Obligation as Optimal Goal Satisfaction

J Philos Logic DOI 10.1007/s10992-017-9440-3 Obligation as Optimal Goal Satisfaction Robert Kowalski1 · Ken Satoh2 Received: 21 October 2016 / Accepted: 23 May 2017 © The Author(s) 2017. This article is an open access publication Abstract Formalising deontic concepts, such as obligation, prohibition and permission, is normally carried out in a modal logic with a possible world semantics, in which some worlds are better than others. The main focus in these logics is on inferring logical consequences, for example inferring that the obligation O q is a logical consequence of the obligations O p and O (p → q). In this paper we propose a nonmodal approach in which obligations are preferred ways of satisfying goals expressed in first-order logic. To say that p is obligatory, but may be violated, resulting in a less than ideal situation s, means that the task is to satisfy the goal p ∨ s, and that models in which p is true are preferred to models in which s is true. Whereas, in modal logic, the preference relation between possible worlds is part of the semantics of the logic, in this non-modal approach, the preference relation between first-order models is external to the logic. Although our main focus is on satisfying goals, we also formulate a notion of logical consequence, which is comparable to the notion of logical consequence in modal deontic logic. In this formalisation, an obligation O p is a logical consequence of goals G, when p is true in all best models of G. We show how this non-modal approach to the treatment of deontic concepts deals with problems of contrary-to-duty obligations and normative conflicts, and argue that the approach is useful for many other applications, including abductive explanations, defeasible reasoning, combinatorial optimisation, and reactive systems of the production system variety.  Robert Kowalski Ken Satoh 1 Imperial College London, Kensington, London SW7 2AZ, UK 2 National Institute of Informatics, Tokyo, Japan R. Kowalski, K. Satoh Keywords Deontic logic · Abductive logic programming · Normative conflicts · Contrary-to-duty obligations · Goals · Preferences 1 Introduction There are two ways to understand such natural language sentences as birds can fly. One is to understand them literally, but only as defeasible assumptions. The other is to understand them as approximations to more precisely stated sentences, such as a bird can fly if the bird is normal, with an extra condition the bird is normal, which is defeasible, but is assumed to hold by default. In this paper, we explore the second approach, applied to natural language sentences involving deontic attitudes. In contrast to modal approaches, which aim to stay close to the literal expression of natural language sentences, our approach uses a nonmodal logic, in which implicit alternatives are made explicit. For example, instead of understanding the sentence you should wear a helmet if you are driving a motorcycle as it is expressed literally, we understand it instead as saying that you have a choice: if you are driving a motorcycle, then you will drive with a helmet or you will risk suffering undesirable consequences that are worse than the discomfort of wearing a helmet. This is not an entirely new idea. Herbert Bohnert [8] suggested a similar approach for imperative sentences, treating the command do a, for example, as an elliptical statement of a non-modal, declarative sentence either you will do a or else s, where s is a sanction or “some future situation of directly unpleasant character”. Alan Ross Anderson [2] built upon Bohnert’s idea, but reformulated it in alethic modal logic, reducing deontic sentences of the form O p (meaning p is obligatory) to alethic sentences of the form N (¬p → s) (meaning it is necessarily the case that if p does not hold, then s holds). A similar reduction of deontic logic to alethic logic was also proposed by Stig Kanger [35]. Our non-modal approach, using abductive logic programming (ALP) [34], is similar in spirit, in the sense that goals in ALP whether they represent the personal goals of an individual agent, the social goals of a society of agents, the dictates of a powerful authority, or physical constraints - are hard constraints that must be satisfied. In the simplified variant of ALP that we use in this paper, an abductive framework is a triple P, G, A, where P is a logic program representing an agent’s beliefs, G is a set of sentences in FOL (first-order logic) representing the agent’s goals, and A is a set of atomic sentences representing candidate assumptions. The logic program P serves as an intensional definition (or representation) of an incomplete model of the world, which can be extended by adding assumptions  ⊆ A, to obtain a more complete model represented by P ∪ . The abductive task is to satisfy the goals G, by generating some  ⊆ A, such that: G is true in the model represented by P ∪ . For simplicity, we consider only logic programs, which are sets of definite clauses of the form conclusion ← condition1 ∧ . . . ∧ conditionn , where conclusion and each conditioni is an atomic formula, and all variables are universally quantified. Any logic program P (or P ∪ ) of this form has a unique minimal model [17]. The logic Obligation as Optimal Goal Satisfaction program can be regarded as a definition of this model, and the model can be regarded as the intended model of the logic program. In ordinary abduction, the goals G represent a set of observations, and  represents external events that explain G. In deontic applications, the goals G represent obligations, augmented if necessary with less desirable alternatives, and  represents actions and possibly other assumptions that together with P satisfy G. In general, there can be many  ⊆ A that satisfy the same goals G. In some cases, the choice between them may not matter; but in other cases, where some  are better than others, it may be required to generate some best . For example, in ordinary abduction, it is normally required to generate the best explanation of the observations. In deontic applications, it is similarly required to generate some best more complete model of the world. However, due to practical limitations of incomplete knowledge and lack of computational resources, it may not always be feasible to generate a best . In some cases, it may not even be possible to decide whether one  is better than another. It other cases, it may be enough simply to satisfy the goals [63] without attempting to optimise them. Nonetheless, the aim of generating a best solution represents a normative ideal, against which other, more practical solutions can be compared. For this purpose, we extend the notion of an abductive framework P, G, A to that of a normative abductive framework P, G, A, <, where < is a strict partial ordering among the models represented by extended logic programs P ∪ , where  ⊆ A. The normative abductive task is to satisfy G by gen (...truncated)