counterfactual reasoning by (means of) defaults

Annals of Mathematics and Artificial Intelligence 9(1993)345-360 345

Counterfactual reasoning by (means of) defaults

J.-J.Ch. Meyer

Free University, Amsterdam~University of Nijmegen, The Netherlands

and

W. van der Hock

Free University, Amsterdam, The Netherlands

Abstract

We show how defaults can be used for counterfactual reasoning. We use a framework of modal l og ic to reason about both defaults and counterfactua]s, in which o n e can

express certainties, possibilities, actualities and (preferred or practical) beliefs in a distinct manner. Firstly, we discuss some properties of our approach in relation to other approaches in the literature.

O. Introduction

Counterfactual reasoning, reasoning "against the facts", has been studied extensively in philosophical literature [29,13,28,23]. This is not surprising since traditionally philosophers have always been interested in (more or less ideal) worlds different from our actual (much less ideal) world, where other propositions hold. However, also in (the philosophy of) the exact sciences, counterfactuals play an important role in describing and applying laws of nature. In fact, every such law can be used in an argument that employs counterfactuals. For example, let us consider the law of gravity. This law can be applied in an academic or engineering argument as follows: "If this building would not be sufficiently secured, it would collapse".

Recently, however, it has been realised [6] that counterfactuals are also important in the area of artificial intelligence. It has uses in questions of data base maintenance, in planning, in diagnosis and in natural language understanding, to mention a few applications (cf. [6]). For instance, if a robot has to plan lines of actions, it needs to examine possibilities which are generally not true at the moment of examination. It should, so to speak, have the ability to perform "experiments of thought". Similar experiments of thought are needed in the other applications mentioned above.

Traditionally, counterfactuals are (a kind of) implications p > q, with intended meaning "if p were true, q would be (have been) true as well", with the underlying presumption that p is not actually true at the moment of utterance. This makes this

�9 LC. Baltzer AG, Science Publishers

346 J.-J.Ch. Meyer. W. van der Hoek, Counterfactual reasoning

kind of reasoning go beyond classical propositional and predicate logic, in which the paradox of material implication holds: false =~ q for any q. Thus, classical logic is obviously not sufficiently sophisticated to treat counterfactuals. In philosophy, counterfactuals are treated by means of so-called conditional logic, which is based upon a more refined (conditional) implication involving worlds that satisfy the counterfactual premise but are "as similar to" actual world as is possible. This is presented in a very abstract setting (cf. [29, 13,23]).

Recently in AI research, however, much effort has been given to the study of a class of reasoning methods (or logics) which is motivated from an entirely different point of view, viz. logics in which defaults may occur (see, e.g. [24,25]). In these logics - generally referred to as nonmonotonic logics - we can express that some formula holds if it is not explicitly known otherwise. Well-known nonmonotonic logics are Reiter's default logic [24], McCarthy's circumscription [15], McDermott and Doyle's nonmonotonic logic (NML, cf. [16, 17]), and Moore's autoepistemic logic (AEL, cf. [22]). Perhaps somewhat surprisingly, a posteriori it has been realised that these logics are related to conditional logic. For instance, Shoham [27] and a fortiori Delgrande [4] have proposed approaches to "implement" nonmonotonic logics by means of conditional logic.

In this paper, we view the relationship the other way around, so to speak, and investigate counterfactuals from the perspective of default reasoning (or nonmonotonic logic, more in general), implementing the "as similar to" clause in the semantic definition of counterfactuals by means of defaults: by default, the hypothetical worlds under investigation are "the same as" the actual one. To be able to express this, we need a logical language in which we can express defaults as well as actualities.

To illustrate the similarity between counterfactuals and defaults on a more concrete level, let us consider (informally) the phenomenon of ambiguity.

Counterfactual reasoning, like default reasoning, is not always unambiguous in the sense that there is a unique conclusion if one assumes a premise that is not actually true. Consider a "classical" example from [26]:

(la) I f Julius Caesar had been a lion, there would have been a lion without a tail.

This statement sounds correct. However, we can also state:

(lb) I f Julius Caesar had been a lion, there would have been a Roman general with a tail.

Although a little more peculiar, (Ib) seems to be correct as well. Whether one asserts (la) or (lb) depends on the "frame of mind" one sticks to. In the case of (la), one considers Caesar - without tail - as a "declared lion"; in the case of (lb), one imagines Caesar to be an actual lion including the property of having a tail. In the first case, the image of Caesar (and the fact that generals have no tails) dominates, in the second the image of a lion (including the fact that lions have tails) does. Moreover, the conclusions of the counterfactual implications (la) and (lb) are inconsistent.

J.-J.Ch. Meyer, W. van der Hoek, Counterfactual reasoning 347

This example of an ambiguous counterfactual implication is similar to examples of default reasoning, where ambiguous conclusions may also occur. A well-known example is that of the Nixon Diamond, of which a compact version reads as follows:

(2) Republicans are typically non-pacifists;

quakers are typically pacifists;

Nixon is both a quaker and a republican;

Is Nixon a pacifist?

As with the Caesar example, there are two mutually inconsistent answers to this question, depending on which "frame of mind", a typical quaker or a typical republican, dominates.

In this paper, we want to investigate this similarity of counterfactuals and defaults in some more depth, using a logic, in which both counterfactual and default reasoning can be represented. In particular, we show how counterfactual reasoning resembles (or, rather, involves a kind of) default reasoning. In order to avoid misunderstandings, this paper does not present new results pertaining to default logics, but rather shows the use of such logics, and in particular the one we have developed in [19,20], for reasoning about counterfactuals. We concentrate on the default-like nature of counterfactuals, which is reflected in the formal treatment by the use of expressions that are related to those that are used to express defaults in our framework. In fact, as we shall explain, our whole approach to counterfactuals is inspired by default reasoning.

Our paper is organized as follows. First, in section 1, we re-introduce our logic S5AP of certainties, possibilities, actualities, and preferred beliefs. This logic was first introduced as a logic for default reasoning in [19,20]. In section 2, we briefly discuss how to represent defaults in this logic, mainly in order to let the reader appreciate the similarity with our approach to counterfactuals rather than aiming at a complete presentation of a new default logic. (The interested reader is referred to [19,20], which such a treatment is given.) Next, in section 3, we discuss how to represent counterfactuals in this framework.

We claim that:

�9 We thus obtain a "less abstract" semantics for counterfactuals as compared to the traditional approach initiated by Stalnaker [29] (cf. also [13]). In some sense, the counterfactual implication is too coarse. In our setting, it is possible to differentiate between "frames of mind" when reasoning about counterfactuals.

�9 A simple S5-1ike modal logic, with all its desirable properties, is sufficient to provide a framework that is adequate for both default and counterfactual reasoning. This is important in view of the design of AI reasoning systems: consider a model of such a system in which several kinds of reasoning modes and mechanisms are performed by different modules. Now a module that performs counterfactual reasoning can use the same logical language as

348 J.-J.Ch. Meyer, W. van der Hoek, Counterfactual reasoning

one that performs default reasoning. Indeed, both kinds of reasoning can in our approach be done in the same logical language by one and the same module.

We thus provide further evidence of the similarities between defaults (nonmonotonic logic) and counterfactuals (conditional logic) by implementing the latter by the former. We expect that this result is not particular for our treatment of defaults, and that it should also be possible to express counterfactuals by means of, for example, Reiter's default logic [24]. It would be interesting to investigate how far our ideas an also be pushed using other default logics as well. For the purpose of this paper, however, we feel more comfortable in the present setting, since in this setting one has a common language to express both defaults and counterfactuals with a clear and simple modal semantics, so that we are able to show our results in a transparent way without having to bother about translations of expressions from the one logical framework to the other.

1. The underlying logic

In [19,20], we introduced an epistemic logic to treat defaults. To do so, we took an S5-1ike modal language extended with a special modal operator P denoting preference or practical belief. (The logic resembles autoepistemic logic (AEL) of [22], but has a much simpler interpretation.)

Assume a fixed collection P of primitive propositions. The language L we shall use is the minimal set of formulas closed under the following rules:

(i) L ~ P.

(ii) if ~p, V E L, then --1 tp, ~p A V, tp v V, ~ D V, tp - V ~ L.

(iii) if q~ ~ L, then Lq~, Mr Atp, P i9 ~ L (i = 1 . . . . . n).

Informally, Lop is read as "~p is certain", Mtp as "tp is (considered) possible", Aq) as "~ is actually true (true in the real world)", and Piq~ as "tp is preferred or a practical belief (within frame of reference i)". As we shall see below, a frame of reference (or mind) refers to a preferred subset of the set of a priori possible worlds that together constitute a body of knowledge or, rather, preferred or practical belief. Formally, expressions of L are interpreted by Kripke structures of the form (5"., ~r, p, Et . . . . . En, ~o, ~o 1 . . . . . ~On), where Z is a collection of possible worlds, ~t: P -+ Z --r {t, f} is a truth assignment to the primitive propositions per world, p ~ Z is the real or actual world, X;i ~ Z are sets (clusters) of preferred worlds, ~o = 2; x Z (and ~ is hence reflexive, transitive and Euclidean), and gai = Z x Zi ~ is transitive and Euclidean. Thus, we have that Vst : ~oist r t ~ Zi. We let SSAP denote the collection of Kripke structures of the above form (the A in SSAP referring to actualities, and the P in SSAP referring to preferences).


Let S5AI ~ denote the collection of pairs ( • , s), with M = (E, tr, p, E1 . . . . . E., to, to t . . . . . ton) e S5AP and s E Z. Elements of S5AP § are referred to as states.

The formal interpretation of the language now reads:

for M = (Z, a', p, Z1 . . . . . Zn, to, to 1 . . . . . ton),

(M, s) ~ p iff tr(p)(s) = L

(M, s) ~ ~ q~ iff not (M, s) ~ ~,

([~, s) ~ q~ A tF iff (M, s) ~ ~p and ([~, s) ~ V,

(M, s) ~ tp v V iff (M, s) ~ tp or (M, s) ~ V,

(l~, s) ~ q~ D V iff (M, s) ~ tp implies (M, s) ~ V, (U, s) ~ ~O - 1V iff [ (~ , s) ~ tp iff (~ , s) ~ ~], (m, s) ~Lq~ iff (k~, t) ~ tp for all t E X,

(1~, s) ~Mcp iff (M, 0 ~ ~P for some t ~ Z,

(l~, s) ~Aq~ iff (1~,/9) ~ q~,

(M,s)~Picpif f (VO, 0 ~ p f o r all t ~Y-i, and

~ q~ iff (M, s) ~ tp for all s ~ E, as usual.

It is possible to axiomatise (the theory of) S5AP as follows (cf. [19,20]): take t l~ $5 system for the modali ty L (and dual M) and use K45 for the P-modali t ies , together with the relating axioms:

LcP D Pitp, Leit~=- Pi~, ~Pi false D (Pi Pjcp= Pjq~), ",Pi false D (Pi Lfp =- Lip);

futhermore, for the A-modality we employ

A(q~ ^ V) - A ~0 ̂ A V, A ---, (p --- - , A(p,

L q ~ A,p,

Aq~ D LAq~,

AP/q~ - P/(p,

--, e~ false ~ (P~Aq~ --- A~o).

Intuitively, these extra axioms express that the tol are subrelations of the relation to, that they reach the same set Z~ no matter where one starts from in Z (including the actual world p), or from another nonempty frame; that to always refers to the whole set Z no matter where one starts in some nonempty frame; that the actual


world p is unique in E, and that with the modality A one always points at p no matter where one starts in Z or in some nonempty subframe (cf. [ 19, 20] for a more technical justification of these correspondences).

2. Defaults

In the language L, we express defaults of the form ~ : ~/X (using Reiter's notation) as

r A MIg D P Z.

The reading of such a formula is "if cp is true and ~ is (considered) possible, then Z is preferred". When V= Z (where " = " stands for syntactical equality), the default is called normal.

Multiple defaults { r : ~ / Z i } i are represented by formulas

q~i A MIll i ~ Pizi.

In [19,20] and [18], we have shown several examples of how to use this formalism for defaults, including the nontrivial Yale Shooting Problem [8] as well as a discussion of problems that are present in other approaches and that can be avoided in this setting (such as the problems with reasoning by cases in default logic and those with transitivity and contraposition in autoepistemic logic). Furthermore, the semantics of the logic [19, 20] is considerably simpler than that of NML, default logic, AEL and circumscription: it uses neither fixed-point constructions as in NML, default logic and AEL, nor higher-order logic as in circumscription. As compared to the other modal approaches such as NML and AEL, and more recent variants (e.g. [ 14]), it is distinct in the respect that special modal operators (the Pi-operators) are used to distinguish assertions that hold by default from assertions that are certain, which solves a number of problems (cf. [19, 20]). Even more importantly, the logic is "as monotonic as possible": the part of the logic that deals with reasoning about preferred assertions is completely monotonic; the only part that is nonmonotonic is that where the possible assertions of the form M~g are derived from the facts that are known, which is based on Halpern and Moses' idea of epistemic states [7] and bears a resemblance to Levesque's logic [12] of "All I know". Since we do not need details for the present purposes, we will not elaborate on this here any further. The interested reader is referred to [21].

Here, we only treat the Nixon Diamond by way of example to get an idea of how defaults are represented in our setting: let r stand for being a republican, p for being a pacifist, and q for being a quaker. Now we can represent the Nixon Diamond as

(1) r A M --,p D Pl - 'P

(2) q AMp ~ P2 P

(3) r A q

(republicans are non-pacifists by default),

(quakers are pacifists by default),

(Nixon is both a republican and a quaker).

J.oJ.Ch. Meyer, IV. van der Hoek, Counterfactual reasoning 351

Given the consistency of both p and --,p, i.e. Mp ^ M ~ p , we can now derive that P1 -',P ^ P2P. That is to say, we have two subframes El and E2 of E: in the one --,p holds, in the other p. This is intuitively correct since there is in this particular case no preference of the one over the other whatsoever. Note also the need for different modalities P1 and P2 (or frames XI and Y-,2) in this case; the use of only one such modality results in an empty frame. In more complicated cases, it may be adequate to put an ordering on the various subframes (cf. [19,20]). As we stated in the introduction, this section is not meant as a complete introduction in our approach to default logic. The reader interested in this should consult [ 19, 20], where also the raison d'etre of this new default logic next to other approaches such as Reiter's default logic [24] and Moore's autoepistemetic logic [22] is discussed in more detail. However, we have included this section to become acquainted with the use of Pi-modalities and to be able to appreciate the resemblance of counterfactuals with defauls, or rather the very idea of dealing with counterfactuals that is inspired by default reasoning.

3. Counterfactuais

3.1. TRYING TO CAPTURE COUNTERFACTUAL REASONING IN S5AP

In this section, we shall discuss by trial and error how to represent counterfactual implications in the setting of S5AP. Here, we use the idea that counterfactuals possess a default nature. In order to arrive at a satisfactory treatment, we start with some experiments trying to capture the exact nature of a counterfactual in our setting. As we go along, we discover several deficiencies that we repair the the next proposal.

Consider the formula

eo(tp A ~ Po ~ IV D Pllg). (*)

This formula expresses that it holds within frame 5:-0 that if q~ holds and ~ is not inconsistent with what is believed in frame Y-,o, qt is preferred within the subframe Yl ~Eo. To this end, we assume further that it holds that Poq~DPoPltp. (By the way, in view of the axioms of S5AP it would be sufficient to use P~ tp instead of Po P1 tp in the above formula and those in the sequel, but we rather employ P0 P1 tp to emphasize that El is contained in the contextual frame Eo.) In order to obtain a better understanding of (*), it is useful to think of the frame Y-,o associated with the modality Po as a collection of imaginative worlds in which we perform our experiments of thought and in which the "counterfacts" in the premises of the concerned counterfactual hold. This is even better seen in a direct consequence of (*):

/'o 0 ^ --,/'o -'1 /'0PLY/. (**)

Here we see in the premise "Potp": "suppose that we have a collection of worlds that satisfy tp". Furthermore, observe the consistency check "--,Po ~ ~ ' in both (*)


and (**), which checks whether the conclusion v tha t holds in the preferred frame Zt is consistent with what is known already in the frame of reference Z0. This is, of course, resembling our treatment of defaults in the previous section, where we encountered similar considerations with respect to the set E of all a priori possible worlds rather than the set Zo of worlds considered in our present thought experiment: we may interpret (*) as "considering a frame of reference in which tp holds, lFholds by default". In fact, the formula (*) is a direct extension of the formula we used for (normal) defaults (substitute L for P0 to obtain our translation of a default rule) and is already a reminder of the counterfactual implication tp > V.

There are, however, a number of problems if we use (*) as a direct translation.

(1) The first problem is that if we take (*) as a representation of q~> V, we obtain, due to the consistency check, that ~p > false, which is obviously incorrect.

(2) Secondly, where in (*) is the connection with the "real" world? We would still like our thought experiments to keep in touch with reality and stay "as similar to" the actual world as possible. (A thought experiment of a completely deranged person with an imagination that has no bearing on reality is easy: just create some wild frame of mind ,v_,0 and reason with the associated modality Po. Here, no relation with the actual world is necessary or even desirable. This is perhaps an explanation of the success of the program PARRY, simulating a psychiatric patient with paranoia [9].)

Problem (2) is solved by using the actuality modality A: When creating a frame ,T,o, we have to take into account as many aspects of the actual world as possible, that is to say, we have to include all literals (i.e. all formulas that are either primitive propositions or negations of these) that hold in the actual world unless we explicitly know otherwise. Clearly, this reminds us of defaults again (and is a kind of law o f inertia that one also encounters in the frame problem in AI, cf. [1]). We could say that in some hypothetical frame ,v_o, a literal vthat is actually true is taken over (i.e. again taken to be true) by default. We can express this idea by the formula (for literals V)

AIF A ~ PO ~ IF D PoPil F (for some i), (A)

which states that if the literal V holds in the actual world and V is not inconsistent within the frame Y--o, then we assume that V holds within some (not required every) subframe Xi of Y-,o. The latter condition allows for several possibilities (like the extensions in default logic), as is best illustrated by an example. We consider the Caesar-lion example again:

EXAMPLE

We take the following propositional constants: g stands for "being a Roman general", E stands for "being a lion", and t for "having a tail". (In fact, this example is best represented in predicate logic, but we consider here a propositional approximation.)


1,

2.

3.

~

5.

6.

7.

Ag

A - , [

Po[

g A-- ,PotD PoPl ~ t

[ A ~Po " , t Z Po P2t

Ag A ~Po ~ g D Po P3g

A -~ I A -~ Po l D Po P4 ~ I

(Actually, Caesar is a Roman general.)

(Actually, Caesar is not a lion.)

(We now hypothesize that Caesar is a lion, creating a frame Eo in which this holds.)

(Normally, Roman generals do not have tails.)

(Normally, lions do have tails.)

(application of (A)).

(application of (A)).

It is interesting to see what we can now derive under the assumption that

(a) ~Po ~ g A ~Po ~ t A ~Pot,

which states that the consistency checks in 4, 5 and 6 are satisfied. In particular, it says that in frame Y,o t is contingent, i.e. there are possible worlds both with t and --, t. In some sense, this merely says that it is possible to organise within Y,o a subframe with t and one with -1 t.

First we note that, by 3, the application of 7 is blocked. This means that we cannot derive that P0/'4 ~ l, which is clearly desirable in this situation. Next, we obtain

8. L(fA ~Po ", t D Po P2t) (from 5, applying necessitation).

9. P o ( f A ~ P o ~ t ~ P o P 2 t ) (from 8, and LtPDPotp).

10. P o f A - - , P o ~ t D P o P 2 t (from 9, by modal reasoning).

11. PoP2t (from 10, 3 and (a)).

Moreover, we also have that

12. Po P3g (from 1, 6 and (a));

and furthermore that

13. P o P 3 g A ~ P o t D P o P l ~ t (from 4 and modal reasoning, of. 8-10).

14. PoP1 --,t

Finally, we obtain

15. POP1[

16. P0 PI(fA --,t)

(from 12, 13 and (a)).

(from 3 and the fact that El is a subframe of Y-o).

(from 14 and 15, by modal reasoning).

The formulas 14 and 11 state that there are two subframes El, 7-,2 within the frame Y,o such that: in the former, Caesar has no tail and (by 16) is a lion without a tail, and, in the latter, Caesar has a tail. This is exactly what is to be expected in this situation. Note, however, that from the above representation it cannot be inferred


that in the frame 7_,2 Caesar is a Roman general with a tail, since although both and E3 are subframes of 7-,0, they need not coincide, or even have some overlap. If one would like to represent the property that 7.,2 and E3 coincide, one should replace /)3 by P2 in 6.

3.2. COUNTERFACTUALS IN S5AP

We are now in a position to solve problem (1) we encountered above. Note that (A) is not yet intended to represent the counterfactual implication itself, as (*) and (**) were. (A) only provides a necessary background for such an implication. So now we continue our search for an adequate representation of ~p > V. If we look at the Caesar example, we should be able to say that in this case both E> t and E> --1 t (even E> Ea--,t). On the other hand, it is clear that these assertions do not hold when implemented according to just either (*) or (**), viz. Po(E^ ~Po ~ IlID Pi IV) (some i) and PoE^ "-,Po-1 V ~ PoPi u/(some i) for V= t or V= ~ t (where i may differ for the cases ~ = t and ~=-- , t).

What we do have, however, is more of an inferential nature, using a background theory. This background theory will at least contain the scheme (A). We use ep ~ r for the meta-statement "from ~p, ~ can be inferred w.r.t, background theory T" (in S5AP using the appropriate consistency requirements, which must hold as well). I"r stands for a nonmonotonic inference relation with background theory T. So ~p ~ r ~ implies that the inference actually can take place (or may "fire" as we shall call it) as far as the consistency (within T) is concerned, which excludes the problematic case q~ I"r false. In this case, firing is blocked since -',Po --, false - false.

We now define

T ~-q~ >o,i 1V as PoOP ~"T PoP/Y//-

TO put it more precisely, we have that T}--~O>o,i ~ r [ T u {Poq~} I " ~ P o ~ ' & Tu{Poq~} &PoPilp'], where I" is some nonmonotonic inference relation with which we may infer the consistency of ~ (within the frame 7.o) with the theory T u {PooP} as well as the consistency checks needed to apply instances of the scheme (A) (cf. [21]). Furthermore, we define

and finally T I- ~o >0 ~ as T J- ~p >o,i ~ for some i,

T [- ~p > tp, as T [-- r >o V for some frame Eo. ( * * *)

(When the theory is understood, and no confusion can arise, we omit the prefix "T [--".) Now, the theory Tcaesar consisting of 1 to 7, together with the assumption (a), indeed yields Tc,esar I-" [>0,t ~ t and Tcaesar }" E>o,2 t, and so/'Caesar I-- f>0 ~ t and Tcaesar I'- E>o t.

Given a collection {E/Ii e l } of subframes of 7-,0, one can also imagine a stronger notion of counterfactual T ~ ~0 >>o,t IV defined as T ~ r >o,i V for all i E I.


This expresses that the consequent ~ must hold in all subframes X;i c_ Eo with i ~ I. This is similar to Lewis' notion of counterfactual [ 13], and also to cautious approaches with respect to extensions in default logic [24]. With respect to the Caesar example above, where l = { 1, 2 }, we can now neither conclude Tcaesar [-"/" ">0d t nor Tcae~r 1"-[~'0,1 "nl, since neither in 5". 1 nor in 7-,2 do the worlds satisfy both t and -~t. We can also imagine that sometimes we need an ordering on frames El, i E I, like in [19,20] to indicate that some frames are more relevant (or preferred) than other ones.

EXAMPLE

We next consider a problem discussed by Ginsberg in [6], viz. the issue of nonmonotonicity of counterfactual implication. As our approach is derived from our approach to nonmonotonic reasoning [19, 20], one may expect that this issue is also captured correctly in our setting. The example of Ginsberg reads as follows: consider the following counterfactual assertions:

i. p > d

ii. p A e > --,d

iii. p A e A n > d

(''If the power had not failed, diner would have been on time").

("If the power had not failed, but I had been elected president, diner would have been late")�9

("If the power had not failed, and I had been elected president, but nobody had bothered to tell me, diner would have been on time").

As Ginsberg argues correctly, this example - which is intuitively completely sound and consistent - demonstrates the nonmonotonicity of > in an evident manner. We now show how this example works out in our framework. We translate i- i i i :

1. Po p b Po t'l d

n . Po(P A e) b 1"o

m . Po(P A e A n) b PO t'3 d

(under the consistency assumption --,Po --, d).

(under the consistency assumption --,Po d).

(under the consistency assumption --,P0 -'1 d).

We can rewrite that as

i*. P o p ~ . P o e l d

11. P o p A P o e F P o P z ~ d

m . P o p A P o e A P o n F P o P 3 d

(under the consistency assumption --,Po ~ d).

(under the consistency assumption ~Po d).

(under the consistency assumption --,P0 ~ d).

Under the assumption that L(e D ~d) , or sufficiently Po(e D--,d), i* and n are consistent, since this implies Poe D Po --,d, which together with Poe yields P0 - ,d ,

�9 �9 �9 . ~ . o ~ * o ~ � 9 1 4 9

blocking apphcauon i . So it is safe the apply n . Likewise, n and m are consistent under the assumptions L(n D d) or directly L(e D --, n). (In the latter case, we have


Po(e D --,n), and so Po --,n by the antecedent of iii* (cq. Poe). However, now assuming the antecedent of nl we obtain Pon A Po " n, i.e. Po(n ^ --, n), i.e. Pofalse, implying Pod, blocking firing of ii*.)

3.3. PROPERTIES OF OUR APPROACH TO COUNTERFACTUALS

Finally in this paper, we shall examine the properties of our setup as compared to G/trdenfors' axioms for conditional logic [5], which are known to be equivalent to the Lewis approach, and which one can use - as Ginsberg does in [6] - as a standard against which one can measure other systems for conditionals. Paraphrased into our language, they read as follows: (Here, we use the notation Lq~ for the meta- statement S S A P I- ~P or, equivalently, SSAP l-L~p, in order to stay in line with the notation of G~rdenfors as much as possible.)

(G1) (q~ >o 90 & (q~ >o Z) =o ((o >o 1//A Z),

(G2) r true,

(G3) L(~ ::) Z) & (q) >o I//) => (q~ >o X),

((34) r >o r

(G5) L(q) -- I//) & (q) >o Z) :=~ (IV >o Z),

((;6) L(r ̂ ~) ~ (r >o ~), ((37) (r >o 9 r) =o (Lop =# L9r

(GS) (r >o X) & (~' >o Z) ~ (r v ~, >o Z), (G9) (~ >o ~) & not (~ >o ", Z) =0 (rp A Z >o ~).

We check which of these hold in our approach:

((31) does not hold. We do not regard this as a flaw in our approach. In fact, the Caesar example was already a counterexample demonstrating that (G1) is not always a desirable property. The crux of the matter is that the consequences 9 r and X may hold in different and disjoint subframes; (G2) holds under the condition that --,Po false, i.e. 7.o ;~ 9 ; (G3) holds. We prove this case by way of an example.

Proof of ((73) Suppose that I - -L(~DZ), and that ~O>o 9", i.e. for some i, [ T u {PooP}

b.--,Po--,gr&Tu{Pocp}k-PoPi~]. Since }--L(~DZ), we also have that [--Po(~D Z), and hence [--Po(--,ZD--,~). Hence, we obtain I-Po --,• D Po --, 1//, and so ['- ~Po "-11//D ~Po ~Z- Moreover, we have I-Po Pi(li/D Z), and so I-Po Pi lied PO PiZ. Thus, we have that [T u {Poq~} b" "-,Po -',Z & T u {PooP} I'-Po PiZ]. Consequently, we obtain r >o Z- vl

A direct corollary of (G3) is


((]3*) L(I//-- Z) & (r >o q/) =r (~P >o 2');

((]4) holds under the consistency condition that --,Po--, ~. So, e.g. false >o false does not hold; (G5) holds, but it is important to note that we do not have the stronger assertion

(G5*) L(~ D ~) & (IV >0 Z) =~ (cp >0 ;f),

which would be a form of monotonicity. As we have seen in Ginsberg's power/diner example, we do not have this, due to the nonmonotonicity in the consistency checks.

Note in passing that (G3*) and (G5) together imply a natural property concerning substitutivity of equivalent assertions

(G35) L(r --- q~2) & L(Igl --- I//2) ~ ( (~ >0 Igl) r (~P2 >0 q/2)).

((]6) holds under the condition that --,Pofaise, i.e. go ;e @. (Admittedly, we have reshaped the original form of (G6), which looks like (~ A 9 r) D (tp >o ~) and which could not be fit in our framework immediately because it is not a well-formed formula; the same holds for (G7).) A related, simpler assertion

(G6*) Lr :=:> (g>o r

also holds under the same condition of--,Pofalse. (G7) (even in adjusted form) does not hold. However, we do have, under the consistency condition that --,Po--, g,

((37*) (cp >o.i IV) =# (Lop =o PoPil//),

which is not so strange: we cannot expect a counterfactural experiment to have an influence on the whole collection of a priori worlds as in (G7), but it has an influence on the subframe 2;i (this is even more intuitive and useful than what is stated in (G7), witness the Caesar example!); (G8) does not hold, due to the fact that Po(r v q/) D PotP v Poq/does not hold; (G9) is related to Ginsberg's nonmonotonicity example treated in section 3.2. It holds almost trivially (without using the second premise) under the assumption that PoZ does not block the validity of the conclusion (which is admittedly very close to the intention of the second premise in (G9), stating something about the consistency of ~ and Z). However, also the consistency of ~ and Z should be considered in our approach. Thus, a version of (G9) that holds in our setting is

(G9*) (r >o 9 r) & T u {Poe, PoZ} k" --, Po --, tV =~ (tp A Z >o ~).

The second premise expresses the consistency (within frame ,Y--o) of 9 r with both ep and Z- Moreover, in order to derive tp A Z >o ~, also tp and Z must be consistent (within Y-,o), since otherwise we have that Poq~ A PoZ l--Pofaise, so that we cannot derive the consistency check - ,Po- , ~.


Also, as discussed in [6], we should not have contraposition of counterfactual implication, i.e. ~p >o I//=~ --, q />o --, tp, which we obviously do not have in our framework (cf. [19 ,20 ] ) .

We finally discuss (non)transitivity of counterfactual implication. Let us consider a (perhaps somewhat outdated) example from [6] again.

EXAMPLE

Consider the following three counterfactual statements:

(a) If J. Edgar Hoover had been born a Russian, he would have been a Communist.

(b) I f he had been a Communist , he would have been a traitor.

(c) If Hoover had been b o m a Russian, he would have been a traitor.

In our setup, we formalise (a) - (c) as

(a*) r > c,

(b*) c > t,

(c*) r > t.

Ginsberg rightly observes that (a) & (b) does not imply (c) by common-sense reasoning. So one would also expect that we do not have transitivity in (a*)-(c*): r > c & c > t :~ r > t. In fact, the reason for the failure of transitivity in the above example can be explained by means of frames. Suppose (a*) concems a frame Y-o, for which obviously Po r holds. So actually (a*) stands for r >o c. However, (b*) concerns a frame Z1, for which P1 a, where "a'" stands for "Hoover is an American". It is this interpretation that makes (b*) reasonable implication. Since we may for simplicity assume that L(r D ---,a), and thus that there is no frame Zi for which Pi(r ^ a), we obtain that Y-,o n El = 0 . (Note that an attempt to derive that Po Pia for some subframe Yi of Eo by an application of Aa ^ --,Po ~ a D Po Pia fails due to the fact that Po r and thus Po --,a holds, blocking the inference to Po Pia.) However, under this assumption it is obvious that we do not have transitivity: r >o c & c >1 t =~r >i t for some i.

It is nevertheless interesting that in our framework we are able to allow for a restricted form of transitivity. Suppose that E2 ~ E1 ~ Eo. Then it holds that ~0>0,1 Ipr• 1//>1,2 ~::=~ ~>0,2Z! (Since in this case P o q ~ . P o P l l g - P l l g and PI ~t I- PI P2X - P2Z imply that Po ~ ~ Po PEX - P2Z; note that the consistency checks work out fine as well!) A n example of such a transitive counteffactual is: if John had broken his leg yesterday, he had stayed at home today, and if he stayed at home today, he would not have been involved in this car accident, which indeed implies: if John had broken a leg yesterday, he would not have been involved in this car accident.

4. C o n c l u s i o n s

In the setting of the logic S5AP, we have shown how to treat counteffactuals by means o f defaults. It is interesting to view our result in connect ion with the


recently discovered relation between preference-based semantics of nonmonotonic reasoning [27] and conditional logic [4]. In some sense, we have established the relation both in the other direction and on a lower (more concrete) level. We have investigated some properties of our approach as compared to Glrdenfors' axioms for conditionals, and discussed the issues of contraposition and transitivity of counterfactual implication. In our opinion, we may conclude from this that our framework is rather intuitive, and that, afortiori, it possesses some very desirable features lacking in other approaches, such as the nonvalidity of (G 1) as exemplified by the Caesar example.

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counterfactual reasoning by (means of) defaults

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