Counterfactuals and Updates as Inverse Modalities

Journal of Logic, Language, and Information 6: 123–146, 1997. 123c 1997 Kluwer Academic Publishers. Printed in the Netherlands.

Counterfactuals and Updates as Inverse Modalities

MARK RYANSchool of Computer Science, University of Birmingham, Birmingham B15 2TT, U.K.E-mail: [email protected], http://www.cs.bham.ac.uk/˜mdr

PIERRE-YVES SCHOBBENSInstitut d’Informatique, Facultes Universitaires de Namur, Rue Grandgagnage 21, 5000 Namur,BelgiumE-mail: [email protected], http://www.info.fundp.ac.be/˜pys

(Received 15 June 1996; in final form 8 September 1996)

Abstract. We point out a simple but hitherto ignored link between the theory of updates, the theory ofcounterfactuals, and classical modal logic: update is a classical existential modality, counterfactualis a classical universal modality, and the accessibility relations corresponding to these modalitiesare inverses. The Ramsey Rule (often thought esoteric) is simply an axiomatisation of this inverserelationship.

We use this fact to translate between rules for updates and rules for counterfactuals. Thus, Katsunoand Mendelzon’s postulates U1–U8 are translated into counterfactual rules C1–C8 (Table VII), andmany of the familiar counterfactual rules are translated into rules for updates (Table VIII). Ourconclusions are summarised in Table V.

From known properties of inverse modalities we deduce that not all rules for updates may betranslated into rules for counterfactuals, and vice versa. We present a syntactic condition which issufficient to guarantee that a translation from update to counterfactual (or vice versa) is possible.

Key words: Updates, counterfactuals, conditional logic, belief revision, inverse modalities, multi-modal logic, correspondence theory, Kripke semantics

1. Introduction

Background. An intuitive connection between theory change and counterfactualswas observed by F.P. Ramsey (1931), who proposed what has become known asthe Ramsey Rule:

To find out whether the counterfactual “if A were true, then B would be true”is satisfied in a state S, change the state S minimally to include A, and testwhether B is satisfied in the resulting state.

(Actually, Ramsey proposed the rule only for non-counterfactual conditionals, butthe term “Ramsey Rule” is now taken to refer to counterfactuals too.)

It was initially hoped that the AGM theory of belief revision (Gardenfors, 1988;Makinson, 1985) would provide the right notion of minimal change. However,the intuitively acceptable AGM postulates for belief revision are known to beincompatible with the Ramsey Rule (Gardenfors, 1986, 1988).

124 M. RYAN AND P.-Y. SCHOBBENS

It turns out that the theory of updates proposed by Katsuno and Mendelzon(1992) is compatible with the Ramsey Rule (Grahne, 1991). Updates, like revisions,are a formalisation of theory change; but whereas revisions are intended to modelchanging beliefs about a fixed world, updates are intended to model a changingworld. The difference between the formalisations of updates and revisions can beseen in terms of postulates; for example, the AGM postulate

A � B = A ^B if A ^B is consistent

is accepted for revisions, but rejected for updates. The difference can also be seenin terms of operations on models; in revision, we measure the distance to themodels of the old theory as a whole, while in update we measure the distanceto them pointwise. The intuition behind this crucial difference (expanded upon inSection 3.1) is that in updates we want to change total states of the world, whereasin revisions we change partial descriptions of it. Though they were discovered morerecently, updates are conceptually simpler than revisions, because of this pointwisecharacter.

Our contribution. We capitalise on the realisation that updates are the rightnotion of theory change for the Ramsey rule. We show that the standard treatmentsof updates (e.g., Katsuno and Mendelzon, 1992) is a system of multi-modal logic.Furthermore, this modal logic bears a particular relationship with the modal logicof counterfactuals (Stalnaker, 1968; Lewis, 1973; Pollock, 1981): it has the inverseaccessibility relation. It is therefore appropriate to present a single modal logic,with positive and negative modalities; the positive modalities (used for updates)refer to the accessibility relationR, and the negative ones (used for counterfactuals)refer to the relation R�1.

The Ramsey rule turns out to be an axiomatisation of the inverse relationshipbetween the two sets of modalities.

We work out the correspondence properties of the accessibility relation for thestandard rules for updates and counterfactuals (Tables IV and VI), and we providea systematic translation of rules for updates into rules for counterfactuals, andconversely (Theorem 14). The paper extends two previous papers (Ryan et al., 1996;Ryan and Schobbens, 1996).

Structure. The paper is arranged as follows. Section 2 contains preliminariesabout modal logic and (inverse) accessibility relations. In Section 3, we show thatupdates and conditionals are systems of multi-modal logic, and that they haveinverse accessibility relations. In Section 4, we show that this is equivalent to theRamsey rule, and we translate the standard rules for update into conditional logicrules, and vice versa. Conclusions are in Section 5.

COUNTERFACTUALS AND UPDATES AS INVERSE MODALITIES 125

2. Preliminaries

This section sets our notation and provides background technical details aboutthe version of multi-modal logic we will use, and about inverse modalities. Themodally-expert reader can skim it quite fast.

2.1. MULTI-MODAL LOGIC

We assume a propositional language L with atomic propositions p, q, r, : : : andconnectives ^;_;:;!;$;2;3;�;3. The unary connectives bind most tightly,then^;_, and!;$ bind least tightly. The modal connectives take two arguments;if A;B are formulas then so are 2AB, 3AB, �AB, 3AB. (The formula 3AB willlater be read as the result of updating B by A; the formula �AB will be read asthe counterfactual “if A were the case,B would be the case”. The other two modalformulas are their duals.) The set L is the set of atomic formulas p, q, r, : : : ; theset L is the set of all formulas over L.

The semantics of multi-modal logic with inverses is given as follows. A modelM = hW;R; V i of the multi-modal language L is a set W of worlds, an accessi-bility relation R � P(W )�W �W and a valuation V : L! P(W ).

The relation of satisfaction between a modelM = hW;R; V i, a world x 2W

and a formula A is defined inductively on A as follows.

x M p iff x 2 V (p)

x M :A iff x 6 M A

x M A ^B iff x M A and x M B

x M 2AB iff for each y 2W , RjAj(x; y) implies y M B

x M �AB iff for each y 2W , RjAj(y; x) implies y M B

The missing connectives_,!,$,3 ,3 are defined by similar (standard) clauses.As can be seen,�;3 are like2;3 except that they refer to the inverse accessibilityrelation.

In the context of a model M , jAj is defined to be fx 2 W j x M Ag. Thesubscript on M will usually be dropped in order to make the notation lighter.

The model M satisfies the formula A, written M A, if x M A for eachx 2 W . A frame F = hW;Ri consists of a set of worlds and an accessibilityrelation. Such a frame F satisfies A, written F A, if for each valuation V , wehave hW;R; V i A. A formula A is valid, written j= A, if it is satisfied by everyframe. A formula A is satisfiable in a model M if jAj 6= ;. The formula A over Lis complete w.r.t. M = hW;R; V i if there is precisely one x 2 W with x A. IfA1; A2; : : : ; An; B are formulas, the rule

A1 A2 : : : An

B


Table I. Modalities, their inverses and duals.

Modality Inverse Dual Inverse dual

2B �B 3B 3B

henceforth B up to now, B eventually B once upon a time,B

tomorrow B yesterday B tomorrow B yesterday B

any execution ofprogram P in thecurrent state resultsin a state satisfyingB

if P has just beenexecuted, the start-ing state satisfied B

there is an execu-tion of program Pin the current statewhich results in astate satisfying B

P could have justbeen executed in astate satisfying B

holds in a frame F if: for all V , if M Ai for each i then M B, whereM = hF; V i. Notice that this is weaker than asserting the axiomA1^: : :^An ! B,but stronger than asserting “if eachAi is valid, thenB is valid”. The double-barred

ruleA=

Bholds in F if for all M = hF; V i, M A iff M B.

This multi-modal logic is not the straightforward one discussed in Goldblatt(1987) and Popkorn (1994), because the indices are formulas. However, our modal-ities are not quite the binary modalities found in Venema (1991), because they treatthe two arguments A and B in quite different ways (and, for example, :2AB isequivalent to 3A:B, but not to 3:A:B as in Venema (1991)). Nevertheless, ourdefinitions also seem natural, and as will be seen, they are right for our application.

2.2. INVERSE MODALITIES

Inverse modalities have already been used in modal logics: in linear temporallogic, they are the past modalities. Table I summarises their intuitive meaning.These inverse modalities should not be confused with the dual modalities, nor withthe inverse of the dual (which is of course dual of the inverse).

Not all modalities have intuitively interesting inverses. Temporal and dynamicmodalities do, but epistemic and doxastic modalities do not. If one interprets 2Bas “I believe B”, then �B seems to say: “in all situations where my beliefs admitthe current situation, B is true”.

In later sections, we discover that the counterfactual modality has an interestinginverse: its inverse is (the dual of) update.


2.2.1. Axiomatising Inverse Modalities

How can we axiomatise the link between 2 and �? There are two ways. The firstis to add a pair of axioms (whose monomodal versions are already known fromtemporal logic (Goldblatt, 1987: ex. 6.1)). The second way is the Ramsey Rule.

THEOREM 1 (Folklore). Assume2;3 are interpreted by the accessibility relationR and �;3 are interpreted by S in a frame F .

The following are equivalent:

1. For each T �W , RT = S�1T .

2. F satisfies the following axiom schemes (each of which is also given in itsdual form):

(1) B ! 2A3AB 3A �A B ! B

(2) B ! �A3AB 3A2AB ! B:

3. F satisfies the following rule (which is also given in its dual form):

B ! �AC

3AB ! CRR

3AC ! B

C ! 2ABRRd:

(The proof of this and all other theorems is given in the Appendix.)Observe that with the reading we will give to � and 3 , the rule RR exactly

expresses the Ramsey rule for counterfactuals. �AC is read as “if A were true,then C would be true”; 3AB is read as the update of B by A; so the rule statesthat the counterfactual “if A, C” is supported in a state B iff the state obtained byupdatingB with A supportsC . Notice that formulas are used to denote both piecesof information and (descriptions of) states.

[Aside. Those knowledgeable about category theory may like to see the rule ofTheorem 1(3) as an adjunction between pre-orders. The pre-order in question isP(W ) (given a fixed model M = hW;R; V i) ordered by inclusion. For any fixedA, �A and 3A may be considered as the following monotonic operators on thepreorder:

3AS = fx 2W j 9y 2 S RjAj(x; y)g

�AS = fy 2W j 8x 2W (RjAj(x; y) ) x 2 S)g

and we have S � �AT iff 3AS � T .]

2.3. OTHER PRELIMINARIES

If (X;�) is a pre-ordered set (i.e., � is a reflexive and transitive ordering on X)and Y � X , then Min�(Y ) is the set of �-minimals in Y , i.e., Min�(Y ) = fy 2

Y j 8x 2 Y: x 6< yg.


If R is a relation in W �W , then R(a) = fb 2 W j R(a; b)g and R�1(b) =

fa 2W j R(a; b)g.

3. Updates and Counterfactuals

Our aim in this section is to show that updates and conditionals are systems of modallogic, having inverse accessibility relations to each other. In Section 3.1 we clarifythe difference between “update” and its cousin “revision”. In Section 3.2 we showthat updates are existential modalities. This enables us to look at correspondenceproperties, in the style of (van Benthem, 1984). In Section 3.3 we recall the knownresult that counterfactuals are universal modalities, and see that the accessibilityrelation is the inverse of the one for updates. We recall some known correspondencetheory, and use it to relate rules for update with rules for counterfactuals.

3.1. UPDATES VS. REVISIONS

This paper concerns the notion of “update”. It is related to the notion of beliefrevision, and shares some properties. Indeed, there has been a historical confusionbetween the two notions. They differ in motivation, however, and they have someimportant technical differences. Before focussing on updates, we briefly recall thedifferences between update and revision.

The need for different notions was pointed out in (Keller and Winslett Wilkins,1985). Updates are intended to model a world which changes, while revisionsare intended to model a static world, about which one’s information changes. In(Katsuno and Mendelzon, 1992) new postulates for updates were proposed. Thesepostulates are similar to those for revisions (Gardenfors, 1988) and are presentedin Table II. In the last column, we have indicated the name used in revision.

Katsuno and Mendelzon use � as an infix operator; q � p means q updated byp. It may appear surprising that the two arguments of � are the same syntactic type(namely, formulas), since one usually expects states to be updated by formulas.The explanation is that q is a formula denoting a set of states, and q � p is theformula denoting the resulting set of states after each of them has been updated byp.

Comparing the update postulates and the standard AGM postulates (Gardenfors,1988), one sees that updates and revisions have much in common, explaining thehistorical confusion. The fundamental difference between the two notions may beseen by looking at the representation theorems associated with each of them, ratherthan the postulates. [In the following two theorems, jAj is the set of valuationsmaking A true; thus, KM and AGM work on specific frames where valuations arein bijection with worlds.]

THEOREM 2 (Gardenfors, 1988). � is a revision operator iff

jB � Aj = Min�jBj(jAj)


Table II. Update postulates according to Katsuno and Mendelzon (1992), using their syntax. Theyuse � as an infix operator; q � p means q updated by p.

Name Postulate Name(Katsuno and (Gardenfors, 1988)Mendelzon, 1992)

U1 q � p! p K*2U2.1 q ! p implies q ! q � p

U2.2 q ! p implies q � p! q K*4wU3 q � p satisfiable, if p; q satisfiable � K*5U4.1 q $ r implies q � p$ r � p

U4.2 q $ r implies p � q $ p � r K*6U5 (q � r) ^ p! q � (r ^ p) K*7U6 q � p! r; q � r! p imply q � p$ q � r

U7 q complete implies (q � p) ^ (q � r)! q � (p _ r)

U8 (q _ r) � p$ (q � p) _ (r � p)

where�jBj �W �W is a preorder of closeness to B (x �S z means that x is atleast as close to the set of worlds S as z is).

THEOREM 3 (Katsuno and Mendelson, 1992). � is an update operator iff

jB � Aj =[

y2jBj

Min�y(jAj)

where �y � W �W is a preorder of closeness to y (x �y z means that x is atleast as close to the world y as z is).

Thus, while revisions measure closeness to jBj as a whole, updates measure close-ness to each element of jBj individually. This is the reason for which, in Katsunoand Mendelzon (1991), updates are referred to as pointwise revisions. The justifi-cation for the pointwise character can be explained in terms of the changing-worldmotivation behind updates. Suppose the world is in a state satisfying p_q. Supposenow a change is made, whose result guarantees :p. Belief revision would tell usthat the resulting state must satisfy :p ^ q, which is equivalent to the conjunctionof these formulas; it says that whenever the conjunction is consistent, then it is theresult of the revision. Updates, however, argue that if all we had was p _ q, thenthe world either satisfied p or it satisfied q. Considering these cases independently,we see that after :p is imposed the world satisfies :p in the first case, or :p^ q inthe second. As we do not know which of the two cases it was, all we can be sureabout now is that it satisfies the disjunction, namely :p.


Table III. Update postulates rewritten as modal logic rules.

Name Rewritten as

(Katsuno and Mendelzon, 1992)

U1 3AB ! A

U2.1B ! A

B ! 3AB

U2.2B ! A

3AB ! B

U3:3AB

:Bif A satisfiable

U4.1B $ C

3AB $ 3AC

U4.2B $ C

3BA$ 3CA

U5 3AB ^ C ! 3A^CB

U63AB ! C 3CB ! A

3AB $ 3CB

U7 B complete implies 3AB ^3CB ! 3A_CB

U8 3A(B _ C)$ 3AB _ 3AC

3.2. UPDATES ARE EXISTENTIAL MODALITIES

Suppose we write 3AB instead of B � A. Looking at the representation theoremfor updates given above, it is easy to see that it can be written

x 3AB iff there exists y s.t. RjAj(x; y) and y B

in the multi-modal model M = hW;R; V i, where W is the set of valuations andthe relation R is given by

RS(x; y) , x 2 Min�y(S):

This fact shows that updates are an existential modality, and justifies the decision towrite 3AB instead of B � A. We can therefore think of Katsuno and Mendelzon’sU1-U8 as multi-modal axioms and rules. They are presented as such in Table III.


Table IV. Correspondence conditions for the update rules.

Name Property of R

R1 RS(x; y) implies x 2 S

R2.1 y 2 S implies RS(y; y)

R2.2 y 2 S and RS(x; y) imply x = y

R3 S 6= ; implies 8y9x:RS(x; y)

R5 x 2 S and RT (x; y) imply RS\T (x; y)

R6 R�1S (y) � T and R�1

T (y) � S imply R�1S (y) = R�1

T (y).

R7 RS \ RT � RS[T

Katsuno and Mendelzon’s theory of updates may be seen as a particular multi-modal logic, the one generated by the axioms and rules of Table III.

To guarantee the classical properties of a modality, we should also have neces-sitation:

B

2AB

THEOREM 4. Necessitation follows from the axioms and rules in Table III.

A corollary of our observation that update is an existential modality is that theKatsuno and Mendelzon theory represents one particular logic in a hierarchy,whose intuitive base level is weaker, being a minimal normal modal logic. Onecould consider stronger or weaker logics, according to applications.

Some of the rules in Table III, namely U4.1, U4.2 and U8, are automaticallyvalid, simply by virtue of the modal semantics:

THEOREM 5. The rules U4.1, U4.2 and U8 hold in any frame.

The other rules are valid if we suitably constrain the accessibility relation. Asusual within the framework of modal logic, we can study the “correspondenceproperties” on R imposed by an axiom or rule.

THEOREM 6. A rule in Table III holds in a frame F = hW;Ri iff R has thecorresponding property stated in Table IV.

Compare correspondence theorems for standard modal logic, e.g. Popkorn (1994:x5.2), Goldblatt (1987: Theorems 1.12, 1.13), and van Benthem (1984). The proofs(given in the Appendix) follow the usual pattern in correspondence theory. In the( direction, we add to the frame hW;Ri an arbitrary valuation V to form themodel M = hW;R; V i, and show that the constraint on R is enough to guaranteethat the rule is satisfied in M . In the ) direction, we make a judicious choice of


the valuation and an instance of the scheme, to show that the constraint on R musthold.

Notice that these conditions are second-order, and (unlike the case for mono-modal logic) none of them can be reduced to first order conditions. This is becausethe R is always indexed by a set.

Note that the “pointwise character” of updates is crucial in their ability to berepresented as an existential modality: it implies to the distribution of 3 over _(U8). Revisions are not existential modalities in this sense. (Revisions may howeverbe analysed as modalities in a rather different sense, for example de Rijke (1994)and van Benthem (1991).)

3.3. COUNTERFACTUALS ARE UNIVERSAL MODALITIES

According to Stalnaker (1968), Lewis (1973), and Pollock (1981), the counter-factual “if A was the case, then B would be the case” may be interpreted by:“In all closest worlds satisfying A, we find that B holds”. It is well-known thatcounterfactuals have the properties of classical universal modalities (Chellas, 1980;Lewis, 1973). The counterfactual “if A was the case, then B would be the case”holds at a world x if B holds in all y in Min�x jAj. But this relation between x andy is simply the inverse of the relation RjAj given at the beginning of Section 3.2. Socounterfactuals and updates are inverses (at the level of accessibility relations), andduals (since updates are existential modalities, while counterfactuals are universal).In terms of our logic, the counterfactual sentence “if A was the case, thenB wouldbe the case” can be written �AB.

Slogan: Counterfactuals are the inverse dual of updates.Updates are the inverse dual of counterfactuals.

One can perform the same analysis as we did for updates, namely, the corre-spondence theory for standard rules for counterfactuals. Much of this is known, buttends to be scattered in the literature, and often the semantic conditions are not putinto one-to-one correspondence with the axiom schemes. We have collated resultsfrom many sources in Tables V and VI.

These tables show that several authors have re-discovered axiom schemes underdifferent names, including ourselves! They also show the correspondence betweensome update rules and counterfactual rules, obtained by noticing that they corre-spond to the same semantic condition on the relation. This is the topic of the nextsection.

Note that we are choosing to work with the Kripke accessibility relationR, whilemany authors cited in Table V use the selection function f . Of course, the relationbetween them is trivial, and which one is used is merely a matter of convention orpersonal preference. RjAj(x; y) expresses that x is closest among jAj to y, whilef(A; y) selects exactly those x’s. We therefore have x 2 f(A; y) iff RjAj(x; y),and may write f(A; y) = R�1

jAj(y).


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Table VI. Correspondence conditions for counterfactuals.

Counterfactual rule Condition on accessibility relation

C1 �AA R1 RS(x; y) implies x 2 S

C2.1 �AB ! (A! B) R2.1 y 2 S implies RS(y; y)

C2.2 A ^B ! �AB R2.2 y 2 S and RS(x; y) imply x = y

C3 :�A ? if A satisfiable R3 S 6= ; implies 8y9x:RS(x; y)

C5 �C^AB ! �C(A! B) R5 x 2 S and RT (x; y) imply RS\T (x; y)

C6 �AC ^�CA!(�AD $ �CD)

R6 R�1S (y) � T and R�1

T (y) � S

imply R�1S (y) = R�1

T (y)

C7 B complete implies�A_C:B ! �A:B _�C:B

R7 RS \RT � RS[T

CCS3 �A?! �C:A CS3 R�1S (y) = ; ! R�1

T (y) \ S = ;

CV �AB ^ :(�A:C)! �A^CB CS5 R�1T (y) \ S = ; _ R�1

T\S(y) � R�1T (y)

A3 �AB ^�AC ! �A^BC RA3 R�1S (y) � T ! R�1

S\T (y) � R�1S (y)

SDA �A_A0B ! �AB cm0 S � T ! RS � RT

CA �AB ^�A0B ! �A_A0B cc0 RS[T � RS [RT

CN0 �?A cn0 R; = ;

MOD �:AA! �BA RMOD if R�1:S(y) � S;R�1

T (y) � S

CEM �AB _�A:B RCEM R�1S is a (partial) function

Tr �AB ^�BC ! �AC RTr R�1S (y) � T ! R�1

S (y) � R�1T (y)

Contr �AB ! �:B:A RContr if R�1S (y) � T;R�1

WnT(y) � S

t3 �AB ! (�AC $ �A^BC) Rt3 if R�1S (y) � T;R�1

S (y) = R�1S\T (y)

t4 �AB ! (�BC $ �A_BC) Rt4 if R�1S (y) � T;R�1

T (y) = R�1S[T (y)

B2 �AB ! �A_C(B _ C) RB2 if R�1S (y) � T;R�1

S[U(y) � T [ U

B4B ! A;A ^ C ! B

�AC ! �BCRB4 S \ U � T � S and R�1

S (y) � U

imply R�1T (y) � U

D0 �A_B:A!�A_C:A _�C_B:C

RD0 if R�1S[T (y) \ S = ; then

R�1S[U(y)\S = ; orR�1

T[U (y)\U = ;

D00 �A_BA _�A_BB RD00 RCEM and R1

D0 :�> ? RD0 R�1W (y) 6= ;

A1 �A^:B(A! B)!(�:AA! �:BB)

RA1 if R�1TnS

(y) \ T n S = ; and

R�1S (y) \ S = ; then R�1

T (y) \ T = ;

A2 �A^:B(A! B)! �AB RA2 R�1SnT

(y) \ S n T = ; ! R�1S (y) � T

RT �A^BC ^�AB ! �AC RRT R�1S (y) � T ! R�1

S (y) � R�1S\T (y)

Triv �AB ! �A^:B(A! B) RTriv if R�1S (y) \ T = ; then

R�1S\T (y) \ (S \ T ) = ;


As a convention, we will use the leftmost names from the table in the remainderof the paper. Note that our assumption that counterfactuals are normal modalitiesrenders some postulates in the table (e.g., ID, RI in the first row) equivalent, thoughthese are sometimes distinguished in the literature.

4. Inter-Translating Systems for Counterfactuals and Updates

4.1. VIA THE CORRESPONDENCE CONDITIONS

We have observed that rules for updates correspond to particular properties of theaccessibility relation R, and similarly for counterfactuals, whose rules correspondto properties of the inverse relation R�1. This gives us a criterion for identifying aparticular rule for updates and a rule for counterfactuals.

EXAMPLE 7. The update axiom scheme U1, 3AB ! A, corresponds to thesemantic conditionRS(x; y) implies x 2 S (Theorem 6). The counterfactual axiomscheme �AA known as ID corresponds to the same semantic condition:

F �AA , 8V; x; y (R�1jAj

(x; y) ) y A)

, 8x; y (R�1S (x; y) ) y 2 S)

, 8x; y (RS(x; y) ) x 2 S)

Indeed, U1 and ID intuitively say the same thing.

Using the criterion, we can look for a counterfactual rule which corresponds to anupdate rule, or vice versa.

EXAMPLE 8. The update rule U2.1

B ! A

B ! 3AB

corresponds to the semantic condition y 2 S implies RS(y; y). As it happens, thissemantic condition is equivalent to the same condition on R�1, so it correspondsto the counterfactual rule

B ! A

B ! 3AB

This rule may be more simply stated as �AB ! (A ! B) (known as MP in theliterature) [for proof, see Appendix].

EXAMPLE 9. U6 , C6. Let us first work out the correspondence condition R6for U6.


F satisfies3AB ! C 3CB ! A

3AB $ 3CBU6 (1)

,

[

b B

R�1jAj

(b) � jCj[

b B

R�1jCj

(b) � jAj

[

b B

R�1jAj

(b) =

[

b B

R�1jCj

(b), all V (2)

,R�1jAj

(b) � jCj R�1jCj

(b) � jAj

R�1jAj

(b) = R�1jCj

(b), all V (3)

,R�1S (b) � T R�1

T (b) � S

R�1S (b) = R�1

T (b)R6 (4)

From 1 to 2: recall F satisfying the rule means that anyM = hF; V i satisfying thetop also satisfies the bottom; recall that M satisfies an implication if the worldssatisfying the antecedent are contained in those satisfying the consequent; and theworlds satisfying3AB are

Sb B R

�1jAj

(b).From 2 to 3: The direction 2 ) 3 is by taking the special case jBj = fbg. For

the other direction, we use the fact that for any sets Si; T , we haveSi2I Si � T iff

for each i, Si 2 T . From 3 to 4: The direction 4 ) 3 is immediate, since, given V ,jAj, jBj are just particular sets. For the reverse direction, if we are given S; T wepick atomic A;C and choose V such that V (A) = S, V (C) = T .

Now we must find a corresponding update rule. Let us “guess” that it is C6 :=�AC ^ �CA ! (�AD $ �CD), and verify that it corresponds to the samecondition on R.

x �AC ^ �CA! (�AD $ �CD)

, (R�1jAj

(x; y) ) y C) ^ (R�1jCj

(x; y) ) y A) )

([R�1jAj

(x; y) ) y D] , [R�1jCj

(x; y) ) y D]) (2)

, R�1jAj

(x) � jCj ^R�1jCj

(x) � jAj ) R�1jAj

(x) = R�1jCj

(x) (3)

3 comes from 2 because D is arbitrary. So U6 and C6 both correspond to R6.

Experience can make the guessing easier! However, in the next section we describea more deterministic way of doing the translation.

THEOREM 10. The update rules U1-U8 translate to the counterfactual rules givenin Table VII.


Table VII. Counterfactual rules corresponding to the update rules U1–U8.

Name Counterfactual rule Name

(Nute, 1984: x12)

C1 �AA ID

C2.1 �AB ! (A! B) MP

C2.2 A ^ B ! �AB CS

C3 :�A ? if A satisfiable

C5 �CÂB ! �C(A! B)

C6 �AC ^�CA! (�AD $ �CD) CSO

C7 B complete implies�A_C:B ! �A:B _�C:B

Notice that this modal logic perspective gives us a spectrum of counterfactuallogics (any selection of the rules defines a logic), and also a spectrum of update log-ics; and these spectra are put into one-one correspondence by the set of constraintsthey imply on R.

In the opposite direction, we have taken standard counterfactual rules in theliterature, and worked out the corresponding update rules in Table VIII.

4.2. VIA THE RAMSEY RULE

The proof of the equivalence between update and counterfactual rules can beperformed� either by going via the accessibility relation R, as in the examples above;� or by working directly with the axioms of Theorem 1(2); or, equivalently, the

Ramsey Rule (or its dual) in Theorem 1(3).Here we do it working directly with the axioms and rules.

EXAMPLE 11. U1 , C1. Using the Ramsey rule, U1: 3AB ! A translatesimmediately into B ! �AA. But this can be further simplified, to �AA. [Note:B ! �AA and�AA are not logically equivalent, but they are “frame” equivalent,as was shown in Example 7. This means they are equivalent axiom schemes.]

EXAMPLE 12. U5 , C5.

‘)’

3C �CÂ B Â! 3CÂ �CÂ B U53C �CÂ B ! (A! 3CÂ �CÂ B) equiv.3C �CÂ B ! (A! B) by (1)


Table VIII. Update rules corresponding to some counterfactual rules.

Counterfactual rule Update rule

CCS3 U CS3:3AB

:(3CB Â)

A3 U A33AC ! B

3A^BC ! 3AC

SDA U SDA 3AB ! 3A_A0B

CA UA 3A_A0B ! 3AB _3A0B

CN0 UN0 :3?>

MOD U MOD3:AB ! A

3CB ! A

Tr U Tr3AD! B

3AD ! 3BD

Contr U Contr3AC ! :B

3BC ! :A

t3 Ut33AC ! B

3AC $ 3A^BC

t4 Ut43AC ! B

3BC $ 3A _ BC

B2 UB2 3A_BC ! B _ 3AC

B4 UB4B ! A;A ^ C ! B;3AD ! C

3BD ! C

D0 UD0:3>A

:A

A1 UA13A^:BC Â! B;3:AC ! A

3:BC ! B

A2 UA23A^:BC Â! B

3AC ! B

RT URT3AC ! B

3AC ! 3A^BC

Triv UTriv3AC ! B

3A^:BC Â! B


�C3C �C^A B ! �C(A! B) Nec,K�C^AB ! �C(A! B) by (2)

‘(’

�A^C3A^CB ! �A(C ! 3A^CB) C5B ! �A(C ! 3A^CB) by (2)3AB ! (C ! 3A^CB) by RR3AB ^ C ! 3A^CB equiv.

Proving the equivalence of update rules and counterfactual rules by the Ramsey rulecan be mechanised to some extent. We present syntactic criteria on counterfactualrules which allow them to be translated to update rules, and conversely, using theRamsey Rule.

DEFINITION 13. We say a formula is “�-simple” if it is formed from arbitrarynon-modal formulas and the operators � and ^, where all subscripts of � arenon-modal.

A rule of inference is �-simple if:

– its premises are implications whose antecedent is non-modal and whose con-sequent is �-simple; and

– its conclusion is an implication whose antecedent and consequent are�-simple.

Dually,3-simple formulas are formed from non-modal formulas and the oper-ators 3 and _, where all subscripts of 3 are non-modal. A rule is 3-simple if:

– its premises are implications whose antecedent is 3-simple and whose conse-quent is non-modal; and

– its conclusion is an implication whose antecedent and consequent are3-simple.

2-simple, 3-simple formulas and rules are defined similarly.

THEOREM 14. Any�-simple counterfactual inference rule can be translated intoa 3-simple update inference rule (and conversely) using the Ramsey Rule.

The proof is an algorithm performing the translation. For readability, we treat �-simple rules; the three other cases are similar. We first massage the conclusionfor application of the Ramsey Rule, by introducing a fresh meta-variable, say X ,and replacing the conclusion S1 ! S2 by the equivalent rule: X!S1

X!S2. Then we

iterate the following steps for each premise and the conclusion, now all of the formN ! S, where N does not contain �:


� if S contains no �, we are done;� if S has an outermost�, we use the Ramsey rule to replaceN ! �MT by the

equivalent3MN ! T ;� if S has an outermost ^, we distribute N ! S1 ^ S2 into N ! S1, N ! S2.

We eventually obtain an equivalent3-simple rule, which can sometimes be furthersimplified.

Note that the conditions of Theorem 14 are quite liberal, since:

� an axiom is a rule with no premises� any formula can be regarded as an implication with antecedent>� it allows nesting of modalities, though not in subscripts.

This freedom is more than is needed, since classical rules contain no modalnesting.

EXAMPLE 15. All of the counterfactual rules of Table VI are �-simple, exceptCV, CEM, D0, D00, C3, C7.

EXAMPLE 16. Let us see how this works in the case of counterfactual axiom Trfrom Table VIII.

�AB ^ �BC ! �AC ()X ! �AB X ! �BC

X ! �AC

()3AX ! B 3BX ! C

3AX ! C

()3AX ! B

3AX ! 3BX

The rules C3 and C7 are simple if we ignore the side conditions; and this is safe ifwe factor them out of the proof. This is done for C3 in the proof of Theorem 10 inthe Appendix.

For the remaining rules that are not simple, namely CV, CEM, D0, D00 onepossibility would be to follow Katsuno and Mendelzon by allowing premises aboutwhether formulas are complete or satisfiable (cf. U3, U7). Then the counterpartscan be written

B compl. 3AB ^ C sat.UV

3A^CB ! 3AB

B compl.UEM

3AB compl.

D compl. 3A_BD ! :A 3A_CD ^A sat.UD0

3B_CD ! :C

UD00 is simply UEM+U1.


5. Conclusions

The link between counterfactual and updates, often considered as esoteric, is onlythe usual link between a relation and its inverse; counterfactuals can be consideredas a universal modality, and update as its inverse existential modality.

Update rules are thereby translated into counterfactual rules, and vice versa.We found that some, but not all, of the counterfactual counterparts to U1-U8 areknown in the counterfactual literature; and, symmetrically, the translation fromcounterfactuals to updates gives us some known and some new rules in that field.

Theorem 14 has proved very powerful, but probably does not cover all the caseswhen translation is possible. We would like to have an “iff” characterisation. It issomewhat fortuitous that it was possible to do all the translations we performed,since it is known in temporal logic that there are easy examples of axioms using the“future” modalities for which there is no equivalent using “past” modalities. Onesuch is 2(A^2A! B)_2(B ^2B ! A), whose correspondence condition isforward-linearity:R(x; y)^R(x; z) ) R(y; z)_(y = z)_R(z; y). This conditioncannot be expressed using �;3; to prove this, we show it is not preserved underp-morphisms of frames by exhibiting two appropriate frames.

Acknowledgements

We are grateful to Odinaldo Rodrigues, who co-authored an early draft of this paper(Ryan et al., 1996). We are also grateful to Johan van Benthem for his continuinginterest and his invitation to contribute to the Special Issue.

Thanks also to the referees for ModelAge’96 Workshop, TARK’96, AAAI’96,ECAI’96, Natasha Alechina, Sofia Guerra, Andreas Herzig for their commentson various drafts of this paper. We also acknowledge Esprit for partial fundingunder the WG ModelAge (8319). The first author also acknowledges the NuffieldFoundation for partial support.

Appendix

A. Proofs of Theorems

THEOREM 1. Assume 2;3 are interpreted by the accessibility relation R and�;3 are interpreted by S in a frame F .

The following are equivalent:

1. For each T �W , RT = S�1T .

2. F satisfies the following axiom schemes (each of which is also given in its dualform):

(1) B ! 2A3AB 3A �A B ! B

(2) B ! �A3AB 3A2AB ! B:


3. F satisfies the following rule (which is also given in its dual form):

B ! �AC======== RR3AB ! C

3AC ! B======== RRdC ! 2AB

:

Proof. Let hW; (S;R)i be a frame, whereR is the accessibility relation for2;3 ,and S is the relation for �;3.

(1 ) 2) is straightforward.For (2 ) 1), suppose ST (x; y); choose the valuation V s.t. V (p) = fxg and

V (q) = T ; thenx p, so by axiom (2)x �q3qp so y 3qp so9z;RT (y; z)^z

p: But by V , z = x, so RT (y; x). So ST � R�1T . The converse inclusion is similar,

but uses axiom (1).(2 ) 3.) The axiom scheme (1) implies downward direction of the rule:

B ! �AC by hyp

3AB ! 3A �A C by K, MP

3AB ! C by (1) dual form

The proof that the axiom scheme (2) implies the upward direction of the rule issimilar.

(3 ) 2.) The upward direction of the rule implies the axiom (1):

3AB ! 3AB

B ! 2A3AB RRd

and the downward direction similarly implies (2).

THEOREM 4. Necessitation follows from the axioms and rules in Table III.Proof.

B

:B $ ?U4.1

3A:B $ 3A?

? ! ?U2.2

3A? ! ?

3A? $ ?

3A:B $ ?

:3A:B

THEOREM 5. The rules U4.1, U4.2 and U8 hold in any frame.Proof. Take any frame F , and any valuation V . Let M = hF; V i.

U4.1. Suppose M B $ C and x 3AB. Then there exists y with RjAj(x; y)and y B. But also, y C , so x 3AC . The other half is similar.

U4.2. Suppose M B $ C and x 3BA. Then there exists y with RjBj(x; y)and y A. But also, RjCj(x; y), so x 3CA. The other half is similar.


U8. x 3A(B _ C) iff 9yRjAj(x; y); y B _ C iff 9y1 RjAj(x; y1); y1 B or9y2 RjAj(x; y2); y2 C iff x 3AB _3AC .

THEOREM 6. A rule in Table III holds in a frame F = hW;Ri iff R has thecorresponding property stated in Table IV.

Proof.

U1 , R1. “(”. Let V be any valuation, and let M = hW;R; V i. Suppose x M3AB. Let S = jAj, and take a y such that RS(x; y) and y B. By R1, x 2 S,so x M A.

“)”. SupposeRS(x; y) in the frame hW;Ri; pick V such that V (p) = S andV (q) = fyg. Since y q, we have x 3pq; so x p by U1, and x 2 S bydef. of V .

U2.1 , R2.1. “(”. Suppose M B ! A and x B. Then x A. We havex 3AB iff 9y s.t. y B, and RjAj(x; y), which is true if we set y = x.

“)”. Suppose y 2 S; we prove that RS(y; y). Let V be such that V (p) = S

andV (q) = fyg. It follows thatM q ! p, and therefore, by U2.1,M 3pq.Since y q, we get y 3pq. Therefore,9z q s.t.RS(y; z). But V (q) = fyg.Thus, z must be y, and therefore RS(y; y).

Another style of proof is also possible, and is more immediate once you seehow it works. The equivalence between U1 and R1 may be seen as follows. LetF = hW;Ri be a frame.

F 3AB ! A, all A;B U1 (1),Sb B R

�1jAj

(b) � jAj, all A;B; V (2)

,Sy2T R

�1S (y) � S, all S; T (3)

, R�1S (y) � S, all S; y (4)

, RS(x; y) implies x 2 S, all S; x; y (5)

Notice the quantification is carefully stated: A and B are “local” to each line, andline 2 works for all valuations V . This quantification on V allows us to pass toline 3: in the ) direction, whatever S; T are we can pick V such that V (p) = S

and V (q) = T , for example; while in the ( direction, it is immediate by settingS = jAj and T = jBj. The passage from line 3 to 4 again works because S; T arelocally quantified.

U2.2 , R2.2. “(”. Suppose M B ! A and x 3AB. Take y such thatRjAj(x; y) and y B. Since M B ! A, y 2 jAj; so by R2.2, x = y; sox B.

“)”. Suppose y 2 S andRS(x; y). Let V be such that V (p) = S and V (q) =

fyg. It follows thatM q ! p, and therefore, by U2.2,M 3pq ! q. Sincey q, x 3pq, so x q, so x = y (by def. of V ).


U3 , R3. “(”. Take any valuation such that A is satisfiable in M . Assume M

:3AB. Take any y; we will show y :B. By R3, existsx such thatRjAj(x; y).Since M :3AB, then x :3AB, so we have y :B.

“)”. Suppose S 6= ; and y is given. Take V such that V (p) = S andV (q) = fyg. So p is satisfiable, and M 6 :q, since y q. Therefore, by U3,M 6 :3pq, so exists x such that x 3pq. Then there is a z with RS(x; z) andz q. But by choice of V , z = y; so RS(x; y).

U5 , R5. “(”. Suppose x 3AB ^C . Since x 3AB, 9y B, RjAj(x; y). ByR5, RjAj\jCj(x; y) and thus x 3A^CB.

“)”. Suppose x 2 S andRT (x; y). Pick V such that V (p) = S, V (q) = fyg,and V (r) = T . Then y q and RT (x; y) imply that x 3rq. Since x p, byU5 x 3r^pq. Therefore, 9y0 s.t y0 q and RS\T (x; y

0). But y0 must be y,

since V (q) = fyg, so RS\T (x; y).

U6 , R6. See Example 9 in the text.

U7 , R7

B complete implies 3AB ^3CB ! 3A_CB U7 (1)

, B complete impliesSb B R

�1jAj

(b) \Sb B R

�1jCj

(b) �Sb B R

�1jAj[jCj

(b) (2)

, R�1jAj

(b) \R�1jCj

(b) � R�1jAj[jCj

(b) (3)

, R�1S (y) \R�1

T (y) � R�1S[T (y) (4)

, RS \RT � RS[T (5)

2 ) 3 by setting B = fbg. 3 ) 2 by completeness of B! 3 ) 4 because wecan choose arbitrary valuations; 4 ) 3 by setting S; T to be jAj; jCj.

EXAMPLE 8. (Last part.) The counterfactual rule

B ! A

B ! 3AB

may be more simply stated as �AB ! (A! B) (known as MP in the literature).


Proof. From left to right:

A ^ :B ! Ahyp.

A ^ :B ! 3A(A ^ :B)

prop. of diamondA ^ :B ! 3A:B

equiv.�AB ! (A! B)

and right to left:

B ! A

B ! A ^B

�A:B ! (A! :B) [hyp.]equiv.

A ^B ! 3AB

B ! 3AB

THEOREM 10. The update rules U1-U8 translate to the counterfactual rules givenin Table VII.

Proof. The proofs for U1, U2.1 and U6 are given in Examples 7, 8, 9, respec-tively. The proof for U5 is given using the method of Section 4.2 in Example 12.The other proofs may be constructed similarly. For C3 and C7, we factor out theside condition. For example, for C3, assume that A is satisfiable; we show the

equivalence between:3AB

:B(U3) and : �A ? (C3).

3A> ! 3A>RRd

> ! 2A3A>eq.

:3A �A ?U3

: �A ?

3AB ! ? hyp.RR

B ! �A? �A ?! ? C3.

B ! ?

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