The Chisholm Paradox L.W.N. van der Torre
Y.-H. Tan
Max-Planck-Institute for Computer Science
Euridis, Erasmus University Rotterdam
Im StadtWald, D-66123 Saarbrucken, Germany
P.O. Box 1738, 3000 DR Rotterdam, The Netherlands
[email protected]
[email protected]
Abstract
In this paper we analyze the Chisholm paradox. We survey the paradox in Standard Deontic Logic, preference-based deontic logic, defeasible deontic logic and temporal deontic logic. We propose to combine preferential and temporal notions to analyze it.
1 Introduction
The Chisholm set consists of the following four sentences.
1. 2. 3. 4.
` ought to be (done),' `if is (done), then ought to be (done),' `if is not (done), then ought not to be (done),' and ` is not (done).'
1'. 2'. 3'. 4'.
` ought to be (done),' `if has been (done), then ought to be (done),' `if has not been (done), then ought not to be (done),' and ` has not been (done).'
2.
`if a certain man goes to the assistance of his neighbors, then the man ought to tell his neighbors that he will come'
The formalization of these sentences in Standard Deontic Logic is either inconsistent or the sentences are logically dependent. The Chisholm set is therefore called a paradox. Temporal deontic logic can consistently formalize the set
For most and the rst set can be transformed to the second one without changing the meaning of the sentences. It has been argued that temporal deontic logic therefore solves the paradox [vE82, LB83]. However, this `solution' does not work for the original set given by Chisholm [Chi63], in which is read as `a certain man goes to the assistance of his neighbors' and as `the man tells his neighbors that he will come.' For example,
means something di erent than 2'.
`if a certain man has gone to the assistance of his neighbors, then the man ought to tell his neighbors that he will come.'
1
The representation 1 ? 4 assumes that the antecedent (condition) occurs before the consequent (conclusion) , but the contrary is the case for these speci c and from the Chisholm set! In this paper we show how the temporal antecedent-before-consequent solution can be extended with preferences on sequences of actions to cover the original Chisholm set. But rst we give a survey of the paradox in several types of deontic logic. 0
0
2 The Chisholm paradox The Chisholm paradox is an important benchmark example of deontic logic, and deontic logics incapable of dealing with it are considered insucient tools to analyze deontic reasoning. Related benchmark examples are the Good Samaritan Paradox [ Aqv67], its strengthened version the gentle murderer or Forrester paradox [For84], and the pragmatic oddity [PS96]. These examples are known as the contrary-to-duty paradoxes of deontic logic. We do not have the space to discuss the related examples, but we leave it implicitly understood that a deontic logic only satisfactorily formalizes the Chisholm paradox if it also satisfactorily formalizes the related examples. We give a survey of the Chisholm paradox in the following deontic logics. SDL (MDL) DL PDL DDL CDL
Standard Deontic Logic TDL Temporal Deontic Logic Minimal Deontic Logic (ADL) Action Deontic Logic Jones & Porn's Deontic Logic PTDL Pref.-based Temporal Deontic Logic Preference-Based Deontic Logic (PADL) Pref.-based Action Deontic Logic Defeasible Deontic Logic (DUS) Deontic Update Semantics Contextual Deontic Logic
They are classi ed in Figure 1, which also represents successor relations beSDL (+MDL+DL) PDL
TDL (+ADL)
DDL (+CDL) PTDL (+PADL+DUS) Figure 1: A classi cation of deontic logic
tween the classes. The classi cation is based on semantic notions (preferences and time) and a proof-theoretic notion (defeasibility). It is tailor-made for the Chisholm paradox and the related problems mentioned above. For example, SDL and MDL are in the same class, because they deal with the paradox in the same way, although they are very di erent in the analysis of, for example, dilemmas. The gure illustrates that SDL is the predecessor of all other deontic logics. It also illustrates that there is a split in deontic logic literature into the two di erent approaches PDL and TDL, which are re-united in PTDL. 2
Finally, DDL is a successor of PDL, because defeasibility is formalized by preferential semantics, and preference-based semantics naturally leads to some type of defeasibility (see below). In this section we analyze the Chisholm paradox in SDL, PDL, DDL (+CDL) and TDL, and in Section 3 we analyze the paradox in PTDL. Due to space limitations we have to be brief, but we also give references to more elaborate discussions. We defer a discussion of a minimal logic to formalize CTD reasoning until the conclusions at the end of this paper.
2.1 Standard Deontic Logic (SDL)
SDL is usually formalized by a normal modal system of type KD1 according to the Chellas classi cation [Che80], although it validates the counterintuitive theorem O>, where > stands for any tautology like p _ :p (see e.g. [Alc93] for an alternative). A conditional obligation ` ought to be (done) if is (done)' is usually formalized by ! O , and sometimes by O( ! ). The Chisholm paradox is also called the contrary-to-duty paradox, because it contains a socalled contrary-to-duty obligation, i.e. an obligation that is only valid in a sub-ideal situation. The conditional obligation ! O or O( ! ) is called a Contrary-To-Duty (CTD or secondary) obligation of the (primary) obligation O 1 when and 1 are contradictory. The condition of a CTD obligation is only ful lled if the primary obligation is violated. The Chisholm paradox contains besides a CTD obligation also a what we call According-To-Duty (ATD) obligation. A conditional obligation ! O or O( ! ) is an ATD obligation of O 1 when logically implies 1 . The condition of an ATD obligation is satis ed only if the primary obligation is ful lled. The Chisholm paradox can be represented in SDL by several di erent sets of SDL formulas, which are either inconsistent or logically dependent, see e.g. [Chi63, Aqv67, Smi94]. We rst give an inconsistent representation.
Example 1 (Chisholm paradox) Let T = fOa; O(a ! t); :a ! O:t; :ag,
where a is read as `a certain man goes to the assistance of his neighbors' and t as `he tells them that he will come.' The second obligation is an ATD obligation and the third obligation is a CTD obligation, see Figure 2. Since SDL allows a kind of so-called deontic detachment, i.e. j=SDL (O ^ O( ! )) ! O , we have T j=SDL Ot from the rst two sentences. Moreover, since SDL also allows factual detachment, i.e. j=SDL ( ^ ( ! O )) ! O , we have T j=SDL O:t from the last two sentences. The paradox is that these two derived obligations are inconsistent, although the set of premises is intuitively consistent.
1 System KD is closed under the inference rules modus ponens and necessitation and it satis es besides the propositional theorems the axioms K: O( ! ) ! (O ! O ) that says that the modal operator is closed under modus ponens and D: :(O ^ O: ) that says that there are no con icting obligations.
3
Oa implies
Oa
inconsistent
O(a ! t) :a ! O:t Figure 2: O(a ! t) is an ATD of Oa and :a ! O:t is a CTD of Oa The SDL analysis of the Chisholm paradox is based on rejection of one of the detachment principles (terminology introduced by Greenspan [Gre75]). However, both seem intuitive in most cases. Rejection of one of the principles because they seem to be problematic in a very few cases is a solution that seems like overkill. For example, an alternative representation of the Chisholm set (see e.g. [Chi63, Aqv67, Smi94]) is to represent the second sentence by a ! Ot. An advantage of this representation is that the second and third sentence are represented by logical formulas with the same structure. Moreover, it is an attempt to solve the Chisholm paradox, because it makes the Chisholm set consistent. However, the following example illustrates that this `solution' misses the point of the paradox. The solution does not have deontic detachment whereas deontic detachment is in most cases intuitive. Thus, the representation O(a ! t) derives too much (always deontic detachment) and the representation a ! Ot derives too little (never deontic detachment). Since the problems are caused by the second obligation of the set, we prefer to call it an ATD paradox. For similar reasons, Prakken and Sergot [PS96] argue that the Chisholm paradox should be called a paradox of deontic detachment.
Example 2 (Chisholm paradox, continued) Consider the SDL theory T = fOa; a ! Ot; :a ! O:t; :ag. Notice that a ! Ot can be derived from :a.
Chisholm argued that this logical dependence is counterintuitive, and several logicians (see e.g. [ Aqv67]) have demanded that a solution of the Chisholm paradox should represent the sentences such that they are logically independent. However, Tomberlin [Tom81] observes that the criterion is a `rather glaring theoretical commitment' which `would be a case of agrant methodological questionbegging.' Moreover, this logical dependence is easily solved by introducing a weaker notion of implication. For example, the two conditional obligations can be represented by a > Ot and :a > O:t where `>' is a so-called strict implication [Che74, Mot73, Alc93]. For example, we can represent the obligations by 2(a ! Ot) and 2(:a ! O:t) where 2 is a so-called alethic modal operator that satis es at least axiom T: 2 ! (re exivity), see [Alc93]. This solves the logical dependence, because the formula : ! ( > ) is in contrast to the formula : ! ( ! ) not a theorem. The main problem of this representation is that the obligation the man ought to tell his neighbors that he will come and go to their assistance O(t ^ a) cannot be derived. In SDL the obligation O(t ^ a) is derived if and only if Ot is derived,
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because SDL has the theorem (O ^ O ) $ O( ^ ). In the SDL analysis, the main problem underlying the Chisholm paradox is whether we allow for deontic detachment or not. We have 6j=SDL (O ^ ( ! O )) ! O , hence Ot cannot be deontically detached from the rst two sentences in SDL. We derive Ot and O(t ^ a) only when a is true. However, it seems that the man ought to tell his neighbors that he will come, because otherwise he will violate an obligation. If he does not tell them and later he goes to the assistance, then he violates the second obligation. If he does not tell them and later he does not go to the assistance, then he violates the rst obligation. If Ot is not deontically detached, then the following intuitive deontic reasoning pattern is not supported, see [Pow67, Han97]. Assume that although the man is able to go to the assistance of his neighbors, he has no intention of doing so. He argues: `I ought to change my mind, tell them, and go to their assistance. So I am obligated to tell them. My present ful llment of this obligation will help to make up for my sinfully not going to the assistance!' In the ideal state the man tells his neighbors that he will come. According to the semantics of SDL, the obligation Ot should be derived.
Many alphabetic variants of the Chisholm paradox have been proposed, see e.g. [vE82, LB83]. However, a crucial distinction with the original Chisholm set is that the consequents of the CTD and ATD obligation occur later than the primary obligation! For example, consider the following example from the United Nations Convention on Contracts for the International Sale of Goods [Smi94].
Example 3 (Convention on Contracts) Section 79 subsection 4 reads as
follows (see [Smi94, p.127]): \The party who fails to perform must give notice to the other party of the impediment and its e ect on his ability to perform. If the notice is not received by the other party within a reasonable time after the party who fails to perform knew or ought to have known of the impediment, he is liable for damages resulting from such non-receipt." Here we have a double contrary-to-duty construction: rst a contrary-to-duty obligation (to give notice), and then a prevision of what the consequences will be if that contrary-to-duty obligation remains unful lled (liability for damages). A typical case between the parties A and B can be formalized by the SDL theory fOA p; p ! OA :n; :p ! OA n; :pg where p stands for `party A performs' and p for `it gives notice to party B .' These SDL sentences have the same logical structure as the SDL sentences in Example 2.
At rst sight it may seem that a solution for Example 3 also solves Example 2. However, the two examples are not the same, because they have di erent temporal references. Their logical representations are only the same, because we left the temporal representation implicit. Later in this paper we show that this makes a fundamental di erence. 5
2.2 Preference-based Deontic Logic (PDL)
Hansson [Han71] argues that the fundamental problem underlying the CTD paradoxes is that the type of possible world semantics of SDL is not exible enough. In these semantics only two types of worlds are distinguished in a model; actual and ideal ones. The ideal worlds have to satisfy all obligations in a deontic theory T. Clearly, if these obligations contradict each other, then a problem arises. As Lewis [Lew74] observes, `a mere division of worlds into the ideal and the less-than-ideal will not meet our needs. We must use more complicated value structures that somehow bear information about comparisons or gradations of value.' In the recent deontic literature [Jen85, Jac85, Gob90, Han90, BMW93] the value structures are loosely called betterness relations or preference relations, the latter probably inspired by terminology in arti cial intelligence.2 In the Chisholm paradox both Ot and O:t are implied. No ideal world can satisfy both t and :t, and this causes the paradox. Hence, ideal worlds are simply not enough. In order to model this paradox properly, we need a notion of sub-ideal worlds, in which some but not all obligations are satis ed. For example, in the Chisholm paradox we could distinguish between two types of (sub-)ideal worlds: (sub-ideal) worlds in which :t is true but not t, and (ideal) worlds in which t is true but not :t. This solves the inconsistency in the ideal worlds. Moreover, having the ner distinction between a hierarchy of (sub-)ideal worlds instead of one type of ideal world, we can de ne a preference ordering on these sub-ideal worlds. Given that it is a fact that the man does not go to the assistance, it is better not to tell the neighbors that he is coming than not going and telling them that he will come. Hence, although the sub-ideal worlds in which :a and :t are true are not ideal, they are better than the worlds in which :a and t are true. We say that Hansson introduced a dyadic deontic logic in which the ideality principle of SDL is replaced by an optimality principle. \The problem of conditional obligation is what happens if somebody nevertheless performs a forbidden act. Ideal worlds are excluded. But it may be the case that among the still achievable worlds some are better than others. There should then be an obligation to make the best out of the sad circumstances." Bengt Hansson [Han71] We distinguish two types of preference-based obligations, which we call minimizing and ordering obligations (see [TvdT96]). In the following de nition we give an example of both types. The minimizing obligations OHL are the traditional Hansson-Lewis dyadic obligations [Han71, Lew74]. The ordering obligations OPdl were introduced in Prohairetic Deontic Logic [TvdT96, vdTT97c],
2 To some this terminology has been confusing, because they associate preferences with desires, i.e. with the internal preferences of a person. This association is obviously wrong. The obligations and therefore the preferences represent the desires of the (rational) authority that promulgated the norms.
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following a recent tradition of preference-based deontic logics [Jen85, Jac85, Gob90, Han90, BMW93, HM97, Han97, PS97]. We refer the reader to other papers for the technical details. See [TvdT96] for an axiomatization of both types of obligations in 2dl and see [vdTT97c] for a study of the ordering obligations including a comparison with several preference-based deontic logics. De nition 1 Let M = hW; ; V i be a standard possible worlds model with W a nonempty set of worlds, a binary re exive and transitive (but not necessarily anti-symmetric or connected) accessibility relation, and V a valuation function of the propositions at the worlds.3 We have M j= OHL ( j ) if and only if for all worlds w1 such that M; w1 j= : ^ , there is a world w2 w1 such that M; w2 j= ^ and for all worlds w3 w2 we have M; w3 j= ! . M j= OPdl( j ) if and only if for all worlds w1; w2 2 W such that M; w1 j= : ^ and M; w2 j= ^ we have w1 6 w2, and M j= OHL ( j ).
The following example illustrates the analysis of the Chisholm paradox in preference-based deontic logic.
'$'$'$'$ &%&%&%&%
Example 4 (Chisholm paradox, continued) A typical preference-based model M of the Chisholm sets T1 = fOHL (a j >); OHL(t j a); OHL (:t j :a); :ag and T2 = fOPdl(aj>); OPdl(tja); OPdl(:tj:a); :ag is represented in Figure 3. This F = f:ag ideal situation ordered sub-ideal situations a; t
a; :t
:a; :t
:a; t
Figure 3: Preference relation of the Chisholm paradox
gure should be read as follows. A circle represents a nonempty set of worlds that satis es the propositions written within them. An arrow represents strict preference for all the worlds represented by the circle. The transitive closure is left implicit. The dashed box represents the set of worlds which might be the actual world. They refer to the circumstances that are xed or settled, see e.g. [Han71] for a discussion on the interpretation of circumstances in preference-based logics. We have M j= OHL (a ^ tj>) and M j= OHL (tj>), because the logic has the 3 We may restrict ourselves to full models, i.e. for all propositional there is a world w 2 W such that M;w j= . Moreover, we may not allow for duplicate worlds. That is, we may assume that the logical language is expressive enough to discriminate all worlds. In that case we could write a model as M = hI; i, where I is a set of interpretations and an ordering on I . If we have an in nite set of propositions and therefore an in nite set of interpretations, then there are no technical reasons to allow for duplicate worlds.
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theorem j= OHL ( j ) ^ OHL ( j>) ! OHL ( j>). We have M j= OPdl(a ^ tj>), but also M 6j= OPdl(t j>)4, because Prohairetic Deontic Logic has the theorem j= OPdl( j ) ^ OPdl( j ) ^ OHL ( ^ j ) ! OPdl( ^ j ).
From a semantic point of view Hansson's preference-based semantics was an important step forward, which was especially appreciated in the non-monotonic and default logic literature. Moreover, Example 4 illustrates that preferencebased deontic logics have a kind of deontic detachment. Unfortunately, from a proof-theoretic point of view this solution is less satisfactory, because it is based on the rejection of the factual detachment principle O( j ) ^ ! O , where O is not necessarily de ned by O =def O( j>) (see [Alc93] for alternatives). This principle is intuitive in many cases, as argued by for example Loewer and Belzer [LB83]. As a consequence, it seems that we have to choose between factual detachment and deontic detachment. Factual detachment is accepted in SDL type of deontic logic and deontic detachment is accepted in Hansson-Lewis type of deontic logic. If we are forced to choose between factual detachment and deontic detachment then we prefer the latter, because logics with deontic detachment can formalize the Forrester paradox [For84] whereas logics with factual detachment cannot, see e.g. [vdT97]. However, we are not forced to choose, because there are several ways in which the two principles can be combined. They are discussed in the following sections.
2.3 Distinguishing two types of dilemmas
In some PDL analyses of the Forrester paradox [For84, Gob91] it has been proposed to distinguish between O ^ O: and O( ^ ) ^ O: . The rst is a dilemma and should therefore be inconsistent, and the second is not a dilemma and should therefore be consistent. In these PDL's the weakening rule O( ^ ) ! O is rejected, otherwise the inconsistent O ^ O: would be derivable from the consistent O( ^ ) ^ O: . The following example illustrates how this PDL analysis can be applied to the Chisholm paradox.
Example 5 (Chisholm paradox, continued) Consider Prohairetic Deontic Logic given in De nition 1, extended with O( j ) ^ ! O , where we do not assume any properties of the monadic obligations except :(O ^ O: ). Moreover, assume the set of obligations T2 from Example 4. The set T2 is consistent, and we have T2 j= O(t ^ a) ^ O:t. The PDL logics are given interesting semantics [Jac85, Gob90, Han90], related to the semantics of Prohairetic Deontic Logic. However, the solution seems like overkill, because O( ^ ) ^ O: normally formalizes a dilemma and should therefore be inconsistent.
4 PDL can be extended to a two-phase deontic logic such that M j= O Pdl(t j >), see [TvdT96].
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2.4 Distinguishing two types of obligations
Jones and Porn [JP85] argue that there are two di erent types of obligations, related to Thomason's distinction between the context of justi cation and the context of deliberation [Tho81]. Deontic detachment gives rise to an `ideal' obligation (Oi t), and factual detachment gives rise to an `actual' obligation (Oa :t). In Section 2.2 we argued that if the man does not tell his neighbors that he is coming, then at least one obligation has been violated; we know that a world in which he does not tell that he is coming is not an ideal world, so in a certain sense there is an obligation to tell. However, it is argued by Jones and Porn (see also [Pra96]) that this `ideal ought' is a di erent kind of `ought' than the one of actual obligations. They provide an analysis of the Chisholm set that satis es the demands of logical independence and consistency, it yields factual detachment from the third and fourth sentences, and it yields deontic detachment from the rst and second sentences. Jones and Porn accept the need for two types of accessibility relations, which gives rise to a division in three types of worlds: ideal (accessible by the rst relation), sub-ideal (accessible by the second relation) and sub-sub-ideal (not accessible) worlds. However, the semantics show that the logic is ad hoc, in the sense that it is developed particularly for the Chisholm paradox, ignoring the related CTD examples mentioned in Section 2. In particular, it is unclear how multiple nested levels of ATD's (add O(q j a ^ t) etc.) and multiple levels of CTD's (O(:k j>), O(k ^ g j k), O(l j k ^ :g), etc.) can be handled by only three di erent states. Moreover, the `pragmatic oddity' creates problems for the logic, see [PS96]. It seems unlikely that the distinction between ideal and actual obligations is a satisfactory formalization of CTD reasoning.
2.5 Defeasible and Contextual Deontic Logic (DDL, CDL)
The last decade several defeasible deontic logics have been proposed, that formalize reasoning about obligations which are subject to exceptions [Hor94, Hor93, Jon93, vdTT95, Mor96, vdTT97b, NY97]. It can be argued that most PDL's are DDL's, because they typically formalize obligations that do not have weakening (of the consequent) [Jen85] or defeasible conditionals, i.e. conditionals that do not have strengthening of the antecedent [Alc93]. In a defeasible deontic logic it is argued that deontic detachment should sometimes hold and sometimes not [McC94, RL93, NY97]. We say that deontic detachment holds as a defeasible rule [vdTT95, vdTT97b]. The following example illustrates that the defeasibility in the Chisholm paradox is analogous to defeasibility in assumption-based reasoning.
Example 6 (Chisholm paradox, continued) Consider the two theories T1 = fOa; a ) Ot; :a ) O:t; :ag and T2 = fOa; a ) Ot; :a ) O:tg, where ) is some kind of default implication. A distinction between T1 and T2 is that Oa
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is a deontic cue in the former and a violated obligation in the latter. Given T1 , most people have a clear intuition that O:t should be preferred over Ot, i.e. we have T1 j= O:t. Given that the man does not go to the assistance, he should not tell his neighbors that he will come. However, given T2 most people would expect that { given that :a is not the case { the preference would be the other way round, i.e. we have T2 j= Ot. The man should tell his neighbors that he will come, otherwise he has to violate an obligation later (the ideal has become unreachable). Given the intuitive reading above, Ot should only be inferred from Oa when Oa is a ful lled obligation or a deontic cue. When it is a deontic cue, Ot is derived on the assumption that a will be true. This reading of the example has a non-monotonic character, i.e. conclusions can be lost by the addition of new information.
A drawback of the analysis of the Chisholm paradox in a defeasible deontic logic is that the use of non-monotonic techniques is counterintuitive. In general the use of non-monotonic techniques has been questioned in the deontic logic literature (see e.g. [PS96]), and for good reasons. The crucial question to be solved is: how can obligations disappear (not persist, be defeated) if it is not as a result of exceptional circumstances? The DDL analysis in Example 6 suggests that deontic detachment assumes that the primary obligation Oa is not violated. This analysis is rejected by Prakken [Pra96], who argues that the assumption that people tend to ful ll their obligations is in many practical applications untenable. The CDL analysis in Example 7 illustrates that the assumptions are not related to ful lling primary obligations and deontic detachment but to weakening (of the consequent). As in Example 4 there is an obligation to tell and go to the assistance. Moreover, the obligation to tell can only be derived under the assumption that the man goes to the assistance. The crucial observation is that this assumption is not introduced as an exception related to deontic detachment, but it is inherent to the weakening of O(a ^ t) to Ot: in CDL, every weakening introduces exceptions. The CDL analysis is based on the distinction between factual defeasibility (FD) and overridden defeasibility (OD) [vdTT95, vdTT97b]. FD formalizes that violated obligations are no longer a cue for action, and OD formalizes that stronger rules override weaker rules. For example, prima facie obligations must have OD to formalize that we may break a promise to prevent a disaster. A contextual obligation is written as O( j n ) and read as ` ought to be (done), if is (done), unless is (done).' The contextual obligations are comparable to Reiter defaults : , where : is the so-called justi cation [Rei80]. Contextual obligations and Reiter defaults only formalize FD, and hence they cannot formalize prima facie obligations or speci city structures. See [vdTT97a] for the details and an axiomatization of Contextual Deontic Logic.5 :
5 It can easily be shown in the semantics that we have M j= O Cdl ( j n ) if and only if M j= OPdl( j( ^ ^ : ) _ (: ^ )). This illustrates in an exact way how CDL follows
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De nition 2 Let M = hW; ; V i be a standard possible worlds model as in De nition 1. We have M j= OCdl( j n ) if and only if for all worlds w1 ; w2 2 W such that M; w1 j= : ^ and M; w2 j= ^ ^ : we have w1 6 w2 , and M j= OHL ( j ). The following example illustrates the CDL analysis of the Chisholm paradox. Example 7 (Chisholm paradox, continued) Consider the premise set S = fOCdl (a j> n ?); OCdl(t j a j?); OCdl(:t j:a n ?); :ag. The model in Figure 3 is a typical model of S . We have `the man ought to tell his neighbors that he will come, and go to their assistance' S j= OCdl (a ^ t j> n ?), we do not have `the man ought to tell his neighbors that he will come' S 6j= OCdl (t j> n ?), but we
have `the man ought to tell his neighbors that he will come, unless he does not go'
S j= OCdl(t j> n :a). The rst obligation does not have an exception whereas
the latter (weaker) obligation does. Somewhat misleadingly, we said that deontic detachment holds as a defeasible rule, because the deontically detached `ought to do t' has an exception clause `unless :a.' However, this exception is not introduced by deontic detachment, which derives S j= OCdl(a ^ tj> n ?), but by weakening of the consequent, which derives S j= OCdl (tj> n :a) from the latter obligation.
CDL is monotonic. The defeasibility can be made explicit when we de ne a variant of factual detachment for contextual obligations analogous to the construction of extensions in Reiter's default logic, see [Rei80, Hor93, Hor94]. If :a is true, then Ot cannot factually be detached from O(tj>n:a), but it can as long as the truth value of a has not been settled.
2.6 Temporal and Action Deontic Logic (TDL, ADL)
Since the late seventies, several temporal deontic logics and deontic action logics were introduced, which formalize satisfactorily a special type of CTD obligations, see for example [Tho81, vE82, LB83, Mak93, Alc93]. In this section we illustrate the TDL analysis of the paradox in a recent proposal by Horty [HB95, Hor96], based on a seeing-to-it-that (stit) operator. A stit-frame hTree; ) because at moment m1 the histories h3 and h4 are bad histories, M j= OABC (stitH : :n j p) because at moment m2 history h2 is bad, M j= OABC (stitH : n j :p) because at moment m3 history h3 is bad. Summarizing, history h1 is the only good history, and history h3 is a doubleviolation history. At each moment it is clear what should be done, because for each moment there is an obligation that prescribes the choice party A has to make. Now consider the action model of the assistance of neighbors example represented in Figure 4.b. Notice that the deontic part of the model (i.e. Ought(m)) has not been speci ed in the gure. First, the agent has to choose between `telling' t and `not-telling' :t, and secondly the agent has to choose between `going to the assistance' a and `not-going to the assistance' :a. Given this action model, the problem is how to de ne the deontic part of the model such that the three obligations OABC (stitH : aj>), OABC (stitH : tja) and OABC (stitH : :tj:a) are true. Intuitively, the obligation `to go to the assistance' marks history h2 and h4 as bad histories, and the obligation of the man `to tell the neighbors that he goes if and only if he goes' marks h2 and h3 as bad histories. However, the crucial observation is that in contrast to the previous example, we cannot use good and bad states at each moment to formalize these intuitions. We can only give a global judgment for history h1 , which is again the only good history. The
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fundamental problem is that the model cannot validate the premises, regardless of the choice of Ought(m). For example, we have M 6j= OABC (stitH : t j a), because once a is settled (moment m2 or m3 ) the man can no longer see to it that t. The truth value of t is already xed.
The previous example illustrates that we cannot use OABC to express conditional obligations like `if the man goes to the assistance of his neighbors, then he should tell them,' despite the fact that the actions of the Chisholm paradox can be represented in a temporal model. In other words, ABC logics cannot be used to analyze the backward version of the paradox [Smi94]. There is an underlying problem in the action model. The man does not see to it that (t ^ a) at moment m1 on history h1, i.e. we have M; m1 ; h1 6j= stitH : (t ^ a), because not all histories m1 -indistinguishable from h1 make (t ^ a) true. However, intuitively the man can see to it that (t ^ a) by rst choosing h1 and h2 at moment m1 , and thereafter choosing h1 at moment m2 . Moreover, the man is able to see to it that the ideal state is reached in this way. If the man can reason about sequences of actions (which we call strategies) then this intuitive deontic reasoning in the Chisholm paradox can be formalized. In the following section we show how to reason with strategies.
3 Pref.-based Temporal Deontic Logic (PTDL) In this section we analyze the Chisholm paradox in Preference-based Temporal Deontic Logic (PTDL).7 We show how the temporal deontic logic discussed in the previous section can be extended with preferential notions by formalizing a suggestion from Horty [Hor96, Section 7.1]. To reason about preferred strategies we adapt the de nitions introduced in the previous section in three ways. stitH First, we adapt the de nition of stitH such that strategies are taken into
account. We de ne an action which not only considers the choices at a moment, but also the choices the agent can make in the future. We replace the de nition of m-indistinguishable by a global de nition of indistinguishable. We consider two histories indistinguishable if there is not any moment in which we can distinguish them. We call the operator stitS , where `s' stands for strategies.
7 A similar construction can be made for PADL. In TDL and PTDL, deontic statements of the type `you ought to do ' are presumed applicable to any proposition . On the other hand, decision theoretic methods treat actions as distinct, prede ned objects. In deontic logic literature, this is observed and investigated by Meyer [Mey88]. In his dynamic deontic logic the distinction between actions and assertions [Cas81] is formalized. In ADL and TADL, there is a separate calculus of actions, and we can de ne a preference ordering on these actions. For example, in such an extended logic we have the theorem O( ; ) ! [ ]O and the non-theorem O( ; ) ! O .
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OABC Second, we adapt the de nition of OABC with a notion of dominance for strategies. We therefore need to adapt Ought of the stitH -frame such that it represents a preference ordering on histories instead of a binary distinction between good and bad [HB95, p.617].8 We change the dominance function to the following one: a set of histories is preferred to a second one if each history in the rst set is at least as good as each history in the latter set, and there is a history in the second set which is worse than all histories in the rst set. This de nition goes back to at least [FS48], see [TH96]. This is an arbitrary choice, and other (more complicated) de nitions may be preferred, see [Hor96, Section 7.2]. OABC Third, we also adapt the de nition of OABC for a conditionalization on such that the antecedent can be later than the consequent. We call a history a -history if is true at some moment of it. It is implicitly assumed that propositions formalize facts that cannot change over time. For example, we cannot write s for `Ron is smoking,' but we have to use `Ron is smoking at moment t.' De nition 4 (stitS and O) A stitS -frame hTree; ); O(stitS : t j a); O(stitS : :t j:a)g and let M be a model of T represented in Figure 4.b with the preference ordering of ought de ned by h1 > h3 > h4 > h2 . Note that the ordering on histories re ects the ordering on
states in preference-based deontic logics, as represented in Figure 3. We have M; m1 ; h1 j= stitS : (t ^ a) and M; m2 ; h1 j= stitS : (t ^ a). Moreover, we have M j= O(stitS : a j>), because we have the obligation `go to the assistance' deontically prefers history h1 and h3 , M j= O(stitS : t j a), because the obligation `tell that you go if you go' prefers history h1 to h3 , and M j= O(stitS : :tj:a),
8 This makes the formal system closer to utility distributions [vNM44] as used in decision theory [KR76].
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because the obligation `do not tell that you go if you do not go' prefers history
h4 to h2. We have that the agent rst ought to see to it that t and thereafter ought to see to it that a, M j= O(stitS : t ^ aj>), because history h1 is preferred to all other histories. We do not have that the agent ought to see to it that t, M 6j= O(stitS : tj>), because at moment m1 history h2 is not as good as history h3 and h4 . Summarizing, taking two moments together in consideration we can
derive the obligation to tell as part of a more complex action, but there is not an obligation to tell simpliciter.
4 Conclusions Makinson [Mak93] argues that `at the present state of play, it would not seem advisable to try to cover all complicating factors [of deontic logic] at once, but rather to get an initial appreciation of them few at a time, only subsequently putting them together and investigating their interactions.' In previous work we introduced preference-based frameworks for deontic reasoning and in this paper we propose a logical framework that combines preferential and temporal notions. In [vdTT98] we discuss an alternative way to combine preferences and time by formalizing prescriptive obligations in update semantics. Moreover, in this paper we used the combination of preferences and time to analyze the Chisholm paradox. In previous papers (see e.g. [vdT97]) we argued that PDL is the minimal logic to analyze CTD's. Moreover, Prakken and Sergot [PS96] argue that temporal and action aspects are not the key to the proper analysis of CTD's by discussing some non-temporal, non-praxiological types of CTD scenarios, and in [PS97] they propose a PDL to analyze CTD's. The analyses given in this paper are in accord with these arguments. In particular, it is clear that the PTDL analysis of the Chisholm paradox in the previous section is analogous to the PDL analysis in Section 2.2. A history is a world in which we added temporal structure explicitly, or alternatively, a world is a history in which we abstracted from time. The PTDL analysis is the PDL analysis in a temporal setting. The PTDL analysis of the paradox does not rely on the temporal representation, but it relies on the fact that the possible histories { and therefore also the strategies { can be compared in a preference ordering.
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