STABLE THEORIES IN AUTOEPISTEMIC LOGIC

We investigate the operator producing a stable theory out of its objective part (A stable theory is a set of beliefs of a rational agent). We characterize the objective parts of stable theories. Finally, we discuss the predicate calculus case. Section 1: Introduction Recent developments in the artificial intelligence and, in particular, strong interest in the formalizations of the common sense reasonings and nonmonotonic logics ([MC], [Re2], [Li], [MDD]) and reasoning about knowledge leads to new interesting developments in the areas of logic previously left almost exclusively to philosophers. These subjects now get attention of computer scientists and mathematicians, raising hopes of applicability and of increased mathematical rigour. In one of these contributions, Moore [Mo], successfully formalizes the ideas of [MDD]; the resulting system of modal logic, called autoepistemic logic, deals with the notion of beliefs of a fully rational agent. In particular, Moore has shown that a stable autoepistemic theory is determined by its objective part. (The same result is proved in [HM]). He has not, however, provided the explicit construction of T out of its objective part. In this note, we investigate the connection between the objective part of a stable autoepistemic theory and the theory itself. In particular, we show an effective construction of a theory out of its objective part. We prove that every propositional theory closed under propositional consequence (in a language without modality) is the objective part of a stable autoepistemic theory. The construction is related to that of [FHV]. These results are shown in the Sections 2 and 3. In the Section 4, we consider the case of predicate calculus (which seems to be the original task of both [MDD] and [Mo], although they restrict themselves to the propositional logics in their papers). We extend the stability conditions by strong properties (similar to ω completeness of Godel). This allows us to prove analogons of Barkan’s formulas and, consequently, various normal forms for the formulas. This in turn allows to extend Moore’s results to the (restricted) predicate calculus case. Below we introduce some basic concepts of this paper. We presuppose the acquaintance with Moore’s paper, even though we repeat some of the basic definitions. Let L be the language of the propositional calculus and LM its corresponding modal extension. A theory T ‘i LM is stable if and only if it satisfies Stalnaker’s conditions: 1) T is closed under propositional (tautological) provability.