A Logic of Deliberation

Deliberation typically involves the formation of a plan or intention from a set of values and beliefs. I suggest that deliberation, or “practical reasoning,” is a form of normative reasoning and that the understanding and construction of reasoning systems that can deliberate and act intentionally presupposes a theory of normative reasoning. The language and semantics of a deontic logic is used to develop a theory of defeasible reasoning in normative systems and belief systems. This theory may be applied in action theory and to artificial intelligence by identifying expressions of values, beliefs, and intentions with various types of modal sentences from the language. While there have been some investigations of the structure of normative reasoning in deontic logic, Bayesian decision theory, and philosophical action theory and ethics, there does not yet exist a general theory of normative reasoning. Such a theory is necessary for the understanding and construction of decision-making systems that use normative principles and policies to form plans, strategies, and intentions. A general logic of the all-purpose “normative reasoner,” or “deliberator,” is needed. Practical reasoning, or deliberation, in which intentions to act are formed from a set of desires and beliefs, may be a form of normative reasoning. Expressions of desires and intentions may be treated as rules (norms) or evaluative judgments (Davidson 1977). There also may be a normative component in belief-systems. It has been suggested, for example, that the rules of thumb that enable a system to form tentative conclusions from incomplete information are expressions of “ratiocinative desires” (Doyle 1983a); and that “epistemic policies” guide an epistemic agent in revising beliefs in the light of new information (Stalnaker 1984). The expressions of desires and policies may be interpreted as norms, and therefore an understanding of normative reasoning would be useful in a theory of reasoning with incomplete or new information. $1. The structure of normative reasoning. Several features of normative systems must be respected by any adequate formal representation of normative reasoning. First, some rules are defeasible, that is, they are generally valid but may have exceptions. Secondly, there is a fundamental distinction between prima facie rules and all-things-considered normative commitments. The prima facie rules of a system, together with a set of facts or opinions determine the system’s all-things-considered commitments. Thirdly, for some set of sentences the all-things-considered (a.t.c.) closure should be non-monotonic, that is, set s is included in set s* but the a.t.c. closure of s is not included in the a.t.c. closure of s*. These features of rules may be illustrated simply as follows. Suppose that Nixon told you a secret after you promised to comply with these requirements: (a) You should not tell the secret to Reagan. (b) You should not tell the secret to Gorbachev. (c) You should tell Reagan if you tell Gorbachev. (d) You should tell Gorbachev if you tell Reagan. Suppose you break promise (b) by a certain time, (e) You told the secret to Gorbachev, and you are trying to decide whether you should tell Reagan. If no other rules or facts are relevant then, to comply with the requests as given, clearly you should tell the secret to Reagan, because of rule (c)--and in spite of (a). The prima facie rule (a) is defeasible because of (c). After you have told Gorbachev you have an all-things-considered commitment expressed by the rule (f) You should tell the secret to Reagan. Rules (a) and (f) conflict, yet correct resolution is possible if we recognize that stipulation (a) is a valid prima facie rule whereas (f) expresses a valid all-things-considered commitment after it is settled that you have violated rule (b) by telling Gorbachev. A prima facie rule may be “defeated,” in which case it cannot reliably be used to draw normative conclusions. In the example, after you told the secret to Gorbachev the rule (a) was defeated. To use a prima facie rule in particular circumstances to detach an all-things-considered normative conclusion one needs to know that the prima facie rule is not defeated in those circumstances. If it is not defeated, then it can be used--as in the detachment of(f) from (c) and (e). It does not appear possible to deal separately with the issues of defeasibility and normative reasoning, for even our simple story cannot be represented satisfactorily without defeasible rules we cannot for instance replace (a) and (b) bv 3s / SCIENCE From: AAAI-86 Proceedings. Copyright ©1986, AAAI (www.aaai.org). All rights reserved. (a’) You should not tell Reagan if you do not tell Gorbachev and (b’) You should not tell Gorbachev if you do not tell Reagan. without omitting from the analysis the significant fact that telling neither is preferable to telling both. $2. The Deontic Logic 3-D. Deontic logic is a branch of modal logic whose main goals are to provide a formal representation of rules typically it does so with modal operators for “ought” and “permissible” and to provide a semantics for such expressions. A satisfactory deontic logic must be able to represent the distinction between defeasible prima facie (p.f.) rules and all-things-considered (a.t.c.) rules. Moreover it should not permit the detachment of all-things-considered conclusions from defeated rules; and it must have principles that state when such detachment is acceptable. The deontic logic 3-D (Loewer and Belzer 1983) meets these requirements. The language of 3-D is a propositional language containing the unary connectives T and F to which are added two dyadic deontic operators 0(-/-) and !(-/-), a necessity operator L, and a dyadic operator U(-,-). The wffs of 3-D are characterized as follows: (a) propositional variables are wffs, and (b) if P,Q are wffs then the following (in addition to the usual truth functional wffs) are wffs: O(Q/P), !(Q/P), LP, and U(Q,P). These statements may be read informally as follows: O(Q/P): it ought prima facie to be that Q, given P. !(Q/P): it ought all-things-considered to be that Q, given P. LP: it is settled that P. U(Q,P): P determines the normative status of Q. For tautology T, let OQ = O(Q/T) and !Q = !(Q/T). A 3-D model structure is a 6-tuple (W,T,H,I,s,F) where W is a set of momentary world stages, T is the set of natural numbers (the set of times), H is a subset of the set of functions from T into W (these functions are possible histories), I is a set (of “perspectives”), I is a function from T x H x I into the set of weak orderings on H, and F is a function from T x H x I into H. Call v=, for time t, history h, and perspective i a temporal perspective. The weak ordering s;v is a ranking of possible histories according to the extent to which the histories comply with the values of perspective i at time t in history h (cf. Lewis 1973, 1974). The most highly ranked histories are those at which no value or rule is violated. As one descends the ranking more and/or more serious violations occur. This allows for the interpretation of prima facie rules. O(Q/P) is to hold relative to the temporal perspective v just in case Q is true at each of the most highly ranked P-histories in the p.f. ranking IV. The set Fv is the set of histories accessible at v. For an objective interpretation we stipulate that F() = F() for all i* (that is, in the objective interpretation the perspective i is not relevant to accessibility).* Let P be settIed at v just in case P is true at each of the histories in the set Fv (cf. Thomason, 1970). LP says that P is settled. Now we want to use the p.f. ranking IV and the set Fv to define a new ranking +‘v with which to interpret expressions of all-things-considered (a.t.c.) commitments !(Q/P). The main idea to be used is that the a.t.c. ranking for v can be defined as the ranking that results when all histories that are inaccessible at v are removed from the p.f. ranking for v. Given an ordering x on H and subset y of H, let the restriction of x to y be the ordering z that results by removing from x each element of H not in y. ** Let c’v be the restriction of IV to Fv. !(Q/P) is to hold at v just% case Q is true at each most highly ranked history in s;‘v at which P is true. An interpretation [ ] on a 3-D model structure is defined as follows: [ ] assigns to each propositional variable a subset of T x H x I where we stipulate that for non-modal P: E [P] iff for all t* ET and i* E I, E [PI. In other words, only histories--and not perspectives or times--are relevant to the evaluation of non-modal propositional variables. Recursion clauses for the truth functional connectives are as expected. Now let [Q/P] be the class of weak orderings 5 on H that are such that: Ej(j E [P&Q;] and (k)(k E [P&-Q] + not&j;)). In other words, [Q/P] is the class of weak orderings on H in which some P&Q-history; is ranked more highly than any P&-Q-history. *** For v=, and j,k E H: v E [O(Q/P)] iff Iv E [Q/P]. v E [!(Q/P)] iff s’v E [Q/P]. v E [LPI iff Fv s [PI. For U(Q,P) let us say first that for xa, f (v,x) is the set of most highly ranked histories in x according to IV. Also for x,y GH, let * Cf. $5 below for a subjective interpretation of L that does depend on i. ** For example, suppose that H = {1,2,3,4,5} and x is the ordering (4s) < (1,2,3) and y = { 1,2,4}. The restriction z of x to y would be the ordering 4 < (1,2). *** The proposition expressed by P sometimes is identified with the set of histories [PI. Analogously, the class of rankings [Q/P] may be identified with the w-m expressed by the sentence “it ought to be that Q, given P,” which contains no p.f. or a.t.c. qualifiers. Planning: AUTOMATED REASONING / 39