interest,notably disjunctive rationality, which in turn implies negation rationality—see for examplethe brief accounts in Makinson [9] and Lehmann and Magidor [7], or the more extensive workin Freund [2] and Freund and Lehmann [3] which provide a semantic characterization of in-ference relations satisfying these two rules. It is natural to ask whether there are any otherrules of interest, stronger than or incomparable with rational monotony, but still weaker thanplain monotony.

Makinson [9] drew attention to one such rule, called determinacy preservation, showingthat it lies between monotony and rational monotony, but without investigating it semanti-cally. Bezzazi and Pino Pdrez [1] began a semantic investigation of two other rules, rationaltransitivity and rational contraposition. In this paper we study these and related conditionsmore systematically, establishing interrelations and providing semantic characterizations.

It turns out, as we shall show, that given the preferential rules, rational transitivity and deter-minacy preservation are equivalent, and are in turn equivalent to the combined force of ratio-nal monotony with rational contraposition, as also to the combined force of rational monotonywith another rule that we shall consider. Rational transitivity alias determinacy preservationthus appears to occupy a rather pivotal position in this region. We show that any inference re-lation satisfying that rule in addition to preferential ones, is determined by a preferential modelthat is in a certain sense quasi-linear. The proof makes use of Lehmann and Magidor's rep-resentation theorem for rational monotony, but also of an important tool of a purely semanticnature. This is the result, implicit in Freund [2], that every ranked preferential model deter-mines the same inference relation as some ranked preferential model that is collapsed, in thesense of being both injective and such that each state is minimal for some formula.

We also consider certain non-Horn rules that are not implied by monotony, and which forthis reason are perhaps intuitively less interesting, but which are nevertheless weaker than thewell-known rule of conditional excluded middle of Stalnaker [13], also called full determinacyin Makinson [9]: ifnotaf~/?thena|-—•/?. We isolate two such rules of particular formal inter-est, which we call disjunction fragmentation and conjunction fragmentation. We prove thatthey are equivalent and then we establish a representation theorem for preferential relations sa-tisfying disjunction fragmentation (conjunction fragmentation), using the same semantic toolas for rational transitivity above.

All of the rules so far mentioned as potential additions to those for preferential inference, arenon-Hom. Curiously, Horn rules appear to be less plentiful as potential additions. However, ina final section we identify some such rules, weaker than monotony but not implied by rationalmonotony, represent some of them semantically, and raise a number of open questions.

We presume some familiarity with the main lines of at least one of Kraus, Lehmann andMagidor [5], Lehmann and Magidor [7], Makinson [9].

Beyond Rational Monotony 607

2 Background

In this section we recall some basic definitions and results from Kraus, Lehmann and Magidor[5] and Lehman and Magidor [7], which will be used in the paper.

We consider formulae of classical propositional calculus built over a set of elementary for-mulae denoted War plus two constants T and _L (the formulae true and false, respectively).Let £ be the set of formulae. If Mir is finite we will say that the language £ is finite. Let Ube the set of valuations (or worlds), i.e. functions v : Mir U {T, 1} —• {0,1} such thatv(T) = 1 and v(L) — 0. We use lower case letters of the Greek alphabet to denote formulae,and the letters v, vi, u 2 , . . . to denote worlds. As usual, I- a means that a is a tautology andv |= a means that v satisfies a where compound formulae are evaluated using the usual truth-functional rules. We consider certain binary relations between formulae. These relations willbe called inference relations and will be written h-

DEFINITION 2.1

A relation h is said to be preferential iff the following rules hold

ah/3 h a ~ 7REF LLE

ah/3 H / ? ^ 7 ah/3 a|~7RW AND

a|~/3 ah7OR CM

These rules are known as the rules of the system P. The abbreviations above are read as fol-lows: REF—reflexivity, LLE—left logical equivalence, RW—right weakening, CM—cautious monotony. AND and OR are self-explanatory.

A relation h is said to be rational iff it is preferential and the following rule (rational mono-tony) holds

ah/3 abt-orRM

DEFINITION 2.2

A structure M is defined by a triple (5, t, -<) where S is a set (of arbitrary items, called states),-< is a strict order (i.e. transitive and irreflexive) on 5 and t : S —> U is a total function (theinterpretation function). If the function i is injective the structure is said also to be injective.

Let M = (S, t, -<) be a structure. We adopt the following notation: if T C S, then min(T)is the set of all minimal elements of T with respect to •<, i.e. min(T) = {t 6 T : ->3f' (*' € Tand t' •< t)}\ mod*<(aO = {s £ S : t(s) =̂ a } ; min^(a ) denotes min(mod.vf(<*))•

DEFINITION 2.3

A structure M = (5, t, -<) is said to be a preferential model iff for any formula a the followingproperty (smoothness) holds

Vs € modju(a) \ minA((a) 3s' 6 min^Ca) s' •< s.


A structure M = {S,t,-<} is said to be a ranked model iff it is a preferential model andthere exists a strict linear order (ft, <) and a function r : 5 —> Q such that for any s, s' G S,s -< s'iffr(s) < r(s').

DEFINITION 2.4

Let M = (5, t, -<) be a preferential model. The inference relation \~M is defined by thefollowing

<=> minju(a) C mo&M{P)-

The following (representation) theorem is due to Kraus, Lehmann and Magidor [5].

THEOREM 2.5

(̂ is a preferential relation iff there is a preferential model M = {S, t, -<) such that (~x = (~.If the language is finite then S can be chosen finite.

The following (representation) theorem is due to Lehmann and Magidor [7].

THEOREM 2.6

(~ is a rational relation iff there is a ranked model M = (S, i, -<) such that [~M — K If thelanguage is finite then S can be chosen finite.

PROPOSITION 2.7

If ]y is a preferential relation then the following rules hold

CUT

If (~ is a rational relation then the following rules hold

DR NR

For the proofs see [5] for S and CUT and see [9] or [7] for DR and NR. The abbreviationsabove are read as follows: S—Shoham rule (this abbreviation is taken from [5]; note that thisrule corresponds to the hard half of the deduction theorem for classical h), DR—disjunctiverationality, N R—negation rationality. The term C UT is self-explanatory, but it should be notedthat this form of cut, which plays an important role in nonmonotonic logic, is weaker than theforms of cut usually studied in Gentzen-style formulations of classical and intuitionistic logic.The latter imply transitivity of the inference relation; the former does not.

Notation: If n is a natural number, rl will denote the set {0 ,1 , . . . ,n} linearly ordered with thenatural order <. If A is a set, the cardinality of A will be denoted by |yl|. When M = (S, 1, -<)is a preferential model, u G 5 and a a formula, if there is no ambiguity we shall write u ^ o,mod(a) and min(a) instead of i(u) f= a, modjn(a) and minx (a), respectively.

OBSERVATION 2.8

It is known that there are preferential models whose inference relation is not generated byany injective one. A simple finite example was given en passant by Krauss, Lehmann andMagidor at the end of section 5.2 of [5]. The language is assumed to have just two elementary


sentences p, q. The states are s,- (0 < i < 3) with so < «2 and «i < S3, and ao |= PA->?,

«I |= -ipA-iq, while s2, S3 N PA?. Kraus, Lehmann and Magidor leave the verification of theexample as an exercise; a verification is sketched by Schlechta in Section 1 of [12]. Becauseof its relation with the theme of this paper, we give the verification in full, using moreover aninfinite language so as to make it clear that the example is not an artifact of a limited numberof elementary sentences.

Let pj 0 € ^) be all the other elementary sentences and make them behave just like p,i.e. put Si |= pj iff Si ^ p. Let f~ be the inference relation determined by this preferentialmodel. Then clearly we have the following: (l)(pAg)v->gf—<q, white (2) (p*q)v(p*-iq)\/'-iq(witness S3) and (3) (pAg)v(-ipA->g)|^-ig (witness S2). Moreover (4) pA->pj (~±. We claimthat any injective preferential model whose inference relation agrees with this one on (1) and(4), disagrees with it on (2) or (3).

Consider any injective preferential model M — (5, t, <) and suppose that (1) and (4) hold.In the case that PA<J|~_L clearly we have (pAg)v(pA->?)(~->g and also (pAg)v(->pA-ig)f~-ig,so we may suppose without loss of generality that p*q\fL. Then there is a state s G S withs |= pAg so s =̂ (pAg)v-ig. By (1) s £ min((pAg)v-i</) so there is a t G S with t < s andt 6 min((pAg)v->g) so by (1) again t ^= -<q. Now either t [= p o r t ^ ->p. Consider thelatter; the argument for the former is similar. Suppose for reductio that (3) holds, i.e. there isu £ 5 with u G min((pAg)v(->pA-ig)) and u =̂ q. Then u ^ p. Moreover for all j G J, wehave u ^ pj, for otherwise there is i G J such that u f= pA-ip; so there is u' with u' < uand u' G min(pA->pj) contradicting (4). Similarly s ^ pj for all j G J. Since s, u (= p*q*pjfor all j G J we have «(s) = t(u) so by injectivity s = u. Thus s G min((pAg)v(->pA-ig)),contradicting t < s and t ^ ->pA->g.

3 Some strong non-Horn conditions

Rational monotony of course is a restricted form of, and thus implied by, plain monotony (M):

M

One of our purposes in this paper is to examine some interesting non-Horn conditions,stronger than rational monotony (or in some cases, independent of it) but still weaker thanmonotony. In other words, we wish to investigate the enclosed area of the following diagram.

M

Four rules that arise in this connection are determinacy preservation, rational transitivity,rational contraposition, and weak determinacy.


Determinacy preservation (DP), briefly considered by Makinson [9], is the rule

DP

This rule evidently is a weak form of monotony. It can also seen as a weak form of Stalnaker'srule [13] of conditional excluded middle (if <t>\£ip then 4>\~4>): a consequence of a formula iseither conserved when we add a new hypothesis or we get the negation of this consequence.

Rational transitivity (RT), introduced by Bezzazi and Pino Pe"rez [1], is the rule

ah/3 /?|~T <*bt-'7RT .

Obviously this rule is a weak form of transitivity. The intuition behind this rule is the fol-lowing: when the premisses of transitivity hold we get the usual conclusion except when its'opposite' holds. Note that this rule is also a weak form of conditional excluded middle.

Rational contraposition (RC), also introduced by Bezzazi and Pino P6rez [1 ], is the rule

RC

Obviously this rule is a weak form of contraposition. The intuition behind this rule is the fol-lowing: when the premiss of contraposition holds we get the usual conclusion except whenits 'opposite' holds. This rule is again a weak form of conditional excluded middle.

Weak determinacy (WD), formulated by Michael Freund in correspondence with the au-thors, is the rule

\WD

This rule says that any formula a that is 'exceptional' in the sense of Lehmann and Magidor[7], i.e. such that T^- ia , is complete in the sense that for every formula, either it or its nega-tion is a consequence of the exceptional formula. Given the preferential rules, this is a specialcase of both monotony and conditional excluded middle.

DEFINITION 3.1

A relation h is said to be determinacy preserving iff it is preferential and the rule DP holds.A relation (~ is said to be rational transitive iff it is preferential and the rule RT holds.

In this section we compare the strength of the rules DP, RT, RC, WD with each other as wellas RM on the lower side and M on the upper side. The general picture turns out as follows:

PROPOSITION 3.2

Given the preferential rules P, the rules DP and RT are equivalent, and are implied by mono-tony. They are also equivalent to the pair {RM, RC} and also to the pair {RM, WD}. More-over, given P, RC implies both WD and N R. However given P, none of the following impli-cations hold: RM to WD. RC to DR, WD to NR.


Recalling from [9] that M quite trivially implies DP but not conversely, and that RM impliesDR which implies NR but neither conversely, Proposition 3.2 gives us the following diagram,where one condition implies another, given P, iff one can follow arrows from the former tothe latter.

• M

J DP = RT = {RM, RC} = {RM, WD}1

»•• W D

NR

This proposition suggests a central role for DP alias RT. We verify the components of theproposition separately. The positive parts are first proven syntactically, the negative parts arethen established semantically.

OBSERVATION 3.3

P + RT<» P + DP

PROOF. (^-) Suppose af~7 and aA/?|/—17. We want to show QA/?|~7. By preferentiality,aA/?|~a. Thus we have aA/?|~a, a|~7 and OA/?^-17. So by RT C*A/?|~7 as desired.

(•*=) Suppose a|~/?,/?|~7 and a\/--<j. We want to show a (^7. Now, a A/?[^->J for otherwisesince a|~/? we would have by cut that a\~-*y contrary to supposition. Hence since /?(~7 wehave by DP OA/?|~7, and so since a|~/? we have by cut that a^y as desired. I

OBSERVATION 3.4

P + RT =>• RM. That is, a rational transitive relation is indeed a rational relation.

PROOF. This is a corollary of Observation 3.3 and the fact that P + DP =>• RM proven in [9].Here we give a direct proof. Assume o|~/? and a \^->f. We will show a^f\^p. First we showaA7^->/3. Suppose that it is not true, i.e. aA7J~-i/?. Then, by S, a|~7 —• ->/?. Since a\~(3,by AND and RW we get 0^-17, a contradiction. Second, we have aAj\~a because of REFand RW. Finally, since aA7[~a, a|~/? and aA7^->/?, we conclude using RT. I

OBSERVATION 3.5

P+DP=>RC

PROOF. Suppose a|~/?; we want to show that either -if3\~-<a or ->P^a.Case 1: Suppose T^/3. Now by preferentiality from a|~/? we have T(~a —*• ft. Hence,applying RM (which we noted follows from DP) we have T A - I ^ O —• 0 so by preferentiality-I/3(~->Q as desired.Case 2: Suppose T \~fi. Then by preferentiality T (~-i/? -+ ->a so by the hypothesis DP eitherTA-I/?|—>/? -+ -.0 or TA-./?|—.(-1/? -»• i a ) .

Subcase 2.1: Suppose TA->/?|~->/? —• ->a. Then by preferentiality ->P^->a as desired.Subcase 2.2: Suppose TA->/?^->(-I/? —• ->a). Then by preferentiality ~"/?(~a as desired. I


OBSERVATION 3.6

P+RC=>WD

PROOF. Let |~ be an inference relation satisfying the preferential rules, and suppose that itfails WD.

Since (>- fails WD, there are a, /? with T|~-ia, a[/•/?, aty-'P. Since Tf^-ia we have bypreferential rules that T(~->av-i/?, and so combining this with T|~-ia again we have by CMthat ->ov->/?(~->a.

On the other hand, since <*(//? we have by preferential rules that atya^fi and so ->->a^->(-iav-i^). Also since a \f->P we have by preferentiality that a ^-lav-i/? and so ->->a j/^av->/?.

Putting these three facts together we see that RC fails. I

OBSERVATION 3.7

M + WD=>DP

PROOF. Suppose a(~/? and aA7^-i/?; we want to show a^y\^/3. If T\—<{CCAJ) then the sec-ond hypothesis with WD give us what we want. On the other hand, if T\/*->(a*i) then notingfrom the first hypothesis that T|~a —» /? we get by RM that TA(c*A7)|~a —• /? so by prefe-rential rules, OA7|~/? as desired. I

OBSERVATION 3.8

P + RM + RC =• DP

PROOF. This follows immediately from Observations 3.6 and 3.7. For another verification,suppose that a(~/?. We want to show that either aA7f~/? or asf\—\(3. Now either 0^-17 or

Case 1: Suppose af-17. Then by the hypothesis RM we have a^y^0 as desired.Case 2: Suppose a|—17. Then by preferentiality (rule S) T|~a —• -17 i.e. T|—i(aA7) so byNR which holds by Proposition 2.7, either /?(~->(OA7) or ->/?f—'(aA7), and in each of thesetwo subcases RC tells us that either aAj\^/3 or c*A7f—1/?, as desired. I

OBSERVATION 3.9

P+RC=>NR

PROOF. We have already that P + RC implies WD (Observation 3.6). So it will be enough toprove the following two facts:

Fact 1. P + RC => RC+, where RC+ is the following rule

Fact 2. P + RC++WD=>NR

Proof of fact I: Suppose RC holds, and suppose a^P\^f. Then by preferential rules, O A / ? ^7v->a, so by RC either -i(7v-ia)f~->(aA/?) or ->(7v-ia)|~aA/?. In the former case we haveby preferential rules aA-i7(~-iav->/?, so by preferential rules again cth-*f\~->fi as desired. Inthe latter case we have by preferential rules OA->7^QA/? so as-<-f\~fi as desired.

Proof of fact 2: Suppose RC+ and WD hold, and suppose a\~j3\ we want to show that eitherOA7|~/?oraA->7|~/?. Since af~/J we have by preferential rules that T(~-iav/?. Hence by WDeither -i(->av/?)|~7 or -i(->av/?)f—17, i.e. either aA-i/?|~7 or aA-1^^-17.


Case 1. Suppose QA-I/?(~7. Then by RC+, either aA-<y\~P or QA-I7(~-I/?. In the former sub-case we are done. In the latter subcase by preferential rules (rule S) a \~yv -'P which combinedwith a\~p gives a\~j. But this again combined with a|~/? gives, by CM, aA7|~/? and in thissubcase we are also done.Case 2. Suppose aA-i/?|—17. Then by RC+, either OA7(~^ or QA7I—<0. In the former sub-case we are done. In the latter subcase by preferential rules we have a\~/3 —• ->y which com-bined with o|~/? gives again by preferential rules a|~- i7. From this, we conclude as aboveusing CM that a*-i-f\~f3 and we are also done. I

Given the above positive parts of Proposition 3.2, it suffices to show the following negativeones: P + RM £ WD, P + RC & DR, P + WD 56 NR.OBSERVATION 3.10

Let M = {S,»,-<) be any preferential model with 5 = {«o,«ii«2,«3}, •< = {(so,«2),(s!,S3)} and t(s2) = t(s3). Then \~M satisfies RC.

PROOF. Assume a(~/3 and ->/?(/—ia. We want to show -i/?f~a. It will be enough to see thatmod(a) = S. Note that min(-i/?) (~i mod(a) ^ 0 because -i/J^-ia. But neither so nor s\can be in min(-i/?) n mod(a) for otherwise as s0 and «i are minimals we would have a\^/3contradicting our assumption or|~^. So, either «2 or 53 is in min^/?) n mod(a), so sincet(«2) = t(s3) we have both «2, s3 € mod(-i/3)fimod(a). This and a p i m p l y by smoothnessthat so, «i G min(a). Thus mod(a) = 5, as desired. 1

Note that the hypothesis i(s2) = 1(53) is necessary in this observation. We can easily finda preferential model M = (S, 1, -<) with 5 = {so,si,s2,S3}, -< = {(so,s2), (s i ,s3)} andi(a2) ^ t(s3) which does not satisfy RC.

OBSERVATION 3.11

P + R C ^ D R

PROOF. Consider a model as in the previous observation with t(s2) = 1(33) = {p,g},t(«o) ={p, r}, t(si) = {q, r} (we give the valuations as for a Herbrand model, that is identifying thesubset of variables with its characteristic function). Graphically

By Observation 3.10 RC holds in M but it is clear that ptyr (witness s3), qtyr (witness s2)and pvg|~r so DR fails. I

OBSERVATION 3.12

P+RM^WD

PROOF. Let £ be the set of all formulae built from the elementary formulae p and q. Considerthe following model M = (5,», -<) where 5 = {s0)si,s2},-<= {(so,si),(so,s2)},i(so) ={p,q}, »(si) = {q} and :(s2) = 0. Graphically


It is clear that M is ranked, so it satisfies RM. However it fails to satisfy WD since T|~pwhile ->p[^g and B

OBSERVATION 3.13

P + WD^NR

PROOF. Let M be the model represented by the following schema;

s4 : -ip, -19 • = • s5 : -"P, -<

s2 : r fi.e. .M = (S,», -<) with S = {so , . . . , s5}, -< the transitive closure of the relation {(so, $2)1(«2,«4),(*i,«3),(«3,«5)}. »(«o) = {p,«}. »(«i) = {p}. *(«a) = «(«3) = {g}and»(s4) =i(«8) = 0-NR fails in this model because T|~p, q\fa> (witness S3), -iqtyp (witness s4). But WD is sat-isfied in this model. Suppose that T(~->o, atyfi, aft-ifi. Then there must be u € min(a) nmod(-'/?) and v £ min(a)nmod(/?) andu, t; e {s2, «3, «4, «s}- But it is clear that each choiceof u, D here gives a contradiction. For example if u = s2 and v = S5 then since t(s2) = i(s3)we have v £ min(a) giving a contradiction. I

Observations 3.11 and 3.13 have been established using non-injective preferential modelsas examples. In the case of 3.11, at least, there is no injective model that does the job. For by3.9, any injective model of P + RC is an injective model of P + NR, and it has been shownby Freund and Lehmann [3], that every injective model of P + NR is a model of DR.

It is immediate that transitivity (T) of f- implies RT. However, the converse does not hold:given that P + R T - o P+DP shown above, and the well-known facts (see [9]) that P + T => Mwhile P -I- DP fi M, we have P -I- RT ^ T. A direct verification can also be made with anappropriate two-state model (see Corollary 5.3).

As already remarked, DP, RT, RC and WD are weakened forms not only of monotony butalso of Stalnaker's rule of conditional excluded middle which, unlike the principles so far con-sidered, is not implied by monotony but has figured in philosophical discussion of counterfac-tuals [4, 8,10]. We shall study some other rules in the vicinity of conditional excluded middlein Section 7.

4 Collapsed models

Our goal in Section 5 will be to prove a representation theorem for P + RT (equivalentlyP+D P). As a preliminary, we shall show in this section that every ranked preferential model isequivalent (in the sense of generating the same inference relation) to one that is both injectiveand parsimonious, in the sense that every one of its states is minimal in at least one formula.This result is indeed implicit in Freund [2] in the more general case of relations satisfyingP + DR, but using different arguments. Our procedure for transforming a ranked model intoone with these characteristics is quite straightforward. We proceed in two steps. First, at eachlevel of the model we identify the states of that level that are labelled with the same valuation.Second, we suppress all states that are not minimal in some formula.


DEFINITION 4.1

A model .M = (S,t, -<) is said to be horizontally injectiveifffor all distinct s,t £ 5, if s / tand t •£ s then i(s) ^ »(t).

Note that for ranked models, being horizontally injecti ve actually means injecti vity by lev-els.

LEMMA 4.2

For any ranked model M = (S,t,~<) there exists a horizontally injective ranked model M' =(5',»', -<') such that \~M = \~M>-

PROOF. We define an equivalence relation on S as follows

s = s' <& r(s) = r(s') and »(s) = t(s')

Put S' = S/ = (the quotient of S by =). As usual let [s] denote the equivalent class of s.Define r1 : S' —• Q, <' C S'xS' and i' : 5 ' —> U as follows: ^([s]) = r(s), [s] -<' [s1]iff s -< fi', and •'([«]) = »(«)• It can be easily verified that r', -<' and «' are well defined, i.e.their definition does not depend on the choice of the representative of [«]. It is also clear thatfor all j € fi, i' restricted to Sj = {[s] € 5 ' : r([«]) = j} is injective. Notice that

[s] •<' [«'] # H / » r(fi) < r(«') O r'([«]) < / ( M ) .

So, the model M' defined by M' = (5 ' , : ' , -<') is a ranked model. Moreover, we have \~M —\~M', since clearly for all 8 £ 5 and all a, /? £ C, s £ modx(^) iff [s] £also s £ minx(a) iff [s] £ mi

DEFINITION 4.3

A model M = (S, t, -<) is said to be parsimonious iff for every state s £ 5 there is a formulaa such that s £ minx (<*).

PROPOSITION 4.4

If M = (S, i, •<) is a preferential model then there exists a preferential model M' = (S',t',•<'} such that S' C S and the following properties hold.

1. M' is parsimonious.

2. If M is ranked so is M'.

3. Whenever s, t £ 5' with neither s -<' t nor < -<' s, if»'(«) = »'(<) then «(s) = i(<).

4. \~M = Kw

PROOF. It is quite simple. It is enough to suppress the states which are not minimals for anyformula Essentially the same trick has been used by Pavlos Peppas [11] in the context ofsystems-of-spheres models for belief revision. Define 5 ' = S\{s : ->3a s £ minx (a)}.Let i' and -('be the restrictions of»and •< to 5 ' . Put.M' = (S',i',^'). By definition of M',it is obvious that

Hence smoothness of M implies smoothness of M'. So, M' is a preferential model which,by its construction, is parsimonious. Clearly if M is ranked so is M!. Property 3 is triviallyverified by definition of M'. Finally, let us verify o(~x/? o a\~M'P-(<=): Suppose that minx'(a) C modx'(/?)- We want to show minx (a) Q modx(/?)- Thisfollows from (•) and the fact that modx'O?) £


(=>): Suppose that n i in^ (Q) Q modx(/?). By definition of 5 ' m i n ^ ( a ) C (P)because modx'(/3) = {s € modjn(^) : 3y s € minjuM}. So, by (•), minA<'(o) Q

IREMARK 4.5

When the ranked model M — (S, t, •<) is finite (notice that by Theorem 2.6 every rationalrelation over a finite language is represented by a finite model), i.e. 5 is finite, then 5 ' in theprevious lemma can be constructed by an algorithm. In order to see this, first remark thatwhen 5 is finite, we can suppose the rank function is of the form r : S —> n. Then defineS'o = {s € 5 : r(s) = 0} and for k = 1 to n, Si = {s € 5 : r(s) = Jfc and there exists awiths € minjn(a)}- Finally put S' = u£_0Si.

THEOREM 4.6 (Collapsing)If M = (5, i, -<) is a ranked model then there exists a parsimonious, injective ranked modelM' = (S',i',<') such that \~M = \~M'-

PROOF. Let AC = (S1, «',-<') be the model obtained from M = (S, i, -<) by applicationof Lemma 4.2 and then Proposition 4.4. Clearly M' is parsimonious and ranked, and also\~M = \~M'- It remains to check injectivity. Now by Lemma 4.2 and part (3) of Proposition4.4, M' is horizontally injective. Clearly parsimony implies that M' is 'vertically injective'in the sense that a •<' t implies t(e) ^ t(f). Finally, horizontal and vertical injectivity clearlimply injectivity.

The model M' obtained from a ranked model M by successive application of Lemma 4.2,and Proposition 4.4 will be called the collapse oiM. Clearly a model is equal to its collapseiff it is both parsimonious and injective. As a corollary of Theorem 4.6 we have the followingresult:

COROLLARY 4.7

Every rational inference relation is generated by some collapsed ranked preferential model.

PROOF. Immediate from the Lehmann-Magidor representation theorem (Theorem 2.6) andTheorem 4.6. I

REMARK 4.8

(i) In the proof of Theorem 4.6 we have applied Lemma 4.2 and then Proposition 4.4 but thesame result is obtained if we reverse the order.(ii) It is not hard to see that in the finite case (finite language), if a model is injective then eachstate is minimal for some formula, so the model is parsimonious. But this is not true in generalfor infinite languages. For instance consider an injective ranked model with two levels: oneworld in the upper level and the rest of the worlds in the lower level; then there is no formulafor which the upper world is minimal.

REMARK 4.9

Theorem 4.6 and its Corollary 4.7 are implicit in Freund [2] but in reverse order of demonstra-tion and by quite a different strategy. Freund shows that if |~ is a rational inference relation(indeed more generally, any preferential inference relation satisfying DR) then we can con-struct its 'associated standard model', which is a model generating |~, that is ranked (or, un-der the hypothesis of DR, that is 'filtered' in the sense of his Definition 5.1) and has additionalproperties including the following:

1. Every state u i s ' |—consistent' in the sense (Freund [2], Section 2.1) that there is a formulao with u (= C(a), where as usual C(a) = {0 : a|~/?}.


2. The model is 'standard with respect to |~' in the sense (Freund [2], Definition 3.2) that itis injective and for every formula a and state u,u^ C(a) iff u G min(a).

Property 2 explicitly implies injectivity, and the two properties taken together clearly implyparsimony. Conversely, parsimony and injectivity together imply properties 1 and 2, if weassume that the model is ranked: property 1 is immediate from parsimony recalling that u Gmin(a) immediately implies u f= C(a) in every preferential model, while to derive property2 it suffices to show that whenever u £ min(a) then u ^ C(a). Suppose u £ min(a). Ifu J£ a then we are done, so suppose that u ^= a. Then there is v -< u with v G min(a). Byparsimony, there exists f3 such that u G min(/?) and thus by rankedness for any v' G min(a),v' \=- -i)3. Thus -i)3 G C(a) and so since u ^ j S w e have u |£ C(a), establishing property 2.

Evidently, each approach has its advantages, depending in part on the purposes for which isused. Since our approach covers only ranked models and thus rationally monotone inferencerelations, unless it can be generalized it is useless for Freund's purpose, which is to representpreferential inference relations satisfying DR. On the other hand, it provides a simple andnatural way of proving representation theorems for conditions such as rational transitivity thatare stronger than rational monotony (e.g. Theorems 5.8,7.18 and 8.6 below) and a very directargument for results of independent model-theoretic interest such as Lemma 4.2, Proposition4.4, Theorem 4.6 and its Corollary 4.7.

5 Representation

The goal of this section is to characterize the ranked models that generate rational transitiverelations. Our argument exploits Corollary 4.7

DEFINITION 5.1

A preferential model (not necessarily injective) M = (S, », -<) is saidtobe^uoji-Zineariff itis ranked and it has at most one state at any level above the lowest. In other words quasi-linearmeans ranked and whenever r -< s, r -< t then either s = t or s -< t or t •< 8.

Quasi-linear models have the following graphical shape:

PROPOSITION 5.2

If M = (S, i, -<) is quasi-linear then the relation (̂ = \~M is rational transitive.

PROOF. M is ranked so ^ is a rational relation. We have to prove that |~ satisfies RT. So, sup-pose a \~f}, PY"f and a(/-i7. We want to show a|~7. We consider two cases. First, supposethat min(a) is contained in the lowest level. As also a|~/?, necessarily min(a) C min(/?).Butmin(/?) C mod(7) because )3f~7. Therefore, min(a) C mod(7), i.e. o|~7.

Second, suppose min(o) is not contained in the lowest level. Then min(a) is a singletonbecause M is quasi-linear; suppose min(o) = {s}. Then 8 \fc -iy because o\/>->y. Thuss (= 7, so af-o'- '


COROLLARY 5.3

There are rational transitive inference relations which are not transitive.

PROOF. Consider a language C built on the prepositional variables p, q and r. Define M =(5,t,-<) whereS = {so,«i}. «o -< «i »(«o) = {q,r}, t(sr) = {p,q}. By Proposition 5.2 therelation (~ = \~M is a rational transitive relation. But we can easily verify, pf~g, q\~r andp\/r. SO f~ is not a transitive relation. I

OBSERVATION 5.4

Suppose that the language is finite and M = (5, t, -<) is an injective ranked model which isnot quasi-linear. Then f~ = ^M does not satisfy RT.

PROOF. AS M is not quasi-linear, necessarily there are three different states «i, «2 and s3 in 5such that «i is in the lowest level, «2 and s3 are in the same level and s\ -< s, for t = 2,3. Leta, /? and j be formulae such that mod(a) = {«2, «3}. mod(/7) = {«i, s7, s3} and mod(7) ={«i, fl2}- By finiteness and injectivity such a, /? and 7 clearly exist. Then, it is clear that a|~/3,/?(~7 while aty-iy and aJ/7. Therefore M does not satisfy RT. I

REMARK 5.5

When the language is infinite the above observation does not hold. This can be seen by thefollowing example. Let M be a ranked model whose states are worlds, with two levels: v0 andt>i in the upper level and the rest of the valuations in the lower level, i.e. the order is v -< v, forall valuations v ^ u,-, t = 1,2. By definition M is not quasi-linear. However, \~M satisfiesRT (and indeed, satisfies transitivity and monotony) because for any formula a, minx (o) liesin the lowest level.

But if instead of injectivity in the Observation 5.4 we require that the model M be collapsedthen a similar argument can be used to extend Observation 5.4 to the case of infinite languages.More precisely we have the following proposition:

PROPOSITION 5.6

Suppose .M = (S, 1, -<) is a collapsed ranked model which is not quasi-linear. Then |~ = \~Mdoes not satisfy RT.

PROOF. AS M is not quasi-linear, necessarily there are three different states *i, s2 and s3

in 5 such that «i is in the lowest level, s2 and s3 are in the same level and si -< s, fori = 2,3. We need to find formulae a ,$, 7 with a\~0, P\~y, a\/<-iy, a\fa. By parsimonythere are formulae fa (i — 1,2,3) with «, 6 min(^j) and by injectivity there are formulaerpij ( t , j = 1,2,3 and i ^ j) with «,- 6 mod(V'«j) and SJ £ mod(^,j). Put a = <f>2v<f>3,(3 = (<^2V^3)V(ÎAV'I2) and 7 = (<ÎAV)I2)V(^3AV)32)- Then it is clear that Q h /? soa|~/?; and using rankedness min(/3) C mod((îAV)i2) so 0\~y; while again using ranked-ness 82 € min(a) but «2 £ mod(7) so atyy and finally s3 £ min(a) but S3 € mod(7) soab*->7. •

Propositions 5.2 and 5.6 immediately imply:

THEOREM 5.7

Let M be a collapsed ranked model. Then M is quasi-linear iff \~M is rational transitive.

This with Corollary 4.7 immediately imply the promised representation theorem for rationaltransitive relations:

THEOREM 5.8

Y* is a rational transitive relation iff there is a quasi-linear model M such that f~ = \~M-


Putting together Theorem 5.8 and Proposition 32 we clearly have:

THEOREM 5.9

The following conditions are equivalent for any preferential inference relation (~ :

1. ^ is determined by some quasi-linear model.

2. |~ is determinacy preserving.3. |~ is rational transitive.4. |~ satisfies both RM and RC.

5. K satisfies both RM and WD.

REMARK 5.10

The above results leave open the question of representation theorems for the weaker postulatesets P-t- RC and P+WD. It may be noted that the techniques used above do not appear to carryover in a straightforward way to those systems. Lemma 4.2 (used for Theorem 4.6 and thusCorollary 4.7 and thus Theorem 5.8) is here proven only for ranked preferential models, andeven if the less direct techniques of Freund [2] are used (cf. Remark 4.9) their scope coversonly postulate systems at least as strong as P + DR.

6 Preferential Orderings and Rational Transitivity

After seeing Bezzazi and Pino PeYez [ 1 ] Michael Freund (personal communication) conjec-tured Theorem 6.9 below, which characterizes rational transitive relations in terms of proper-ties of their preferential orders defined as in [2]. The purpose of this section is to prove thatcharacterization. This gives us another way to obtain the representation Theorem 5.8. Theresults of the subsequent sections do not depend upon this one.

DEFINITION 6.1

Let I- be a preferential inference relation. Let a and /? be formulae. The preferential orderassociated with |~ is defined by

a < /? •«• av/?|~-i/?.

The relation < is not a strict order because irreflexivity does not quite hold (for instance-L < ±). Nevertheless by tradition we conserve the name of preferential order for i t

The following lemma from [2] helps understand better the meaning of this relation:

LEMMA 6.2

1 . Q < / ? o a[—i/? and av/?|~a.2. Q(~/3 *> a < aA-i/?.3. a < /? iff in every preferential model M = (S, t, -<) defining |~ the following property

holds: for every element s G mod(/?) there exists t G mod(a) such that t -< s.

It is easy to show the following corollary of point 3 of this lemma:

LEMMA 6.3

Let f~ be a rational relation defined by a preferential ranked model M = (S, t, -<) with r :5 —• Q the ranking function (Q linearly ordered by <). For any formula a define its level,£(a), as oo if a\~±. and otherwise its level is the unique a £ fi such that there exists s £

) with r(«) = a. Then, the level is well defined and a < /? iff t(a) < I{j3).


We remark that the relation < of [7] (definition A3, first defined in [6]) is equivalent to thatof Definition 6.1 in the case of rational inference relations; the idea behind these 'orders' hasroots in Lewis [8]. In [2] Freund called preferential order any relation < on formulae satisfy-ing the following four properties:

Po: a < -LPi: If a h p, then

(a) a < 7 => /? < 7(b) 6 < 0 => 6 < a.

P2: a < 7 and a < 6. implies a < jvS.P3: av/? < /? implies a < /?.

Freund proves that the 'order' associated with a preferential inference relation by Definition6.1 satisfies these properties. Conversely the inference relation f~ associated with a relation< satisfying these properties by putting a|~/? iff a < aA->/? is a preferential inference re-lation; moreover the order associated with this inference relation by Definition 6.1 coincideswith <. Thus < satisfies properties P0-P3 iff it is the preferential order associated with somepreferential inference relation in the sense of Definition 6.1.

We recall the definition of modular relation (see [7]):

DEFINITION 6.4

A relation < on E is said to be modular iff there exists a linear order -< on some set ft and afunction r : E —• ft such that a < b o r(a) -< r(b).

The following characterization of modularity is well known and easy to verify.

LEMMA 6.5

An order < on E is modular iff for any a, b, c € E if a and 6 are incomparables and a < cthen b < c.

The following proposition is due to Freund (personal communication).

PROPOSITION 6.6

(~ is a rational relation iff the preferential order between formulae associated by Definition 6.1with |~ is modular over the set of |~-consistent formulae, i.e. those formulae a with atyL.

PROOF. The only if pan follows from the representation Theorem 2.6 and Lemma 6.3. Moreprecisely, by the Representation Theorem 2.6 there exists a ranked model M = (5,», -<) withr : 5 —* ft the ranking function (ft linearly ordered by <) such that f~ = | ~ x . Define thefunction I mapping a formula a to its level £(a). This mapping and Lemma 6.3 prove that <is modular.

Conversely, suppose that the preferential order between formulae < is modular. Assumea(~/? and a|/—17. We want to show that a*7|~/?. By part 2 of Lemma 6.2, this last expressionis equivalent to QA7 < a\fh->f} and the assumptions are equivalent to a < OA-I/? and a £c*A7. Note that either aA7 < a or QA7 ̂ a. In the first case we use Freund's property Pi (b)to obtain OA7 < aAjA->f3. In the second case, a and aA7 are incomparables because a £<*A7 using part 2 of Lemma 6.2 again. So by modularity, £*A7 < QA->/? because a < aA-i/?.Now, as before using the property Pi (b), we obtain c*A7 < QA7A->/?. I

The following lemma will be useful:


LEMMA 6.7

Let |~ a preferential relation and < its associated preferential order. For any formulae a and0 if a </? thenT < 0

PROOF. Note that a h T. Suppose a < 0. Then by Pi (a) we have T < 0. I

The quasi-linear property, QLP in short, for an order < associated to an inference relation(~ is the following property: for any formulae a and 0, if T < a then either a < 0 or 0 < aor a is |—equivalent to 0, i.e. a\~0 and /?|~a.

PROPOSITION 6.8

Let M = (S, i, ^> be a ranked collapsed model and put ^ = \~M- If the preferential orderassociated with f~ satisfies the property QLP, then the model M is quasi-linear.

PROOF. Suppose that M is not quasi-linear. We want to show that the preferential order doesnot satisfy QLP. As M is not quasi-linear, there are three different states «1( «2 and S3 inS such that s\ is in the lowest level, «j and S3 are in the same level and 8\ -< s, for t =2,3. By parsimony there are formulae fo and $3 with S2 G min(<fo) and s3 6 min((̂ >3). Byinjectivity there exists a formula C such that s? G mod(() and S3 G mod(-iC)- Put a = 4>iAC.et 0 = <faA->£. Note that a £ 0 and 0 ft a because their minimal states lie in the same level.Moreover, si G minx (a) but s2 £ minx(/?). so a and 0 are not |~-equivalent. But it isclear that, T < a. Hence the preferential order < does not satisfy QLP. I

Note that the argument in this proof 'translates' the one of Proposition 5.6.

THEOREM 6.9

(Conjectured by Freund, personal communication): Let ^ be a preferential relation. Then |~satisfies RT iff the preferential order < associated with (̂ satisfies the property QLP.

PROOF. The only //part is deduced from Theorem 5.8 as follows. Suppose that (~ is rationaltransitive. Then by Theorem 5.8 there is a quasi-linear model M such that |~ = \~M- Weknow that M is ranked, i.e. M — (S, t, •<) with r : 5 —• ft the ranking function (ft linearlyordered by <). Now, if a ^ 0 and 0 •£ a we have by Lemma 6.3 £ (a) = £(0) = a G fi. Butif T < a then by quasi-linearity and Lemma 6.3 there is at most one state s £ S such thatr(s) = a. So, min;u(a) = minx(/3) = {s} orminx(a) = min^(/?) = 0. Hence, in eithercase, minju(a) = minx (0), i-e. a is (^-equivalent to 0.

Now we prove the//part. Suppose that \~ is a preferential relation which satisfies QLP. Wewant to show that |~ satisfies RT. By Theorem 5.8, it will be enough to see that |~ is repre-sented by a quasi-linear model. In order to do that, we first show that < is modular. Supposethat a ff 0,0 ff a, and a < 7. We want to show that 0 < 7. By Lemma 6.7, T < 7. So, byQLP, 0 <yoTj<0oTj and 0 are ^-equivalent. But we shall see that the last two caseslead to a contradiction.Case 1: Suppose that 7 < 0. Then by transitivity of < we have a < 0, a contradiction.Case 2: Suppose that 7 and 0 are |—equivalent. Then, in any model M representing (~ wehave minx (7) = rninx(/3). So by the Lemma 6.2 using a < 7 we conclude that a < 0.We find again a contradiction.Therefore the only possibility is 0 < 7 as desired. As < is modular, by the Proposition 6.6,the relation ^ is rational. So there is a ranked model M representing it, and by Theorem 4.6we can suppose that M is collapsed. Thus by Proposition 6.8, M is quasi-linear. Therefore(- satisfies RT. I


REMARK 6.10

We can give a different proof of Theorem 6.9 which does not use the representation Theorem5.8. Moreover this proof provides an alternative argument for Theorem 5.8.

PROOF. Here we give only a sketch. The argument uses Freund's notion of 'standard model*.The only if pan, i.e. that RT implies QLP, is proven as follows. Suppose that T < a, i.e.T|~-.a. If T ^ j J , i.e. T ^ 0, then 0 lies at the lowest level. So 0 < o. If T|~-./? wehave the following situation: av/?f~T, T(~->a and T\—>(3. Then, by RT we have av/?f~->a,or av/?|~->/?, or (when these two possibilities fail) we have both av(3\~a and av/?|~/?. So,/ ? < a o r a < / ? o r a and /? are f~-equivalents.

The if pan, i.e. that QLP implies RT, is proven as follows. By Proposition 5.2 it is enoughto show that (•- is represented by a quasi-linear model. The relation f~ is rational because < ismodular as remarked earlier. So, by the Theorem 6.3 of [2], f~ is generated by its associatedstandard model (cf. Remark 4.9) which is ranked. Moreover, this canonical model is quasi-linear suppose the canonical model is not quasi-linear, i.e. there are two different worlds, mand n, both in a non-minimal level. We want to show that the condition QLP does not hold.By standardness (the canonical model is standard), there are formulae a, /? (not necessarilydifferent) such that m \= C(a), n (= C(/?), m e min(a) and n € min(/3). But, m ^ nimplies that there is a formula 7 such that m ^ 7 and n |= -17. Put »i = OAJ and /?i =/?A-I7. It is clear that m £ min(ai) and n 6 min(/?i), so the minimal elements of a i and /?iare at the same level. Therefore ori £ fix and /?i ^ 01. But it is also clear that m £ min(/?i)so a i and /?i are not (^-equivalent. Thus to see that the property QLP does not hold for a iand P\ it is enough to observe that T < ai because the minimal elements of a are in a levelabove the lowest one. I

7 Some non-Horn rules incomparable with monotony

We consider some non-Horn rules that are stronger than rational monotony, but are not impliedby monotony and for this reason are perhaps less interesting than those we have consideredso far. We show how they may be characterized by certain subclasses of quasi-linear models.

DEFINITION 7.1

A preferential relation |~ is said to be completely determinated iff the following rule holds

CEM

In other words for any a and /?, a^0 or a^*->0.

This rule is called conditional excluded middle in Stalnaker [13] and also called full deter-minacy in Makinson [9].

REMARK 7.2

1. CEM=»DP.2. P + M A CEM.3. P + CEM^M.

PROOF. 1. This is immediate. Note that, as a consequence, by Proposition 3.2 (see diagram),CEM + P implies each of RM, DR, NR, RC, WD.


2. This is well known. Take, for instance, |~ to be the classical consequence relation. Thisrelation obviously satisfies P and M but does not satisfy CEM.3. Also well known. To recall: take the preferential structure with just two states, one less thanthe other. Every model on this structure satisfies P and CEM, whilst an appropriate model onit (e.g. the one used in the proof of Corollary 5.3) fails to satisfy M. I

DEFINITION 7.3

A preferential model (not necessarily injective) M — (S, i, x ) is said to be linear iff it isranked and has at most one state at each level, i.e. iff it is of the shape:

The following theorem can be extracted from work of Stalnaker and Lewis on the logic ofcounterfactual conditionals, but we give a direct verification here.

THEOREM 7.4

A preferential inference relation |- is completely determinated iff there exists a linear modelM such that \~M = (~.

PROOF. The if pan is evident. We prove the only if pan. Suppose that |~ is completely deter-minated. Then by Remark 7.2 and Proposition 3.2, ^ satisfies RM. By the Lehmann-Magidorrepresentation Theorem 2.6, f~ can be represented by a ranked preferential model, which byTheorem 4.6 we may suppose collapsed. To show that the model is linear, it suffices to showthat there are no two distinct states on the same level. Suppose for reductio that s, t are on thesame level and s ^ t. By parsimony there are formulae a, /? with s G min(a)andi G min(/?).By injectivity there is an elementary formula p with s G mod(p) and t £ mod(p). Thenclearly, using rankedness of the model, we have {s,t} C min(av/?), so av/?j/-p and av/?|^-ip,contrary to complete determination. I

DEFINITION 7.5

A preferential model is said to be almost linear iff it is ranked and has at most one state atany rank above the lowest and at most two states at the lowest level. In other words, iff it isquasi-linear and has at most two states in the lowest level.

One may wonder whether these models satisfy any interesting new rules. And if so, whetherwe can characterize those rules by almost linear models. Both answers are positive, as we shallnow show. We consider the following two rules

Fragmented Disjunction:

FD

that is, if Q|~/?V7 then either a]^0 or a\^j or ->P\~~f. Its 'dual' rule is

Fragmented Conjuction:

FC

that is, if aA/?f~7 then either a\~f or /?|~7 or a|~-i/?.


PROPOSITION 7.6

P + F D = » F C

PROOF. Suppose FC fails, i.e. that c*A/?f~7, QJ /T , 0\£J and a (/•-•/?. From the first we have(by S and RW) o|—>0vy and from the third (by LLE) we have ->-<0\fa. But these togetherwith the second and fourth show the failure of FD. IPROPOSITION 7.7

P+FC=>RM

PROOF. Suppose a\~0, aj^-17. We want to show that a*i\~0. We consider two cases: a\~jand a\/>y. In the first case we have a/\y\~0 by CM. Now consider the case a\/^f. By prefe-rentiality (REF and RW) we have

(aA^)A(aA7)(-aA/?A7. (7.1)

We cannot have a*0\~aA0Aj, otherwise by preferentiality (CUT and RW) we have af>-7 acontradiction. Thus

aA/?[âA/?A7. (7.2)

We cannot have a*0\—'(0^7). otherwise by preferentiality (RW) we would have ah0\~a -+-17 and again by preferentiality (CUT.REF.AND and RW) we have a|—17 a contradiction.Thus

c*A^-.(aA7). (7.3)

By FC, it follows from 7.1,7.2 and 7.3 that aA7âAÂ7 so by RW a*y[~0. I

PROPOSITION 7.8

P + F C = > F D

PROOF. Suppose that FD fails, i.e. a|~/?v7, a\^0, a\fa and ->0\^f. From the first two hy-potheses (by RM which holds by Proposition 7.7) we have aA-<@\~0v-f. By REF and RWwe have QA->0\—>0 so by AND and RW we have a*-i0^»y. We have also, from the second,a\/'-<-<0. But these last two together with the third and fourth hypotheses show the failure ofFC. I

From Propositions 7.6 and 7.8 we have immediately:

COROLLARY 7.9

Given P, FD o FC

PROPOSITION 7.10

If M is an almost linear model then \~M verifies FD and FC.

PROOF. By Corollary 7.9 it suffices to consider FD. Let M = (S, t, •<) be an almost linearmodel and put (- = f"M- Suppose that (~ does not verify FD, i.e. that a(~/?v7, o( / -â | /7 and-<0\/'j. From the last three, there are states «i, s? and S3 such that «i e min(o) D mod(-<^),«j G min(a) n mod(-i7), s3 e min(->/?) n mod(->7). Then «i and «2 are on the same leveland are distinct (for using a\~0vy we have «i G mod(7)), and so, by quasi-linearity, theyare on the lowest level. Since si 6 mod(->/?), necessarily min(-</9) is included in the lowestlevel. So 83 is also on the lowest level. Since a\~0vy we have also «a G mod(/?) and S3 Gmod(->a). Hence «i, «2 and S3 are mutually distinct states on the lowest level, contradictingalmost linearity. I


REMARK 7.11

Clearly, P + M does not imply FD; we need only note that classical consequence, which satis-fies monotony fails not only CEM as already observed in Remark 7.2 but also FD. The samealso follows from the fact that there are flat models (no states less than any other) that fail FD.For instance consider the model M = (5, t, -<) consisting of just three states «i, «2 and S3 allof the same and hence lowest level. Choose a, /? and 7 three distinct elementary propositionsandputt(si) |= OA-ipAj, 1(82) |= a\(i*-i~f and i(«a) ^ -ictA->/?A-i7. Then clearly FDfails. Actually we can put this observation in more general form:

PROPOSITION 7.12

If M is an injective quasi-linear model which is not almost linear then FD (and FC) fails.

PROOF. Assume that M = {S,», -<) is an injective quasi-linear model which is not almostlinear. Then there are three different states «i, «2 and S3 on the lowest level. Using injectivity,there are formulae c*i, 02, Q3 such that a, 6 mod(sj) iff i = j (for i,j = 1,2,3). Nowtrivially a iva 2 (- aiva2 so aiva2f~QiVQr2- But aiva^j/ 'ai (witness s2) and a iva2^*2(witness si), and ->ai \f*ai (witness S3), so that FD fails. I

We now compare the strength of the rules FD and FC with those implied by monotony thatwere studied in Section 3.

THEOREM 7.13

Given P we have CEM =>• FC, FD ^ DP, but neither converse holds.

Before proving the theorem, we combine the information that it contains with Corollary 7.9,Remark 7.11, and Proposition 3.2, to get the following diagram:

RM

WD

NR

As before, we verify the positive parts of the theorem syntactically and the negative partssemantically. We begin with the positive parts

PROPOSITION 7.14

PROOF. Suppose aA(3\~y but atyj and /?b^7; we want to show a\^->fi. Suppose for reductiothat aty-if). Then by CEM, a(~/3 and so by the first premiss using CUT, a\^/ contradictinthe second premiss.

gI

Note that we also have an easy semantical proof of this proposition using Theorem 7.4 andProposition 7.10.


PROPOSITION 7.15

P + FD=>DP

PROOF. We have already shown (Proposition 7.7 and Corollary 7.9) that P + FD => RM, andwe know from Theorem 3.2 that P + RM + RC =$> DP, so we need only show P + FD => RC.

Recall that by preferentiality, whenever 4>\~1. then <j>\~rl> so by CM </>AV>|~-L SO by S i>\~4> —»±so V'h'""̂

Assume P + FD. Suppose that RC docs not hold, i.e. a(~/?, -i0\f>-<a and -<P\f>a. Wewant to get a contradiction. By supraclassicality (i.e. the rule a h /? => a(~/3, derivablefrom the preferential postulates REF and RW) we have -i/?|~(-i/?A-ia)v(-i/?Aa). We havealso -</?1/-I/?A-IQ and -i/?[^-./?Aa. So by FD -.(->/?A->a)|~-'/?Aa. So by LLE ov/?|~aA->^.Thus avj?(va and av/?|—./?. So, by CM (av/?)Aah->/9. So,byLLEa|—•/?. But this togetherwith a|~/? implies (by AND) a|~_L By the fact recalled at the beginning of the proof, we get->/?^-ia, a contradiction as desired. I

PROPOSITION 7.16

P+FC96CEM

PROOF. Take an almost linear model of only one level, containing two states both of whichsatisfy a but just one of which satisfies /?. Clearly this fails CEM, but by Proposition 7.10 itsatisfies FC. I

PROPOSITION 7.17

P + D P ^ F D

PROOF. Immediate form Remark 7.11 and the fact that trivially M =>• DP. I

THEOREM 7.18

Let |~ be a preferential relation. Then (~ verifies FD (or FC) iff there exists an almost linearmodel M = (S, i, -<) such that |~ = (~x.

PROOF. The if part is Proposition 7.10. We prove the only if part. By Proposition 7.15 andTheorem 5.8, there exists a quasi-linear model M such that |~ = \~M • By collapsing, we canassume that M is injective. So by Proposition 7.12, M is almost linear. I

8 Some Horn rules between preferential inference and monotony

Up to now all the rules studied as potential additions to those of preferential inference, arenon-Horn. One may wonder if there are 'interesting' Horn rules beyond those of preferentialinference, but still weaker than monotony. One such rule may be called Conjunctive Insis-tence:

Cl

PROPOSITION 8.1

Monotony implies Cl but the converse does not hold even if we suppose CEM. The preferen-tial rules plus CEM do not imply Cl. Moreover Cl does not imply NR or WD.

PROOF. Clearly Cl is implied by monotony. To see that monotony is not implied by Cl evenwith CEM recall again the model in the proof of Corollary 5.3 and of Remark 7.2 (3). Weknow that this model satisfies P and CEM but fails M. We complete the proof by showingthat every model with this structure satisfies Cl.


Suppose that aA/?^7 in a model with this structure; we need to show that either a\^y or. By the hypothesis there is a state s G min(aA/?) satisfiying -17.

Case 1: Suppose s is of level zero. Then s G min(a) n min(/?) and so in fact we have both

Case 2: Suppose s is of level one. Since s G min(c*A/?) the unique state t -< s of level zerofails to satisfy at least one of a, /?. But in this case s G min(a) or s G min(/?) giving us inthis case atyy or fifty.

Cl is not implied by preferential rules plus CEM—consider for instance a ranked preferen-tial structure with just three states at three levels, with an appropriate distribution of truth val-ues as in the following figure:

81 : -ip, q, T

Clearly here p\~r, gj-r but p^q\^r. Note that this model is also linear, so that indeed P plusCEM does not imply Cl.

We prove now that Cl does not imply NR (which is enough to show that Cl does not implyany of DR, RM.RC, DP, FD, CEM). Consider the model defined by the following figure:

s2 : p , g , - r » • 53 : p,-iq,->r

s0 : p,-^q,r* • si : p,q,r

Clearly this fails NR, for p\~r whilst p^qtyr (witness S2) and PA-19 ̂ r (witness S3). However,it satisfies Cl. Suppose for reductio that a^y, P\~y but aA^y. From the last assumption,there is a s G min(aA/?) such that s \= -*y. Since 0(^7, 0\~y, s ( = a , s(=/3, 5 ^ 7 there areu,u' -< s with u G min(a), u' G min(/?). But there is at most one state less than «, so « = u'sou^= OA/? contradicting the minimality of s in mod(aA/3).

The same model shows that P -I- Cl £• WD; we have T^-i(pA-r), pA-r^g (witness S3),pA->r\fi-iq (witness s2). 1

We suspect that there are not 'very many' Horn rules which, like Cl, are implied by preferen-tial rules with monotony but are not implied by the preferential rules alone. There are some,however, of technical more than conceptual interest. Consider the infinite series of rules ofn-monotony (n > 1), n-M in short, constructed as follows:

1-M

2-M

and in general


n-M

where each <r, (0) is either <j> or -><£ according as i is odd or even, and noting that the conclusion-rule uses (rn rather than <rn+i- This rule is evidently reminiscent of the alternating sequenceof statements in the party example in Section 1.2 of Lewis [8]: 'If Otto had come, it wouldhave been a lively party; but if both Otto and Anna had come, it would have been a drearyparty; but if Waldo had come as well, it would have been lively; but...' Of course, that isan infinite list of conditional expressions, and not a list of Horn rules about them. The rulenumber n is a scheme, one of whose instances in effect takes the first n Lewis statements asits n premisses, and puts as conclusion a statement that is like Lewis' statement n + 1 but withopposite consequent.

Clearly, 1-M is plain monotony. Moreover we have the following:

OBSERVATION 8.2

1. For all n, P + n-M implies (n + 1)-M.2. For all n, P + (n -f 1)-M does not imply n-M, even if CEM is also assumed.3. P + Cl implies 2-M.4. P + 2-M does not imply Cl, even if FD is also assumed.

PROOF. Here we give only an outline.

1. Simply treat aiAaj as a single formula in the premisses of the rule of (n + 1)-M, relabelletters, and apply the rule of n-M to the last n premisses.

2. It is easy to see that (a) any ranked model with at most n levels satisfies n-M, (b) there isa linear preferential model with n + 1 levels that fails n-M. These two facts give the desiredresult, using Theorem 7.4.

3. Let h* be a preferential relation that fails 2-M; we want to show that it fails Cl. Sinceit fails 2-M, there are formulae a ! , a 2 , <*3,4> with cti\~4>, oiAa2 |~-i^, aiAajAaa^-x^. Weneed to find formulae /?, j , 6 with 0\~6, -j^6, Pîtyb. Put 0 = aiââz), j = ax AQ2,6 = -IOJV-I^.

To show that /3\~6, i.e. that aiA(->a2va3)(~-ia2v-><^ note that since O I A O ^ - ^ we haveaif^-iajv-i^, so since ai\~4> we have «i|—>a2 by preferential rules, so that by furtherpreferential rules, a\ |~-ia2va3 and alsac^ (•~->a2v-i^, so finally by the preferential rule CM,aiA(-ia2va3)|~->a2v-!0 as desired. To show that J\~6, i.e. that aiAQ2(~-ia2v-i^ simplyapply the preferential rule RW to the assumption QiAa2|—><j>. Finally, to show that I3^j^6,i.e. that QiA(-<a2va3)A(aiAa2)|/'->a2v-i(£, suppose the contrary and apply LLE to geta1Aa2AO3^—'a2v-i<£ so that aiAa2Aa3^-><^ contrary to hypothesis.

4. To prove this part, it suffices by Proposition 7.10 and 2(a) above, to find an almost linearmodel with two levels that fails Cl. Clearly the following model will do: the language is builtover the elementary formulae p,g,r ; the lowestlevel has two states «i ,s2 with »(«i) = {p,r},i(a2) = {q, r}, whilst the next level has one state only S3 with i(s3) = {p, q}, so p\~r, q\^r,

I

We note that since by Proposition 8.1, P + Cl ^ NR|WD, points 1 and 3 above tells us thatP + n-M 7^ NR|WD, whenever n > 1.

We may thus extend our diagram as follows:


• CEM

FD = FC

• WD

Contrasting with point 4 of Observation 8.2 we have:

OBSERVATION 8.3

P + 2-M + CEM=>CI

PROOF. It is possible to verify this semantically, but the following is a direct syntactic proof.Suppose P + 2-M + CEM. Let a, /?, 4> be formulae. Suppose a\~4>, /?|~<£; we want to showa A / ? ^ . We divide the argument into two cases.

Case 1. Suppose a v ^ a «-• <j> and av/?|~/3 <-• <f>. Since o(~< ,̂ 0\~4> we have by ORav/?|~</>, so by preferentiality, av/?f~aA/?. Now using CM we have (av/?)A(aA/?)|~</>, i.e.aA/?|~^ as desired.

Case 2. Suppose av/?|^a *-+ <j> or av/?fc£/? <-• <j>. We consider the former, the latter issimilar. By CEM, av/?|~-i(a •-+ (/>). But also since a\~4> we clearly have a(~a «-+ 4>< i-e.(av/3)Ao(^a *-> <j>. Hence by 2-M we have ((av/?)Aa)A/?|~a •-+ <j>, i.e. aA/?(~a *-> <j>, so

^ as desired. I

8.1 Semantics for n-monotony

Let M = (S, i, -<) be a preferential model. Define the height of M to be the maximal lengthof any chain of states si,...,sn with «i -< s2 -< < «n- If there is no maximal length, putthe height of the model to be oo. The following observations generalize points 2(a) and 2(b)in the proof of Observation 8.2.

OBSERVATION 8.4

Every preferential model of height < n satisfies n-M.

PROOF. Consider a preferential model that fails n-M. We show that it has height > n + 1.Since n-M fails in the model, there are formulae « i , . . .an, a n + i , <j> with

(1)i> (2)

(n)


where each <r, (<f>) is <j> or -K£ according as i is odd OT even. From (n + 1) there is a state a n + iwith «n + i 6 min(aiA . . . AOn+1) but t(«n+i) £ <rn(<£). But by (n), s n + i £ min(aiA . . . Aa n ) , so there is sn -< sn+i with sn 6 min(aiA . . . Aan) and t(«n) ^ crn{4>). Continuingdown like this we get a sequence sn+i y «„ >-•••>- «i of length n + 1 of states of themodel, so that its height is > n + 1. I

OBSERVATION 8.5

Every parsimonious ranked model of height > n fails n-M.

PROOF. Consider any parsimonious ranked model. Let [~ be the inference relation generatedby the model. Since the model is of height > n there is a sequence of states sx,..., sn, sn+iwith «i -< «j -<•••-< «n+i. By parsimony there exists a sequence of formulae 71, - -., fn+\with Si € min(7,) for i = 1 , . . . , n + 1. We define the formulae a, for t = 1 , . . . , n + 1 and4> as follows:

n+l 2i<n n+1

k=i k>0 i

Then we have:1. For every i = 1 , . . . , n and for every state u at the same level as «<, u ^ -17* for any

i = » + l , . . . , n + l.2. min(o,) = min(7,) for any » = 1 , . . . , n + 1.3 . h a i A O 2 A • • - A a , - <-> a,- f o r e v e r y »' = 1 , . . . , n + 1 .

The points 1 and 2 are easy consequences of rankedness and point 3 is evident by definitionof a,-. From points 1 and 2 is easy to see that a,^<T,(^) for any i = 1 , . . . , n + 1. Thus wehave

From this it is evident that to show that n-M fails it is enough to see that <*IA . . . AonAan+ibi<Tn(<^)- From points 2 and 3, ajA .. .AanAan + 1 is (^-consistent and since O^A .. .AanAan+i\^an+i(<f>), necessarily C*IA . . . AanAan+i|Mi(<£), as desired. I

THEOREM 8.6

Let f~ be any preferential inference relation. Then the following conditions are equivalent:1. h satisfies n-M and RM (resp. RT, FD).2. |~ is generated by some ranked (resp. quasi-linear, almost linear) preferential model of

height < n.

PROOF. For the implication 2 => 1, apply Observation 8.4 together with Theorem 2.6 (resp.5.2,7.10). For the implication 1 => 2, apply observation 8.5 together with Corollary 4.7 (plus5.6, 7.12 respectively). 1

Conjecture and open questions

We conclude with a conjecture and some open questions.


Conjecture: There are no Horn-rules with a single premiss which, given the preferentialrules, are implied by monotony, but do not imply monotony, and are not implied by the pre-ferential rules alone (recall that Cl has two premisses, and n-monotony has n premisses).

Open questions:1. Determine whether the non-implication P + WD ^ NR can be witnessed by injective

preferential models (cf. Observation 3.13)2. Determine whether the construction used to prove Lemma 4.2 can be adapted for a class

of preferential models broader than the ranked ones, e.g. to the class of all models that arefiltered in the sense of Freund [2] (cf. the discussion in Remark 4.9).

3. Find appropriate classes of preferential models to provide representation theorems for RC,WD, Cl (cf. the discussion in Remark 5.10).

Acknowledgements

We would like to thank Michael Freund for very penetrating comments on several versions ofthis work and for the unpublished observations cited as personal communications in the text.

References[1] H. Bezzazi and R. Pino Perez. Rational Transitivity and its models. In Proceedings of the 26th International

Symposium on Multiple-Valued Logic, pp. 160-165. IEEE Computer Society Press, 1996.[2] M. Freund. Injective nwdels and disjunctive relations, yourna/o/Logic a/u/Computo/ion, 3,231-247,1993.[3] M. Freund and D. Lehmann. On negation rationality. Journal of Logic and Computation, 6,1-7,1996.[4] W. L. Harper, R. Stalnaker and G. Pearce (eds). Ifs: Conditionals, Belief, Decision, Chance and Time. Reidel,

Dordrecht, 1980.[S] S. Kraus, D. Lehmann and M. Magidor. Nonmonotonic reasoning, preferential models and cumulative logics.

Artificial Intelligence,^, 167-207,1990.[6] D. Lehmann. What does a conditional knowledge base entail? In Proceedings of the First International Con-

ference on Principles of Knowledge Representation and Reasoning, R. Brachmann and H. L. Levesque, eds.Toronto, 1989.

[7] D. Lehmann, and M. Magidor. What does a conditional knowledge base entail? Artificial Intelligence, 55,1-60,1992.

[8] D. Lewis. Counterfactuals. Blackwell, Oxford, 1973.[9] D. Makinson. General patterns in nonmonotonic reasoning. In Handbook of Logic in Artificial Intelligence and

Logic Programming, \bL 3: Non Monotonic Reasoning and Uncertain Reasoning, D. Gabbay, C. G. Hoggerand J. A. Robinson, eds. pp. 35-110. Clarendon Press, Oxford, 1994.

[10] D. Nute. Conditional logic. In Handbook of Philosophical Logic, VoL 11, D. Gabbay tod F. GUnther, eds. pp.387-439. Reidel, Dordrecht, 1984.

[11] P. Peppas. Well behaved and multiple belief revision. In Proceedings of the 12th European Conference on Arti-ficial Intelligence, W. Wahlster ed. pp. 90-94. John Wiley and Sons, Budapest, 1996.

[12] K. Schlechta. Some completeness results for stoppered and ranked classical preferential models. Journal ofLogic and Computation, 6, 599-622,1996.

[ 13] R. C. Stalnaker. A theory of conditionals. In Studies in Logical Theory, N. Rescher, ed. Blackwell, Oxford, 1968.Reprinted in [4].

Received 12 July 1996

Beyond Rational Monotony: Some Strong Non-Horn Rules for Nonmonotonic Inference Relations

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