Autoepistemic Modal Logics

A modM approach to nonmonotonic reasoning was proposed by Drew McDermott and Jon Doyle in 1980-82. Almost immediately some disadvantages of that approach were pointed out. Robert Moore (1983) proposed his atltoepistemic logic, which overcomes these difficulties. Later, some authors (Kurt Konolige, Paul Morris and others) found peculiarities of different kinds in Moore's logic and proposed rather complicated solutions to these problems. A careful mathematical analysis of Moore's and McDermott's approaches shows that Moore's logic is merely a special case of McDermott's logic, at least formally. The problems that arose in Moore's logic may find a simple and uniform solution by going back to McDermott's original concept. I N T R O D U C T I O N Moore [1] introduced autoepistemic logic for formalising reasoning of an agent which may contain references to the agent's own knowledge (or belief). This kind of reasoning, which Moore calls autoepistemic, has the nonmonotonicity property: The set of "theorems" does not increase with the set of "axioms." (Mool"e attributes this observation to Stalnaker's work [2] which is not available to the author.) The language of Moore's logic is the usual propositional language augmented by the modal operator L. The intended interpretation of L¢ is: "the rational agent believes (or knows) ¢". Because of nonmonotonicity, the set of autoepistemic consequences of a given premisses cannot be defined as the set of sentences obtained from the premisses by applying some axioms and inference rules. Instead, Moore [1] introduced the following fixed point construction. Let A be any set of sentences in the modal propositional language. A set of sentences T is said to be a stable expansion of A iff T = {¢ : At2 {L¢: ¢ C T} U {-~L¢: ¢ ~ T} ~¢}. (1) The sign ~denotes here the usual tautological consequence relation. The stable expansion of A may be described informally as the set of beliefs of an ideal rational agent on the basis of the premisses A. Two sets of formulas added in (1) to A are produced by "positive introspection" ({L¢: ¢ e T}) and "negative introspection" of the agent. Moore's work was preceded by McDermott and Doyle's work [4], who attempted to formalise default reasoning, another important form of non-monotonic reasoning. The informal interpretation of L¢ in [4] is "¢ is provable." (As a primary modal operator, the dual operator M = -~L-~ is