An Update Semantics for Defeasible Obligations

The deontic logic DUS is a Deontic update semantics for prescriptive obligations based on the update semantics of Veltman. In ous the definition of logical validity of obligations is not based on static truth values but on dynamic action transitions. In this paper prescriptive defeasible obligations are formalized in update semantics and the diagnostic problem of defeasible deon­ tic logic is discussed. Assume a defeasible obli­ gation 'normally a ought to be (done)' together with the fact '-,a is (done).' Is this an exception of the normality claim, or is it a violation of the obligation? In this paper we formalize the heuris­ tic principle that it is a violation, unless there is a more specific overriding obligation. The un­ derlying motivation from legal reasoning is that criminals should have as little opportunities as possible to excuse themselves by claiming that their behavior was exceptional rather than crimi­ nal. 1 THE LOGIC OF NORMS Computer scientists use the logic of obligations, prohibi­ tions and permissions called deontic logic since the early eighties to represent and reason with legal knowledge (McCarty, 1994), and recently it has been used to spec­ ify and analyze security issues about electronic networks (Conte and Falcone, 1997), to represent norms in qualita­ tive decision theory (Pearl, 1993; Boutilier, 1994; Lang, 1996) and to represent rights, duties and commitments in multi-agent systems (van der Torre and Tan, 1999b). A fur­ ther increase may be expected now recently developed pre­ scriptive deontic logics (Makinson, 1999; van der Torre and Tan, 1998a) have delivered some promising approaches for the following long-standing problems in normative reason­ ing and their notorious deontic paradoxes. Yao-Hua Tan EURIDIS Erasmus University Rotterdam P.O. Box 1738 3000 DR Rotterdam The Netherlands [email protected] Contrary-to-duty. The conceptual issue of the contrary­ to-duty paradoxes is how to proceed once a norm has been violated. Clearly this issue is of great practical relevance, because in most applications norms are vi­ olated frequently. In electronic contracting the con­ tracting parties usually do not want to consider a vi­ olation as a breach of contract, but simply as a dis­ ruption in the execution of the contract that has to be repaired. Dilemma. The conceptual problem of the dilemma para­ doxes is to determine the coherence conditions of a normative system. For example, when drafting regu­ lations a coherence check indicates whether they have this desired property, or whether they should be fur­ ther modified. In this paper we introduce a deontic update semantics for defeasible obligations and we show that the dynamic ap­ proach not only gives a better analysis of the traditional deontic problems, but it also gives a better analysis of the problems discussed in defeasible deontic logic. An exam­ ple of reasoning with defeasible obligations is that nor­ mally you have an obligation not to have a fence around your cottage, but this obligation is defeated in the excep­ tional circumstances when your cottage is next to a cliff (the examples in this paper are taken from the cottage hous­ ing regulations discussed in (Prakken and Sergo!, 1996)). Defeasible obligations can be overridden or cancelled by other, stronger obligations. It has been argued that more specific defeasible obligations are stronger than more gen­ eral defeasible obligations, and therefore override them in case of conflict (Horty, 1993; van der Torre, 1994; Asher and Bonevac, 1996; Morreau, 1996). Unfortunately, the analysis of the specificity principle in logics of defeasible reasoning does not apply to defeasible deontic logic, be­ cause it may interfere with the violability of norms. In other words, the combination of reasoning about uncer­ tainty and contrary-to-duty reasoning leads to new com­ plications. This interference is illustrated by the following diagnostic problem. 632 van der Torre and Tan 1. There should not be a fence around the cottage. 2. If the cottage is next to a cliff, then there should be a white fence. 3. If there is a fence, tlien it should be white. 4. There is a white fence. Is the fact 'there is a white fence' a violation or an ex­ ception? Obviously, this is a crucial question for legal knowledge-based systems. If the cottage is next to a cliff, then there should be a white fence according to the second line and the first obligation is cancelled. Moreover, if there is a fence, then there should be a white fence according to the third line, but the first obligation is not cancelled. In this paper we formalize the heuristic principle that a defeasible obligation 'normally a ought to be (done)' to­ gether with the fact '•a is (done)' is a violation, unless there is a more specific overriding obligation. The underly­ ing motivation from legal reasoning is that criminals should have as little opportunities as possible to excuse themselves by claiming that their behavior was exceptional rather than criminal. In absence of a cliff you have to pay a penalty for having a fence, because in that case the first obligation is a violated actual obligation. The difference between the an­ tecedent of the second and third obligation is represented in the deontic states of the update semantics by two different orderings: the second gives rise to levels of exceptional­ ity (inspired by preference-based approaches to defeasible reasoning) and the third gives rise to levels of ideality (in­ spired by preference-based approaches to deontic reason­ ing). Summarizing, if there is a fence without a cliff, then the first obligation is overshadowed (by the third obliga­ tion) but not cancelled (by the second obligation). it is still in force thus it is violated. The defeasible obligations discussed in this paper should not be confused with prima facie obligations (Ross, 1930; Asher and Bonevac, 1996; Morreau, 1996; van der Torre and Tan, 1998b). The typical example of prima facie obli· gations is that you have a prima facie obligation to keep your promises, but this prima facie obligation does not turn into an actual obligation when it leads to a disaster. The dis­ tinctive property is that the obligation 'there ought not to be a fence' is completely cancelled if your cottage is next to a cliff. However, if you have to break a promise to prevent a disaster, then the obligation to hold promises still holds as a prima facie obligation. Consequently, prima facie obliga­ tions have properties defeasible obligations considered in this paper do not have, such as reinstatement (van der Torre and Tan, 1997). The layout of this paper is as follows. First, we intro­ duce prescriptive defeasible obligations in update seman­ tics. Second, we show that the logic formalizes the speci­ ficity principle without introducing an irrelevance problem. Third, we show how test operators can be introduced in the logic. Due to space limitations we do not discuss per­ missions, a first -order base language, nested conditionals, background knowledge, authorities, agents, actions, and time. In the context of deontic update semantics some of these have been discussed in (van der Torre and Tan, 1999b). 2 DEFEASIBLE OBLIGATIONS INDUS In this section we define prescriptive defeasible obligations in update semantics. The logic handles conflicts of hier­ archic obligations, which normally exist, but might be dy­ namically re-evaluated. Two characteristic properties of the logic are that obligations are overridden by more specific and conflicting obligations, and that unresolvable strong conflicts like 'p ought to be (done) and •p ought to be (done)' are 'inconsistent' in the sense that they derive all sentences of the deontic language. We start with the basic definitions of Veltman's update se­ mantics (Veltman, 1996). To define a deontic update se­ mantics for a deontic language L, one has to specify a set E of relevant deontic states (called information states in (Velt­ man, 1996)), and a function [] that assigns to each sentence ¢ an operation ( ¢] on E. If u is a state and ¢ a sentence, then we write 'u( ¢]' to denote the result of updating u with ¢. We can write 'u(¢1] ... [1/>n]' for the result of updating u with the sequence of sentences 1/>1, ... , 1/>n. Moreover, one of the deontic states has to be labeled as the minimal deontic state, written as 0, and another one as the absurd state, written as 1. Definition 1 (Deontic update system) A deontic update system is a triple (L, E, [ ]) consisting of a logical lan­ guage L, a set of relevant deontic states E and a function [ ] that assigns to each sentence ¢ of L an operation. E contains the elements 0 and 1. Veltman explains what kind of semantic phenomena may successfully be analyzed in update semantics and he gives a detailed analysis of one such phenomenon: default rea­ soning. To define obligations in update semantics we have to define the deontic language, the deontic states and the deontic updates. The deontic language is a propositional language with the dyadic operator oblige( a I !3), read as 'normally a ought to be (done), if /3 is (done).' Definition 2 (Deontic language) Let A be a set of atoms and L� a propositional language with A as its non-logical symbols. A string of symbols ¢ is a sentence of Lf if and only if either ¢ is a sentence of L� or there are two sen­ tences 1/JJ and 'lj;2 of L� such that¢= oblige(1/JJI1/J2). We write oblige 'lj; for oblige( 'lj; IT), where T stands for any tautology. A deontic state is a possible worlds model written as u = An Update Semantics for Defeasible Obligations 633 (W, ::;I, ::;;N, V), where W is a set of worlds, �I is an ac­ cessibility relation for ideality, �N is an accessibility rela­ tion for normality, and V a valuation function for proposi­ tions at the worlds. For propositional¢ and world w E W we write rT, w f= ] is defined as fol­ lows. • if c;l> is a factual sentence of L� , then ifW' = {w E W* I a, w f= ¢} f. 0, then a[¢]= ( W, W', ::;1, ::;N, V); otherwise, a[c;i>] = 1. • ifc;l> = oblige( al/3), then if a f. 1 and there are WI, w2 E W such that a, WI f= -,a II /3 and a, w2 f= a II /3, then * ifpref(aoblige(al/3), /3) =a then a[¢] = aoblige(al/3) * otherwise if pref(a N oblige(al/3)oblige(al/3), /3) = a then a[¢] a-N oblige(al/3) oblige(al/3) otherwise, a[¢] = 1 otherwise, a[¢] = 1. A crucial notion of update systems is acceptance. The for­ mula c;l> is accepted in a deontic state a, written as a II-¢, if the update by c;1> results in the same state. In that case, the information conveyed by ¢ is already subsumed by a. Acceptance is the counterpart of satisfaction in standard se­ mantics. Definition 8 (Acceptance) Let a be a deontic state and¢ a formula of the deontic language Lf . a II¢ if and only if a[c;i>] =a. If an update is accepted, then the deontic state usually has a specific content. For example, it is easily checked that a fact a is accepted if all the worlds of W* f. 0 satisfy a, or a= 1. The notion of acceptance is used to define notions of valid­ ity. An argument is I h valid if updating the minimal state 0 with the premises t/!I, . . . , t/Jn, in that order, yields a deontic state in which the conclusion is accepted, and an argument is I f-. valid if all deontic states constructed by updating the minimal state 0 with the premises t/J1, . . . , t/Jn in some order such that the premises are accepted, also accept the conclusion (van der Torre and Tan, 1998b). Note that the order of the premises is only relevant for lh, not for I f-. . Definition 9 (Validity) Let t/JJ. . . . , t/Jn and c;1> be sentences of the deontic language Lf. The argument of¢ from the premises tPI , ... , tPn is valid, written as tPI , ... , t/Jn 11-1 ¢, if and only if O[tf;I] ... [t/Jn]llc;l>. The argument of c;l> from the premises tPI, ... , tPn is nonmonotonically valid, written as tPI, ... , tPn I f-. c;i>, if and only if for all permutations 1r of 1 . . . n such that t/J,(I)• ... , t/J,(n) ll-1 tf;; for 1 ::; i ::; n we have t/J,(!), ... , t/J,(n) ll-1 c;l>. It is clear that checking entailment for all possible orders in which the obligations are taken into account leads to a factorial number of entailment problems. The complexity of the inference is therefor very high. Below some simple examples of the validity relation are given that do not create exceptionality levels, see (van der Torre and Tan, 1998a ). oblige(pi T) If-oblige(plq) oblige(plr), oblige(qlr) If-oblige(p II qlr) oblige(plqllr), oblige(qlr) If-oblige(pllqlr) oblige(pi T) llfoblige(pv qi T) oblige(plq), oblige(plr) I If oblige(plq v r) In the following section we illustrate that one of the fea­ tures of I f-. is that more specific and conflicting obliga­ tions are only accepted if they are later than more general ones. Hence, more specific and conflicting obligations cre­ ate exceptionality levels and override more general ones. Moreover, we also illustrate how the logic deals with the diagnostic problem.